EVALUATION OF INTEGRALS AND THE MELLIN TRANSFORM A. P. Prudnikov, Yu. A. Brychkov, UDC 517.3+517.44

and O. L Marichev

1. INTRODUCTION 1o. Various problems, for the solution of which there arises the necessity of evaluating integrals, have stimulated the appearance of a large number of works regarding both the development of general methods for the evaluation of integrals and the determination of analytic expressions for concrete integrals occurring in applications. The fundamental methods for the evaluation of integrals are: 1. The direct evaluation of integrals with the aid of a change of variable, integration by parts, differentiation, limiting process with respect to parameters. 2. The expansion of the integrand into a series, t e r m - b y - t e r m integration, and the summation of the obtained series. 3. The application of operational relations of the type of the Efros formula. 4. The construction and the solving of a differential equation, satisfied by the considered integral. 5. The use of the properties of the differential equation, satisfied by the functions occurring in the integrand. 6. The reduction of the evaluation of the integral to the solving of a functional equation, satisfied by the integral. 7. The representation of the integral in the form of a composition of integral transforms, in particular Laplace transform, and the use of existing tables of integral transforms. 8. The application of contour integration and residue theory. 9. The use of the properties of integral transforms, for example the Parseval equality, for concrete functions. 10. The evaluation of more general integrals with the subsequent substitution in the result of particular values for the parameters. 11. The use of an operational approach, based on the Mellin transform. The last method is sufficiently universal, it is closely related to the methods 8-10, and allows the evaluation of a wide class of integrals, containing elementary and special functions of hypergeometric type and arising in various problems of mathematical analysis and applications. We mention that it has been used in an essential manner for the creation of a series of encyclopedic manuals on integrals and series [ 162]-[ 164], exceeding significantly in informativeness the well-known books [13], [14], [45], [678], [679], [1024], [1026], [1028], [1029], [1194]. Below, in Sections 2-8 we give a sufficiently detailed exposition of this method with numerous examples. The list of references contains works on the indicated methods, as well as on the evaluation of actual integrals, In it we have included papers and monographs reflected in the review.journal "Matematika", starting from 1953, and also works of an earlier period or not reviewed in this journal. Such a detailed and sufficiently complete list of works regarding evaluation of integrals is published for the first time. The majority of the papers is devoted to the evaluation of concrete integrals or of integrals containing certain classes of functions, for example, Bessel functions, Legendre functions, generalized hypergeometric functions, G- and H-functions. etc. From them, as consequences, one obtains simpler integrals, containing various elementary or special functions. Due to space insufficiency and the extensiveness of the bibliography, it is not possible to give the explicit form of the integrals evaluated in the cited works. Therefore, in the survey we indicate basically the handbooks and monographs containing extensive tables of integrals, as well as works referring to general methods for the evaluation of integrals. In many cases the content of the papers is sufficiently well reflected in the reviews from the Referativnyi Zhurnal Matematika;* the corresponding data are given in the list of references. It should be noted that isolated integrals or classes of integrals can be encountered in works on integral transforms and also in various investigations of an applied character; these are not given in the references.

*RZhMat data have been omitted in the translated list of references -- Publisher. Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 27, pp. 3-146, 1989.

0090-4104/91/5406-1239512.50 9

Plenum Publishing Corporation

1239

2~ Extensive tables of indefinite integrals of elementary functions are contained in [45], [50], [125], [162], [182], [187], [190], [346], [396], [679], [844], [958]. Tables of definite integrals of elementary functions can be found in [45], [116], [125], [162], [186], [272], [392], [575], [680], [846], and, moreover, many integrals are contained in tables of integral transforms of Fourier [13], [22], [137], [1024], Laplace [13], [22], [63], [108], [477], [551]-[553], [840], [1029], [1194], Mellin [13], [116], [476], [477], [908], etc. A significant number of integrals, containing only elementary functions, can be found in works on special functions as examples in the case of concrete selections of the parameters of the special functions. For example, integrals involving sin x and cos x can be obtained from integrals of the Bessel function J~,(x), taking into account the formulas

J1]2(x)= -V ~~ slnX,

J_1]2(r = V ~ _ ~ c o s x,

Tables of indefinite integrals of special functions are contained in [125], [163], [164], [187], [199], [932], [1075], [1559]. Extensive tables of definite integrals of special functions can be found in [163], [164], [272], [575], [579], [854], [1030]; they are contained also in books on integral transforms of Fourier [13], [22], [137], [1024], Laplace [13], [22], [63], [108], [551]-[553], [840], [1029], [1194], Mellin [13], [116], [476], [477], [908], Bessel [14], [1028]. Tables of double and multiple integrals can be found in [59], [162]-[164], [1547]. Some integrals, involving elementary and special functions, are computed in various works of analysis, theory of differential and integral equations, mechanics, physics, where they play the role of auxiliary results. A series of works contain formulas of a genera/character, in which the integrand involves an arbitrary function. In the case when the right-hand side is simpler than the initial integral (in particular, the multiplicity of the integral is reduced) or it gives the possibility to make use of already known results, such formulas turn out to be extremely useful. From works of this kind we mention [72], [102], [438], [509], [562], [620], [621], [632], [681], [687], [831], [834], [838], [953], [1008], [1041], [1080], [1253]. In particular, results from [33], [75]-[77], [105], [133]-[135], [153], [184], [702], [853], [1408]-[1410] allow us to reduce multiple integrals to simple ones. Some general methods for the evaluation of indefinite and definite integrals can be found in [21], [67], [116], [127]-129], [196]-[199], [220], [271], [393], [1041], [1580]. A series of methods for the evaluation of integrals are based on the methods of operational calculus and the properties of convolutions for various classes of integral transforms, first of all for the Laplace transform [21], [67], [127]-[129] (see also the surveys [26], [64]). Tables of integrals and integral transforms involving generalized functions can be found in [22], [476], [477], [840]. 2. MELLIN TRANSFORM 1% Several integrals that have been obtained by various methods can be evaluated with the aid of a general and, to a significant degree, formalized method, based on the Mellin transform [954]

oo

,Tg'*(s)--S ~ (x)xS-Idx, s:~-~ir.

(2.1)

0

Elements of the theory of the Mellin transform can be found in [13], [22], [56], [67], [116], [908], I1 t92] (see also the surveys [5], [64] and the references therein). The relationship of the Mellin transform with other transforms is reflected in the most complete manner in [21], [56], [I13], [116], [908]. In particular, in [21] with the aid of the Mellin transform one has shown that all the other classical transforms, such that the Hankel, Stieltjes, Hilbert, Riemann--Liouville, Weyl, Meijer, etc. can be represented in the form of compositions of Laplace transforms. The modern theory of special functions of hypergeometric type is based on the use of the Mellin transform and of the related Mellin--Barnes integrals (see [10], [I07], [164], [194], [578], [579], [853], [909], [917], [918], [1411]. The theory of Mellin--Barnes integrals has been developed in [547], [954] and has found applications in [191], [194], [273], [533]. However, special interest in it has arisen after the publication of a series of works by Meijer [944], [948], [949], [951], in which the G-function has been introduced and investigated, becoming fundamental among func-

1240

tions of hypergeometric type. The theory of Meijer's G-function, defined by the Mellin--Barnes integral

:=i

Yci

j=,

P

q

j=n+I

j =m+l

][ I'(~zj+s) ]I

=Gr

z-Sds

J~'1,. . . . .., %)~

(2.2)

r(1--l~j--s)

is set forth in [10], [116], [122], [164], [855], [908], [917], [1418]; the most complete list of its properties and extensive tables of special cases are contained in [164]. More general is Fox's H-function, introduced in [600] by means of the Mellin--Barnes integral

i 2~ti I

t~

~[ P (bj + ~js) ~[ F ( 1 - - a i - - c z i s )

n

j=l z _ s d S =i_]m n { Z ] (Oh, 0q), ..., (ap, ~p)] p q Pq ~ I(bl, 1~1), , (bq, ~q)] r[ P(ai+cxfi) IT l'(l--bf--[3js) j =i

j=n+l

(2.3)

]:m+l

with arbitrary positive coefficients of s. Its theory is presented in [164], [415], [918], [1452] and other works. If there exists a substitution s = as 1, after which all the coefficients of s 1 become rational, then such an H-function can be transformed into a G-function, but with larger dimensions m, n, p, q. With the aid of the theory of residues [44], [116], [908], a G-function can be expressed in terms of linear combinations of generalized hypergeometric functions pFq(al,..., ap; b 1..... bq; z) [I0], [107], [164], [271]-[273], [304], [599], [855], [917], [1418] or, in the singular (logarithmic) cases, in terms of the limiting values of these combinations [5], [915], [917] (see also Sections 5, 6). In the general case, an Hfunction is expressed in terms of series with coefficients depending on the Gamma function (see Section 6, Subsection 40). Problems regarding asymptotics in the theory of pFq-, G-, and H-functions are very difficult; they have been investigated in [122], [414], [599], [855], [917], [1038] and presented in [i0], [107], [116], [122], [166]-[168], [915], [918]. In [120], [122], [164] one has investigated the behavior of the pFq- and G-functions near a third singular point in the cases when it exists. In [273], [446], [578], [1217] one has defined multiple hypergeometric series and functions, while in [122], [425], [617], [ 1452] one has introduced and investigated their corresponding G- and H-functions of several variables, defined by means of multiple Mellin--Barnes integrals, similar to (2.3) (see (2.8)). We mention that in the case of the vertical rectilinear contour La _- (~/- ioo, ~/+ ioo), 91s -- 7, the integrals (2.2), (2.3) perform the inverse Mellin transform of the corresponding ratios of the products of Gamma functions. Thus, under certain conditions [121], [918], formula (2.1) holds, where instead of Yf(x) one has to take the right-hand sides from (2.2), (2.3) (for z -- x), while instead of Y~* (s) the mentioned ratios from the left-hand sides. This property, as well as the known convolution theorem for the Mellin transform (Theorem 2.3) and the theory of residues, are fundamental for the method presented here for the evaluation of integrals of products of several G- and H-functions. Moreover, we shall use Slater's [1418] convenient symbolism A

i,[(a)+s, ( b ) + s ] L(c)+s, ( d ) + s ] =

B

l I P (aj+s) H r(b/--s) j=i j=l c D I[ (Cj+ S) 1~ P (dj--s) j=i

.['

(2.4)

j=!

for the notation of the ratio of products of Gamma functions. We mention that, in the applications of the method in various concrete situations, one has separated two approaches: a constructive one (Sections 3-6) and a formal one (Sections 7, 8). Regarding the properties of the Gamma function and notations, see, for example, [164].

1241

2 ~ S o m e Properties o f the Mellin Transform. The Mellin transform o f a f u n c t i o n (0, oo), is defined by the f o r m u l a {Yd (x); s} -~ X'* (s) = J X' (x) x~-~dx, 0 If the image

Yd*(s)

is known, then the original

Yd(x)

Yd (x), given on the semiaxis

s = ~, + i'c.

(2.5)

is obtained by the inverse Mellin t r a n s f o r m f o r m u l a

y+i~

-Us

(s) x - ~ d s ,

x>O,

Re s =~t.

(2.6)

y--ioo

In a similar m a n n e r , the equalities

a~* (sl . . . . .

')~('1

sn) . . . . ~. J q.' ( x. l , . 0 0 '~'a+i~ Yn+ic~ 1

.....

Xn)~-'(22Ii);'

S

"'"

gt--ico

~ d~* "~n--ioo

xn) x~~-1 . .. x ,~ - 1a- x l . . . dx~,

(S 1 . . . . .

(2.7)

Sn) OClS~ . . . XnSnds1 . . . a s n,

(2.8)

"%= Re sk, k = 1, 2 . . . . . n, define, respectively, the direct and the inverse multiple Mellin transform of the f u n c t i o n Yd(Xl . . . . . x~). of cases, the formulas o f type (2.5) and (2.7) will be written also in the f o r m

In a series

(x) +-~X ~* (s). For the Mellin transform (2.5), by the convolution of the functions we mean the integral 0o

oo

0

0

Ydx(x)

and

,Yd2(x), defined for x > 0,

(2.9)

x>O. To the multiple Mellin t r a n s f o r m there corresponds the multiple convolution of the f o r m co

n+l

0

j=2

We give some results w h i c h will be used in the sequel [116]. T H E O R E M 2.1. Let YY(x) be a function, absolutely integrable on any finite interval (s, E), 0 < ~ < E < ~ , and assume that in the neighborhoods o f the points x = 0, x = ~ we have the estimates [5~(x) [ < A x -a, O < x ~ e , 1"7{(x) I < Ax-b, x ~ E , A being a constant, while a < b. Then, in the band a < ~ s < b o f the plane o f the variable s, the Mellin transform ~ * (s) o f the f u n c t i o n a~ (x) exists and it is an analytic function, while at each point x, where a~(x) is continuous, the f o r m u l a (2.6) for the inversion of the Mellin transform holds, in which the integral at e~ is understood in the principal value sense and the integration is carried out along any vertical line f r o m the band a < 91s = "/< b. At the points x where ,7d(x) is discontinuous but there exist b o u n d e d o n e - s i d e d limits ~ ( x - 4 - 0 ) , the function

3V(x)

in the l e f t - h a n d side o f (2.6) has to be replaced by the expression

1 -~ [Yd(xq-0) + 3 ~ ( x - - 0 ) ]

. In the

case o f an exponential decrease of Yd(x) at x --* oo (or x--* 0), the f u n c t i o n oval*(s) is analytic in the h a l f - p l a n e ~ s > a (91s < b) and one has to set b = -oo (a = -oo). T H E O R E M 2.2. Let Y d ( z ) , z = r e ~~ , be an analytic f u n c t i o n in the sector -cz < 0 0, for large z, u n i f o r m l y in any angle

1242

interior to -o~ < O < fl, and, moreover, a < b. Then the f u n c t i o n ~ * (s), defined by the f o r m u l a (2.5), is analytic with respect to s in the band a < 9r < b and satisfies the conditions (e-(~-e)~,~), a~t'* (s) - ( 0 (e(~-~)I'~'),

Im s-~ ~- oo, Im s-> - - oo,

for each e > 0, u n i f o r m l y in each band interior to a < ~ < b, and, moreover, f o r m u l a (2.6) holds, where x -- z. Conversely, if ff~*(s) is a given function, satisfying the indicated conditions, then the f u n c t i o n ff{(z), defined by the f o r m u l a (2.6), x = z, satisfies the above conditions imposed on it and f o r m u l a (2.5) holds. T H E O R E M 2.3. If the functions x ~ - t ~ ( x ) , j = 1, 2, are absolutely integrable on the ray (0, oo), then the convolution (2.9) exists, also the f u n c t i o n x r-~ (/7E~o~2) (x) is absolutely integrable, and for each s such that ~ s = % the Mellin transform of the convolution (2.9) exists and it is equal to the p r o d u c t o f the images o f the convoluted functions:

{(X'~oX'2) (x); s}-- X'~ (s) X'~ (s) = X'* (s),

Re s = ~.

(2.11)

Moreover, the indicated conditions on the functions Se~ are sufficient. P r o p e r t y (2.11) is preserved in certain cases also when one o f the integrals .7~j* (s) converges conditionally or is understood in the principal value sense. R e m a r k 2.1. If for each o f the functions W j ( x ) the conditions o f T h e o r e m 2.1 hold with order exponents aj and bj, j = 1, 2, then the intervals (al, bl) and (%, b 2) intersect and 7 has to be taken f r o m their c o m m o n part: max(a 1, %) < "t < min(bl, b2). F r o m the last inequality there follow the conditions a 1 < b 2 and a 2 < bl, ensuring the convergence of the integral (2.9) f o r x = 0 and x = oo. R e m a r k 2.2. F r o m the equalities (2.11), (2.6), (2.9) there follows Parseval's k n o w n f o r m u l a for the Mellin transform: oo

3r (x) ~

y+ioo

9g'1(t)

ff-{'l(s),)~2(s)x.-ds.

2 ~T ] 7 =2zx--7

0

(2.12)

~--ioo

The latter can be true also in those cases when one of the functions )~'j(x) increases exponentially at 0 or at oo and its image ff~j*(s) is understood in a certain generalized sense (see 3 ~ f r o m Section 6). R e m a r k 2.3. A p p l y i n g the operator o f the multiple Mellin transform (2.7) to the convolution (2.10), we obtain the equality n+l

if/'* (sl . . . . .

s,,)-~ 3~*~(s1-1- s2q- 9 9 9 q- s,) ]-[ ,Tt*i (s j_l),

(2.13)

.i=2

which is the analogue o f the relation (2.11). T H E O R E M 2.4. Assume that the function

.TE (x)

is continuous for x > 0, decreases at oo faster than any power co

of x, while in the n e i g h b o r h o o d o f the point x = 0 can be expanded into a c o n v e r g e n t series

3,

fit(x) =Y. ckx ~ , O < k=0

x~e, 0~-~0~.t~ ... and An --* oo, n ---, oo. Then the Mellin transform ff{*(s) o f this f u n c t i o n can be continued analytically to the entire plane s, with the exception of the points s = -A k, k = 0, 1,2 ..... where ~ * (s) has poles o f first order. R e m a r k 2.4. Setting c k = (-1)k/k!, Ak = k, we obtain function f i t ' ( x ) = e -x , whose Mellin transform ff/'*(s) is equal to the G a m m a f u n c t i o n F(s). F r o m T h e o r e m 2.4 there follows that F(s) can be c o n t i n u e d analytically into the plane s, with the exclusion o f the points s = 0, -1, -2, ..., where it has simple poles. As it is k n o w n , the residues o f r(s) are obtained by the f o r m u l a

resF(s)=~,

S~--k

k~0,

1,2 .....

(2.14)

1243

which follows f r o m the expansion r { s ) - -( ---3-W-/l I ) ~ ' " -? eq)(k + 1) + e-~[ a ~ + 3r 2 ( k + l ) - - 3 q / (k + 1)116 + ea [r~2, (k + 1) + ~pa(k + 1) - - 3q~ (k + 1) q / ( k + 1) + q~" (k + 1)1 / 6 "~- O (84)}, g ~ S -1- k, k = 0 , 1,2 . . . . .

(2.15)

valid in the neighborhoods of the points s = -k. In the neighborhoods of the remaining points the G a m m a function can be represented by the Taylor formula in the form r ( s ) =r(So){l+s,(So)+e2[,' (So)+,2(So)]/2+ -}-ea[* a (So) +3q~ (So), ! (So) -)-~" (so) ]/6-I-, O (e 4) },

(2.16)

8 = S - - S 0.

The properties of the Mellin transform and tables for it can be found in [13], [1 16], [476], [477], [958], [1028].

3. A L G O R I T H M FOR T H E E V A L U A T I O N OF I N T E G R A L S OF P R O D U C T S OF F U N C T I O N S OF H Y P E R G E O M E T R I C TYPE

Assume that there is given the integral b ,~--I

n~2, 3.... a

(3.1)

j=i

For its evaluation one can use the following general scheme: I) By means of the change of variables and functions

e , C~ ~

X J-1 --

j~2,

I (cl . . . . .

3,

" " "'

n--l,

x~_l -~bcl,

x.~acl,

(3.2)

c._1, a, 0):~ c{1~ (Xl . . . . . x,,)

and the consideration (for a # 0 and b # 0) of the Heaviside step functions Jd~,~(q) = H ( ~ I - - 1 ) , 28~+1 (n) = H ( 1 - - ~ I ) ,

(3.3)

where H07) = 1 for r/>_ 0, H(t/) = 0 for ~ < 0, this integral reduces to an integral of the multiple convolution type (2.10) for the Mellin transform. 2) To this convolution one applies the multiple Mellin transform (2.7) which, by the convolution theorem, reduces to equality (2.13), allowing us to find the value of 3f* (sl . . . . . s•) of the image of unknown integral 28 (xl . . . . . x~) in the form of the product of the images of the known functions- 3fj, j = 1, 2 . . . . . n + 1. 3) From the known image 3~* (sl . . . . . s,~) one restores the original 28(x~ . . . . . x~), leading, after the inverse substitutions of (3.2), (3.3), to the value of the desired integral (3.1). One makes use of the inversion formula (2.8) of the multiple Mellin transform. We mention that for a = 0 and b = oo it is not necessary to introduce one Heaviside function, while for a = 0, b = o~ both Heaviside functions and, instead of n + 1 functions 285 , the convolution (2.1 0) will contain n or n - 1 functions 28j . In many cases the convolution (2.10) has the form of a one-dimensional convolution for the Mellin transform (2.9) and, moreover, if 3 ~ (x/t) = H ( x / t - - 1), then this integral is indefinite ( 28' (x) = x-ljg'2 ( x ) ) . After applying the Mellin transform (2.5) we obtain the equality (2.11), where 28j* (s) are Mellin transforms of functions 28j(x), ]= 1,2 . From (2.1 1) we can obtain the function o~f"(x) with the aid of the inversion formula (2.6) of the Mellin transform

1244

or its equivalent Parseval formula (2.12). The latter may be preferable in the case of the exponential growth of one of the functions 5~j(x). The expressions of 5~ (x), YC (x~ . . . . . x~) from the mentioned formulas are obtained with the aid of the residue theory or with the aid of the previously established general formulas which, in turn, are also obtained with the aid of the residue theory. Moreover, we automatically obtain the restrictions on the parameters, ensuring the convergence of the desired integrals. The presented scheme is most effectively implemented in the case, important for applications, when the functions SfYj belong tO the class of functions of hypergeometric type, defined as linear combinations of the Mellin--Barnes integrals

(3.4)

j,~,t,,,~

where LPis some infinite contour, c~j, ilk, "re, 6m are real positive parameters, aj, bk, cs d m are complex parameters, and z is a complex variable. In this case, also their convolution ,Ze , if it exists, belongs to the same class with respect to each of the variables. In general, the value .~ (z) of the integral (3.4) is expressed in terms of the sum of the residues at the poles of the integrand, representing a linear combination of series of the form oo

;.~ ]-[, r ( ~ j + ~ w ) r ( t k - ~ (-;)'~ n=o ),k,t,ra r (~t +~,'~n) I" ('dm--'6mn) n! '

(3.5)

whose parameters are computed in a well-defined manner in terms of the parameters of the integral (3.4), while the prime denotes the absence of one of the factors of the numerator. If all the coefficients c~j, ilk, "/l, 6m are rational, then (z) can be represented in a somewhat more convenient form in terms of a combination of generalized hypergeometric series. If the integrand in (3.4) contains multiple poles, then, in general, ~ ( z ) can be expressed in terms of series including lnPz, p = l, 2 ..... which are obtained from the linear combinations of the series (3.5) by the continuity of the parameters. We mention that in such cases, which, for the sake of brevity, will be said to be logarithmic, the representations for ~ ( z ) may have a very complicated form (see Section 4). Many elementary and special functions belong to the functions of hypergeometric type; the basic ones are enumerated in Subsection 8.4 of [164]. At the implementation of the presented scheme, the greatest difficulty is met at the evaluation of the integral (3.4) or of its analogous Mellin--Barnes integral (2.8). If in the formula (3.4) we set aj = flk = "Ys = 6m = 1, then ~ ( z ) is expressed in terms of Meijer's G-function (see (2.2)). The general case leads to Fox's H - f u n c t i o n (see (2.3)). The theory of multiple Mellin--Barnes integrals is far from being complete. However, formally, in actual cases the multiple integrals can be evaluated by analogy with the simple ones and one obtains results in the form of n-fold generalized hypergeometric series, in particular, Appell--Horn and Lauricella series (see [273], [477], [578], [1217] and also Subsection 7.2.4 from [164]. Examples of such evaluations are examined in part 6 ~ of Section 6. 4. DEFINITIONS, NOTATIONS, AND SOME AUXILIARY FUNCTIONS 1~ Let (a) ~ a l , a2,. .. , aa; (a) + s = ai-t-s, a 2 + s . . . . . a~t+s; k (a) = k a l , ( a ) ' - - a j - - - - - a l - - c t j , . . . , a j _ l - - a j,

ka2 . . . . . kaa;

a j + l - - a j , . . . , a A - - a i,

The symbols (b), (c), (d), (a), (b), (~), (d) denote analogous vectors, having B, C, D, .4,/~, C,/~ complex components, respectively. Let [a] be a vector obtained from (a) such that after a renumbering and a permutation of the component places we should have the relations

1245

.4

N"

q,

r [(a)+ s l - I I r (a,+ s ) ~ H II r ((~,:§ s ) - r IIal+ sl, j=l

q j > l,

j~ l .....

j=l

r,

q~--l,

i

1

j = r § l . . . . . . V,

r=O .....

N

(4.1)

where [l~]'-~-/~ll~ 0~21, 9 ' ' ,

99

air, a2r, . . . ,

aq~ll

/~12~ ~22~ 9 9 .~ ~q22~ 9 9 9

~qr r, a l , r + l ,

O~],r+2, . . . ,

ct~+~,/--ctt/~mq,

m u ~ O , 1, 2 . . . . . j ~ l . . . . . r, +_1, +_2. . . . . j@rt,

au--ak~@O,

(4.2)

al,\%

i=l ..... j,n=l

qj--1,

.....

N.

A s s u m e also that

too~--O,

m q f l : oo.

(4.3)

O b v i o u s l y , we have N

r

q'=X j=l

qj+N--r--A,

(4.4)

j=l

while in the j t h g r o u p the p a r a m e t e r s aij , i = 1 ..... q j, will d i f f e r by integers, b u t o n l y a m o n g t h e m s e l v e s (and not f r o m the p a r a m e t e r s o f a n o t h e r g r o u p ) , a n d the series o f points s= --a~--l, l=m~_~,~. . . . . rn,~--i will be poles o f m u l t i p l i c i t y i f o r the f u n c t i o n F[[a] + s]. We d e n o t e b y [a]' - aij the (A - / ) - d i m e n s i o n a l v e c t o r [a] - aij, f r o m w h i c h one has r e m o v e d the first i c o m p o n e n t s in the j t h group: [a]

t --

aij~

all --aij,

9 9 9

gaij,

aqi_,,i-1

cti+l

,j

--

alj, . . . .

lT.qjj - - a i f i

(4.5)

al.j-71 - - aij~ 9 9 9 aqNN - - ai].

In a d d i t i o n , let

Rill:

(--1)Hi/zaH IX (l +ail--al~j)[

r [[a]'--aH' ( b ) + a H ] X / ( c ) - - a i / , (d)+ai/A (4.6)

(1-- (c)+ aij)t ((0)+ aij)t ((__ 1)C_az)t ' X ( l - - [ a ' 1+ ail)z((ct)+ aij)t i~l

.....

qj,

j~:l .....

N,

l=0,1,2

.....

where we have used Slater's s y m b o l (2.4) and

L//2I

Hij ~ X

(4.7)

t'~2#-1,1'

k=l

B

((o) + a,j)~-- II (o~ + aij).

(c)t = c (c + 1)(c + 2) . . . (c + l - - 1) =

k=l

r (c + l) r (e) '

(c) o ~ 1.

(4.8)

O b v i o u s l y , f o r j = r + 1 ..... N we have the e q u a l i t y oo

Mj = "~ 17,1.t : : za,iF [ [ a l ' - - a , i, (0)+ a , j ] X l=0

J

k(~)--a,i,

(d)+a~i

((b)+a,j, ll-}-at]--(e); (--1)C--A•), X B+cFA+o--1 \ ( d ) + a,i, + aaj--[a]' /

1246

(4.9)

where the symbol B+cFA+D-I ( ' " ; x) denotes the generalized hypergeometric function pFq(...; ...; x) [164] with dimensionsp:--B+Candq=A+D1. Further, let A

B

C

D

j~l

k=l

l=1

rn~l

(4.10) (4.11) j =1

k=l

/=1

m=l

A=A+D--B--C, A = A q - D - - B - - C ,

(4.12)

E=A+B--C--D,

(4.13)

6=rA--A, 1 [~= ~

E=Aq-B--C--D,

q0=~-1 ( r E q - E ) ,

r - - qD,

(4.14)

(E-I- E)-- v--'~--m~-- 1,

~=R-qRppt~aq-qaoclo3-p,

(4.15)

z=o-Uro3,

(4.16)

z o~- R R - ' / ' r ~ exp (-- r-l~r~i).

(4.17)

Here a, a, w are complex quantities, while R, R, p, q, r > 0 are real quantities. In the case of a rational number r we assume that the numbers p and q are relatively prime integers. Let 8 - = 1 / 2 for ] h ] = l ;

~ = 1 for ] A I > I ;

~_-1]2 for I~l=a;

~ = ] for 1s

~'c~ A

[~-I ':~c~

(4.18)

[ arg a 14- (D ~B) n

A

+ , ~ I R I I/EcOs ,arg~,+(b--~)~.~..

L~= A}_~_i~tasgn(argo)sin larg(,I+(D--B)u a E

(4.19)

+

, argo -arg co# 0 ,

(4.20)

X ~ = l i m X , for argo=/=0, a r g o - + +_0, ~ { = l i m ~ , s for argo-+ + 0 , argo0@0, X+~-----limX~ for argo-+ + 0 , argco~ T-0; s=

,,s~++~,+-s for argo = ark co.~=O.

~s~+~s

(4.21) (4.22)

for a r g o @ 0 , argr

Ls ~ ~+~,~- for argo = 0, arg o) :~ 0. ] = 1 , 2 . . . . . A for 7X<0,

(4.23)

2Re ( ~ - - bkrA q--v) < 1 --k E, k ~ 1, 2 . . . . . B for A > 0,

(4.24)

2Re(cx~A-}-a:-A+-v)(l+E,

2 . . . . . -,4 for A < 0 ,

(4.25)

2Re (aA--'bkA + rv) < (1 + E) r, k = 1, 2 . . . . . B for A > 0,

(4.26)

2Re(czA-q-~iAq-r,c)((lnt-E)r,

j~l,

1247

I A - - r'-A I + [2Re (c~5~A+ r~A + v A ) - - A E - - rAE] > 0,

(4.27)

rAE] > O,

(4.28)

I A - - rA 1-- [2Re (c~AA+ r ~ 5 + ~A) - - AE - larg~ l
(4.29)

Iarg(rl=~E/2,

(4.30)

I arg co I <

nEl2,

(4.31) (4.32)

largo) I = ~ E / 2 . If6=0and~o_<

r, then [ arg (1 - -

zgz-P) I < n,

(4.33)

and, moreover, for ~/3 > 0 the value z = z o is assumed. A

C

r(a)] ~*,. ,v [(c)J = ~= v (at)- ~=1 ~ * (c~), A

(4.34)

C

~, r( a)] =~.= 5:**.,., l(c)J w (at)- j=l ~ , ' (c). Here the and r while the nor f r o m

*

(4.35)

~]**

symbols ~ , mean that if the vector (a) contains nonpositive integer c o m p o n e n t s aj, then the terms ~b(aj) have to be replaced by r - aj) and -r - aj), respectively. For example, if a 2 = a a + m, rn = 0, 1, 2 ..... remaining p a r a m e t e r s as, a 4..... a A of the vector (a) (A _> 2) do not d i f f e r by integers, neither a m o n g themselves the p a r a m e t e r aa, then

[(a)--a2--l, (d)-+a~-q-l] L(c)--a2--l, (b)+a2+lJ = q ~ ( l + l ) + ~ A

D

(1 +re+l)+ C

(4.36) j=l

j=s

]=1

B

- - X ~ (b]+a2+l). ]=1

We introduce the notation A

B

I I r (a i + s) H r (bk--s)

[(a)+ s, ( o ) - s] = j=,c

r[(c)+s'(d)--sJ

~=t~

= fly* (s).

I I r (c~+s) H r (a~--s) l =1

m=l

Here the e m p t y p r o d u c t has to be replaced by unity; for example, for A = 0 we set

(4.37)

0

I I I" (a t-t- s) = 1. ]=1

The function

flY* (s)

at the points

s=--aj--m,

1248

] = 1 , 2 . . . . . A (s=bk+m, k = l , 2 . . . . . B), m = 0 , 1,2 . . . . .

(4.38)

has poles of certain multiplicities if the singularities of its n u m e r a t o r at these points are not c o m p e n s a t e d by the singularities of the denominator; c o m p e n s a t i o n m a y take place when the quantities [ aj - c e l , [ d m - b k [, - c t - b k, -aj - d m contain nonnegative integers. The points s = - a j - m, s = b k + m will be called, respectively, the left and the right poles of function 3tg* (s). Assume that the contour L_oo (L+ce) represents a left (right) loop, situated in some horizontal band, starting at the point - ~ + i~o1 (+ce + i~ol), leaving all the left poles of the f o r m s = - a j - rn of the f u n c t i o n aY* (s) on the lefthand side, while all the right poles of the f o r m s = b k + m on the r i g h t - h a n d side of the contour, and terminating at the point -oc + i~o9. (+ce + i~o2), where ~o1 < ~o2. Here w e a s s u m e that all the left poles are separated f r o m the right ones, i.e. there is a curve which separates them. For this it is sufficient if the conditions aj + b k ~ 0, - l , - 2 .... are satisfied. We denote by L i ec an a r b i t r a r y infinite contour, starting at the point ~ / - ice, terminating at the point "t + i ~ , and separating all the left poles f r o m the right ones. If 3t(aj + bk) > 0, then, in particular, for the contour L i ec one can take the vertical line Lip o = (7 - lop, 7 + lop), -glaj < ,'/< 9tb k. In the general case, the c o n t o u r L i ec = ('7 - ice, ,/+ ice) is a curve going f r o m 7 - i~o to 7 + ice, which m a y be obtained by a continuous u n b e n d i n g of the loop L_+oo, without passing through the poles of the function 3r ~* (s). We introduce the Mellin--Barnes integral of the f o r m

L

(4.39)

L(c)+s, (d)--s] z-'ds.

By the contour L we shall always mean any of the infinite contours L+ce, L_ce, Lioo, along which the integral converges under a p p r o p r i a t e conditions on the parameters. If several contours of this f o r m are assumed, then the contour can be arbitrary but to each choice there will correspond its own condition o f c o n v e r g e n c e for the integral. The function IL(z) is c o n n e c t e d w i t h M e i j e r ' s G - f u n c t i o n

Ore" '/zl(%)i '' (2.2) by the relation Pq

\ i /(%)

I L (Z) - - I'~A'B ----B+C,A+~D

Z

i 1-@), (a),

(c)/

(4.40)

1--(d)/"

Let f,(s)=NRss

,

f (ax)=~

N

[(a)+s, (b)--s], L(c)+s, (d)--sJ

(4.41)

+s, l(c)+ s,

(4.42)

@)-s]

l r (ox'R-1)-Sds, L(c)+ s, ( a ) - s J L

T

r > O,

(4.43)

(4.44)

s, (a)-sJ

where/~, by analogy with L, is one of the contours of the types/~_+ce, Lice;

(4.45)

where s = s

or ~q?ioo is an infinite contour, similar to L, separating left poles of function

from the right ones. T h e left poles are a m o n g the points among the points

s

bk+ m

q

a

p

S=

p

s = _ _ a j +qm ____~, p s=

~ - +p m

f*(qs+-;)i (-ps) while the right ones are

m = O, 1 2 ....

1249

a~'* (s)z -s at the left and right poles,

We denote by NA(z), EB(1/z) the sums of the residues of the function respectively,

a ]~a ( z ) = ~_j ~_]

res

{a~**( s ) z - q ,

(4.46)

res {9~'*(s)z-q,

(4.47)

j ~1 /z=O s=--ai--k oo

~]8(1/z)=--

~

k=l I=0 s=b,~+l

under the assumption that these series converge. By P,,__(a) and I3__,(1/6) we denote the analogous sums of residues of function q f * (qs+=/r) f * (--ps) z ps at the left and right poles, respectively. In this case the variables z and b will be assumed to be connected by the formulas (4.16). As it follows f r o m the definition, the orientation on the contours L+oo and La+oo is clockwise. The residues at the right poles are computed in the usual manner, i.e., counterclockwise; therefore, their sums ~3B(1/z) and E_~(1/~) will be taken with the minus sign in order to make it consistent with the orientation of the contours L+oo and LP+oo. 2 ~ The above defined auxiliary functions P~A(z), En(1/z), P.,_.(6) and E._,(1/~) depend continuously on the parameters occurring in them. For certain values of these parameters (in the logarithmic cases), in the analytic expressions of these functions there may arise indeterminacies of the oo - oo type. In the sequel we shall assume that in the logarithmic cases the functions EA(z) and their analogues are equal to the corresponding limits of the regular functions, given below, when the parameters tend continuously to the considered singular values. We formulate some statements on the representation of these functions in the regular and logarithmic cases with double, triple, and n-multiple poles [5], [116], [908]. T H E O R E M 4.1. Assume that all the poles of the function Y{* (s) (see (4.37-4.38)) are simple, which holds when the parameters of the vectors (a) and (b), where A _> 1, B >_ 1, do not differ by integers. If, in addition, aj + b k 0, 1, 2 ..... then we have the equalities

res {~P* (s) z - q = f f * ' ( - - a / - - k) z~J+~ ( - - l ?

s=-aj-k

le!

res {fig'* (s) z-s}=Yg '*'

s=Ok+t

(b~+l) z % - z (-1)z-~ l[

' '

(4.48) (4.49)

(the primes in .No*' denote that from Yf* one has removed the factors I'(-k), I'(-e)), the summation of which with respect to all the admissible values of j , k, e lead to the representation of the functions (4.46) - (4.47) in the form A ]~A ( Z ) = ~

Z a J r ](C/,)" - - a,j,

[(c)--aj,

(b)+aj]

.

(d)+a/ ] ""

(4.50)

((b)+ aj, (1 - - ( c ) + a j ; ( - 1)C-az), X B+cFA+D--1~ (d)+aj, 1--(a) + a i ] B

"

~

-~k" V(b~'--bk, ( a ) + G ] v ,

XA+DFB+c--I\

(C)'+-bk,

(4.51)

1,D--B\

1 - - ( b ) +Ok

J"

If A = 0 (or B = 0), then ~A(Z) = 0 (EB(1/z) = 0). T H E O R E M 4.2. Let a 2 = a 1 + m, m = 0, 1, 2,..., and assume that the remaining parameters a 3, a 4..... a A of the vector (a) (for which A >_ 2) do not differ by integers among themselves or from the parameter a v Assume, in addition, that the vector (c) - a 2 does not contain integer components, while the vector (d) + a 2 does not contain nonpositive integer components. Then the function EA(z) has the representation

A--~ Za(Z)= x

j =2

1250

m-1 Mj+ x

l =0

oo

(

R,~t-t-~]/;?2,, X --lnz--l-~" 1~0

[(a)--a2--l, (d)+a2+~]} (c)--a2--l, (b)Wa2+ '

(4.52)

w h e r e the v a l u e s Mj,

Rile (i

= 1, 2) are o b t a i n e d f r o m the f o r m u l a s (4.1) - (4.9) i f we set i n t h e m r = 1, ql = 2, N - A -

1, a l l =

a 1, a2.1 = ag., a l j = aj+ 1, j = 2, 3 . . . . . N, r o l l = m. T H E O R E M 4.3. L e t a 2 = a 1 + m , az = a z + n, m, n = 0, I, 2 ..... a n d a s s u m e t h a t t h e r e m a i n i n g p a r a m e t e r s a 4, a 5..... a A o f the v e c t o r (a) ( f o r w h i c h A ___ 3) do n o t d i f f e r b y i n t e g e r s a m o n g t h e m s e l v e s or f r o m the p a r a m e t e r a v A s s u m e , i n a d d i t i o n , t h a t the v e c t o r s (c) - a 2 a n d (d) + a 2 do n o t c o n t a i n c o m p o n e n t s e q u a l to 0, _+1, +2 . . . . a n d 0, - 1 , - 2 ..... r e s p e c t i v e l y . T h e n t h e f u n c t i o n r.A(z ) has t h e r e p r e s e n t a t i o n A--2

~_~A(Z)= x

m--1

Mj-q-~

) ~2

~,,11,-]-

t =0

n--1

(4.53)

+,=

L(c)-a3-/, (d)+~+

:

w h e r e the v a l u e s Mj, Ril s (i = 1, 2, 3) are o b t a i n e d f r o m the f o r m u l a s (4.1) - (4.9) f o r r = 1, ql = 3, N = A - 2, a l l = a l ,

@1 = a2, a31 = a3, a l j = aj+2, J = 2, 3 ..... N, r o l l = m, rng.l = n; f o r ~[::], ~'[::] see ( 4 . 3 4 - 4 . 3 6 ) . T H E O R E M 4.4. A s s u m e t h a t the f o r m u l a s (4.1)-(4.3) h o l d a n d the v e c t o r s (c) - azj a n d (d) + ag.j, j = 1, 2 ..... r, do n o t c o n t a i n c o m p o n e n t s e q u a l to 0, _+1, _+2.... a n d 0, - 1 , - 2 ..... r e s p e c t i v e l y . T h e n t h e f u n c t i o n ~A(z) has the representation

r

q] rntj--I

j=l

IV"

2; t2;= o

2; M..

i--1

(4.54)

]=r+t

where /--1

Xt-I~---Z tn~O

A~ ~

(--Inz)m rat

OmV~_.,,

.Ai_m-l,

V~ ~ ~ OmF._m,

tn~O

m~O i

l

+== k t + . .Z II .+. .+l~t=m s=l

(4.55)

p~0

P

N p k = "~ (--1) "-' ~r r(s)(l+l+aii~aki) ~=1 ~ l~o-,,k F]'i:t--Tq-~ " W2,,,+1 = 0 ,

W ~ m = 2 ( - - 1)m+12'"~-'-I (2m)l

N~

~x2mB2m

[Bm are the B e r n o u l l i n u m b e r s ] ,

F s = -- ~ TkF~_it, F o x 1, It Tk == X

(-- 1)'~Ck-"D~ '

,'TZ~0

kl-t-...+/~c~m s ~ l

II

,~str

(c~--a~i--i) ,

ks[F (ds + atl + 1),

1251

(3,= ~

(-- 1)"A,~.,,.,i3,.,,,

ti"Z=0

IIr(",'(o,+,,,+,) let+...+kB~m s~l

kslF(bs+au+l

qi

) ,

F(US)(asj~aji~l)

s=i+l

hi+l+...+~r

IV "rXS,kS~ ka+...+kN=m s=l ~1=0

s~]

qs

A(a) 3,kS~

rn

II nt!P(ats--aH--ll, r("P(at'--ai i--t) Am=X

ha+...+nqa~k $ t = l

.ll k zltn--la. k(1)a(2)

k--0

Remark 4.1. In Theorems 4.2-4.4 the restrictions on the vectors (c) - a2j and (d) + a2j can be removed by defining in these cases the functions (4.52)-(4.55) by continuity or taking into account the reduction of the multiplicity of the poles of the function (4.37) after the use of the multiplication formula for the Gamma function. Remark 4.2. Obviously, EA(z) and Es(1/z) turn into each other if one interchanges the vectors (a) and (b), (c) and (d), z and z -1. This property allows us to express the representation of EB(1/z) in the corresponding logarithmic cases. THEOREM 4.5. In the regular (non-logarithmic) cases we have the equalities

; . ,2: , , , : o ' @ ) + a / - h , (~)+;s, (~

~r~ ~ oo

(a)+as+h, (c)+;j,- (a)- ;sJ--;- S=l ~ h=o ~ ~(-1)~R-~ x L(c)--0j, ( d ) + 0 / , ( c ) + b s + h ,

(4.56)

(d)--bs--h J

Y~_>.(II~)=_/VJVi 2 --'hT-(--l)'~ ~t'i+"z(\R; z-~lri'/X y =l/z=0 x r [(a)+oj+/z, (o)'-bs-tz, (h)-m, (z;)+ns]+ [(c)+bFJ-h, (d)--bs--h, (cS--ni, (cl)+n] J

+-.7-~ 2 ~(-,? R~s ( - ~ ) %+hX j = l /z=0 •

(4.57)

[(a)+~j, (o)-~s, @'-,Ts-h, (~)+a~+~] [(c)+~ s, (a)-~, (c)-as-tz, (8)+~s+h ' ;i=(as+h)r+a,

~}~-(bs+h)r--=, o j = @ + h - ~ ) r -I, gs=(Ej+h+~)r -~.

(4.58)

In the logarithmic cases, i.e., in the cases when the vectors (a)' -- a / - - h, (5)--~s, (b)' -- /~j-- h, ( a ) - - 0 i or (b)'--b/--~- (a)--111, (~z)'--[zs--h, (b)--~j contain nonpositive integer components, the values of E,__(5) and E_+(1/5) are defined from (4.56), (4.57) by continuity. If, in addition, r = p/q, where p and q are relatively prime numbers, then E,_(~) and E._.(I/~) can be written in terms of generalized hypergeometric functions or in terms of the limits of these functions with respect to parameters in the following manner: in regular (not logarithmic) cases

1252

y,+ (~)= NAT ~ q-' -:=1 h=o

L(c) -- aj-- h,

(O)+aj+h, (a)+~y, (b-)--~j].+iF,,,(t, A(q, t.-(c)+ay+h), (d)+aj+h, (c)+~j, (d)--~jJ \A(q, 1--(a)+ai+h),

A (q, (b) + a / + h), A (p, (a) + ;/),

A (q, (d)+ a: + h), A (p, ~) + ;:), A (p, 1 -- ( 2 ) + ;j); (-- 1)q(c-A~+p~o-"~ ~ a (p, 1 - (g) + ;:) + NN

(h:) k

-- ~"

[(a)--0j,

~=o

(b)-JcO], ('~)-Jl-b]-~]z,

[ ( c ) - 0 j , (d)+0:,

(~)+~s+h,

(b)'--b/--h] {1, A(q, 1--(c)+01),'A(q, (b)+Oj), (d)--bj--h J"+'Fm\A (q, 1-- (a) + 0:), A(q, (d)+0j), A (p, ( a ) + b j + h ) , A (p, 1 - - ( d ) + b j + h ) ; (-- 1)q(c-a)+~176 "~ -

A (p, (c~) -+- b:+ h), A (p, 1 -- (b) +

-

b:-[-h)

) ,

(4.59)

B q--I

:=~=o

[(c)+b:+h,

"

(O)'--b:--h, (a)--~U, (b)-kBj] F (1, A(q, (a)+bj+h), (d) -- b1 - h, ~)--*li, (d) + ~I/ ] m+, . \A (q, (c) + b~+h), a(q, 1--(d)+O:+h), A(p, ~)+~lj), A(p, 1--(c)+~11); A(q, 1--(b)+b/+h), A(p, (d)+~l~), A(p, 1--(a)+~l/)

[ ( c ) + ~ , (d)--~:, (c)--as--h, (d)+a~+h J X.~+aF~ (1, A(q, (a)+~/), A(q, 1--(d)+~:), A(p, 1 - - ~ ) + ~ + h ) , \A(q, (c)+~fl, A(q, 1--(b)+~j), A(p, 1--(a)+a:+h),

(4.60)

A (p, (~ + ~ + ~); ( - ~)~o-~)+~-~) o--~ A (p, (d) + aj + h)

]

where m ----q (A + D) + p (/~ + C), n = q (B + C) +_ p (.4+/9), while in the logarithmic cases ~,__(a) and ~3_,(1/a) are determined from (4.59), (4.60) by continuity. In particular, if, for example, a~=at +n, bz-----biq-m,m, n = 0 , 1, 2 . . . . . then the points s~-- --a2--h, s = --b~--tt, h = 0 , 1, 2 . . . . are, in general, double poles of the function qf * (qs+c~/r)f ~ (_ps)zOS . If, in addition, the equation rk=h_ho"

(4.61)

ho~a_~+a.~r"

a~+k ~z, s = has nonnegative integer solutions k, h, then, in general, the poles s -~+h -, whose indices satisfy equation (4.61), are also double. If all the remaining poles are simple, tt~en the ocorresponding .p function ~ ( e ) for A_> 3, B_> 3hasthe form

( A

~

\

n-1

\j=4

2~4

/

h=0

~o { [(a.)- a,--/z, (d) -1742 -l-/'t ] +Xt,=0 (--1Y+h(n+h)!S~,a In( R/~-'zr)+~F [(c)--a2--h, (b)Wa2+h + 1253

{

+ r-' n=0 ~ ((re+h)! - l)m+--------~n T'~, ~ in (RL?-rz ") + qr [ (c) -- 0~_, (b) + O~J+ (4.62)

, (,:7)+ ~ 4- ~ j]/, l-t- ,,( q a~,)

-}-rWL(E)--~2--h

~(a~)

-i-F9 r~,~

2: sh) ,o+ #(73

[(c)--Oz, (b)+0~ J +

Jr-r~ k(d)--b3--h, ( ~ ) + ~ + h A where Pj are the terms of the sum X P1 , while Qj are the terms of the sum ~ Q1 1=1

in the formulas (4.56) or (4.59),

1=1 ~

h[

X

/(c)--aj--h, (d)+aj+k, (d)--; i, (~)+;H T i 'n ~ - F -

~z j

X

hi

(o)+% L(c)-% (d)+% (d)--Oj--h, (c)+Oj+h I here by S'2,n, T'2,h and T~,~ we have denoted the quantities $2, h, T2,h, and T3,h in which the following negative components have been omitted: a ~ - - a ~ - - h , "bl--bg_--h and ~ - - 0 3 = ~ k , respectively (for h = h o + rk). The symbols

L

l(~k ) h( ,,, ) h )

denote summation with respect to all nonnegative h, for which there do not exist (there exist)

nonnegative integers k, satisfying the equation (4.61). R e m a r k 4.3. The relations (4.56), (4.57) can be obtained formally from (4.59), (4.60) by replacing the functions n+lF m and m+lFn by 1, while p and q by c~. R e m a r k 4.4. In the formulas (4.59), (4.60) the vectors of type A(p, I - - ( a ) + a l - } - h ) contain a unit component and, therefore, the order of the functions n+lFm and m+lFn can be always lowered to nFm_l, mFn_l . R e m a r k 4.5. The function Y:_.{1/~) is obtained from ~L-(6) by replacing a, b, c, d, A, B, C, D, fi, tg,...,/), R,/~, w, o, c~ by b, a, d, c, B, A, D, C, b, h ..... C, R -1, R -1, co-1, a -1, -a. Performing this substitution, one can find a representation of ~._,(1/~) in the logarithmic case of the indicated form. 5. MELLIN--BARNES INTEGRALS The functions I]A(z), P,B(I/z), ~,__(a), P~__,(1/e), defined in Section 4, have a general character in view of the large number of independent parameters. This follows from the connecting formulas between the mentioned functions and the Mellin--Barnes integrals (4.39), (4.45), given here and in Section 7. THEOREM 5.1 (Slater). Assume that the function a~* (s) has the form (4.37). If the conditions --Reaj
] = 1 , 2 . . . . . A, k = l , 2 . . . . . B,

(5.1)

and one of the conditions

A+B>C+D, A+B=C+D,

1254

A+D=/=B+C, Re s ( A + D - - B - - C )

(5.2) 1

< ~---Re v,

(5.3)

(5.4)

A=C, B=D, R e ~ ' < O , where u is d e f i n e d in (4.10), hold, then for such s there exists inverse Mellin t r a n s f o r m having the f o r m [.~A(X) f o r X > 0 , ' J-f A @ D > 2 3 @ C , /~A(x) for 0 l , 3r (x) = / if A @ D = B @ C , [~.~z~(l/x) for . v ~ O , i f ' A 4 : - D < B q - C ,

(x) o f f u n c t i o n

ae*(s),

(5.5)

and, moreover, 9Y ~ ( 1 ) = E A ( 1 ) = E B ( 1 ) , • A+D=B-t-C, Re v + C - - A + I < 0 , A>~C.

(5.6)

Remark 5.1. If for certain parameters one or several of the conditions aj = cs + n (or aj = - d m - n) [b k = d m + n (or b k = -cg - n)], where n = 0, 1, 2 ..... hold and, moreover, the vectors (a)" - aj, (b)" -- b k do not contain integer components, then f r o m the group o f conditions (5.I) one can remove (or weaken) the conditions referring to these parameters. For aj = c~. + n [b k = d m + n] the corresponding conditions ~ ( s + aj) > 0 [~R(bk - s) > 0] are removed, while for aj = - d m - n [b k = -cs - n] they are replaced by the weaker ones 9t(s + aj) > - n - 1 [9t(b k - s) > - n - 1]. If the vectors ( a ) ' - a j , ( b ) ' - b k contain integer components, then the question regarding the relaxing o f the restrictions (5.1) requires a special investigation. Remark 5.2. The restrictions (5.1)-(5.4) ensure at least the conditional c o n v e r g e n c e o f the integral (2.5), (5.5) at 0, c~ (and 1). If they are violated, then, in general, this i m p r o p e r integral diverges. H o w e v e r , in certain cases it may exist in the principal value sense (see Example 12). Remark 5.3. I f the conditions (5.1) (taking into account R e m a r k 5.1) are not satisfied and the line ~Rs = 7 does not c,pntain poles of the f u n c t i o n (4.37), then T h e o r e m 5.1 remains valid if the functions I2A(X), ~2n(1/x ) are rep!aced by 2A+(X) - P , B _ ( I / x ) a n d ~B+(1/X) - ~A_(x), respectively, which are introduced in the following manner. We assume that the line 9ts = 7 dissects both series of points s=--a~--nj, s=b~-l-nk, n~, nh=O, 1, 2 , . . . , so that f r o m the l e f t - h a n d series the points with the indices nj -- 0, 1, 2,..., p j a r e cut off, while f r o m the r i g h t - h a n d series those with indices n k = 0, 1, 2 ..... qk, where j , k are some o f the 1, 2 ..... A, 1, 2 ..... B. Thus, f o r these indices we have the inequalities Re b k + q k < ~ y = R e s . < - - R e aj--pj, (5.7)

~ R e aj--p~--I < ~ y ~ R e bk+qkq- 1, while for the remaining indices j , k we have the corresponding inequalities f r o m the g r o u p (5.1). N o w we denote by ~A+(X) and ~ B + ( I / x ) the functions obtained f r o m (4.46) and (4.47) by r e m o v i n g the terms with the indices k = 0, 1 ..... pj and g = 0, l ..... qk, for which pj and qk satisfy the inequalities (5.5). By ~ A _ ( x ) a n d Z B _ ( l / x ) we denote the sums f r o m r e m o v e d terms: Y,A-(X)=Y,a(X) - - ~ A + ( X ) , ~ - - ( 1 / X ) - - - ~ B ( 1 / X ) - - ~ B + ( 1 / X ) . With these modifications, T h e o r e m 5.1 remains valid. For example, f r o m what has been said there follows that the orin

ginal o f the f u n c t i o n F(s) for - n - 1 < 9ts < -n, n = 0, l, 2 ..... is the f u n c t i o n

e-*--~ /z=0

(_x)k kl

(see 8.4.3.3 f r o m [164]).

A more complicated case is considered below in Example 13. We have the following statement regarding the representation o f the Mellin--Barnes integral in terms o f series, more general than T h e o r e m 5.1. T H E O R E M 5.2. Let aj + b k ~: 0, -1, - 2 ..... j = 1, 2 ..... A, k -- 1, 2 ..... B. T h e n the Mellin--Barnes integral (4.39) represents an analytic f u n c t i o n o f the variable z and complex parameters (a), (b), (c), (d), which can be written in the form & (z) = ~ ( z ) =

EA(Z), A + D ~ B q - C , L=L_~, Jz[<~; A@D=B+C, L--L_o~, I z ] < l ; = ~ ( 1 / z ) , A q - D < B @ C , L-=L+~, [ z l < o o , z=/=O; A-q-D=Bq-C, L=L+~, ]z]>l;

l

(5.8)

1255

~,A(Z)=Y,B(1/Z), A + D = B - t - C , R e v W C - - A + I C .

[zl=l,

T h e r e p r e s e n t a t i o n (5.8) r e m a i n s valid also in the case o f the c o n t o u r Lioo, going f r o m 7 - ioo to "y + ioo and s e p a r a t i n g the l e f t - h a n d series o f poles o f the f u n c t i o n (4.37) f r o m the r i g h t - h a n d one, i f f o r A + B > C + D one has (5.9)

l a r g z I , < (A + B - - C - - D ) r t / 2 .

If, in a d d i t i o n , A + D = B + C, then the f u n c t i o n s EA(z) and ~ B ( I / z ) a r e the a n a l y t i c c o n t i n u a t i o n s o f each o t h e r in the a n a l y t i c i t y sector (5.9) a n d , m o r e o v e r , in the cases A + B - C - D _> 4 the cut j o i n i n g the points, z = ( - 1 ) A - c and z = (-1)A-Cr can be r e m o v e d since at the p o i n t z = ( - I ) A - C the f u n c t i o n ~2A(z) does not have s i n g u l a r i t y w h e n A + B - C D _> 4. I f A + D -- B + C, A + B = C + D a n d ~tv < 0, then, in g e n e r a l , the f u n c t i o n s EA(z), E B ( 1 / z ) are not analytic c o n t i n u a t i o n s o f each o t h e r , b u t on the axis 0 < z < oo t h e y are c o n t i n u o u s at the p o i n t z = 1: EA(I) = EB(I) i f also ~lv + 1 < 0. T h e a b s o l u t e c o n v e r g e n c e o f the i n t e g r a l ILi~ (Z) is p r e s e r v e d also on the b o u n d a r y o f the s e c t o r (5.9), i.e., for ] a r g z ] = (A + B - C - D)~r/2, i f the f o l l o w i n g c o n d i t i o n s hold:

A-ff-B>~C-t--D, ,I (A-}- D - - B - - C ) < - - 1 - - R e ~;+ 89 (A-t- B - - C - - D).

In the case A + B = C + D, f o r the c o n v e r g e n c e o f the integral Then, u n d e r the c o n d i t i o n

"f(A+ D - - B - - C ) <

I L ~ (z)

it is n e c e s s a r y that arg z = 0, i.e., z = x > 0.

--Rev

the equalities (5.8) hold, w h e r e z = x 4= l , L = Lio o. 6. A P P L I C A T I O N O F T H E M E L L I N T R A N S F O R M T O THE EVALUATION OF INTEGRALS 1~ Simplest Examples. Example 1. With the a i d o f the s c h e m e c o n s i d e r e d in Section 3, we o b t a i n the r e l a t i o n oo

(6.1)

0

The integral r e p r e s e n t s the M e l l i n c o n v o l u t i o n (2.9) o f the f u n c t i o n s

,.7tl ( t) = t~e-t, ,7{2(rl) =e -1/". T h e i r M e l l i n t r a n s f o r m s are o b t a i n e d f r o m the f o r m u l a s 8.4.3.1, 8.4.1.5, 8.4.3.2 o f [164]:

,Tg'l*(s) =s162 The Mellin transform

Re s < 0 .

,7/'* (s) o f the integral (6.1) has the f o r m ~ * (s) = F (r

1256

Re(~xq-s)~0; ~2*(s)=s

F ( - - s ) , - - R e c~< Re s = ~ < 0.

(6.2)

The integral (6.1) is evaluated with the aid of the formulas (2.6), (5.5), (4.47), (4.49), (4.8) and 1Fo(a; z) - - ( I -

z)-~: 7+ioo

oo

y--ioo

k-0 s=~

=--

F(~+k)x-k(--O~-'=r(~)--~-L-Vk! /~=0

----

(6.3)

=

/z=0

= r (o0( 1 + ~ )1 -~ , I x l > l . In front of the sum we have the minus sign since all the poles s = k, k = 0, 1,2 ..... are to the right of the integration line 91s = 7. The residue of I'[a + s, -s]x -s is obtained by a simple substitution into this function of the value s = k with the subsequent replacement of F(-k) by (-l)k-1/k! since the function r ( - s ) has a minus sign in front of s. The obtained series converges for I x I > 1. Since for the considered function Yd*(s) we h a v e A - - B = 1, C = D = 0 ( f r o m w h e r e A +D-B-C=0),it follows that the corresponding Mellin--Barnes integral (4.39) can be evaluated, not only in terms of the residues at the right poles s = k, but also in terms of the residues at the left poles s = -cz - k, this time with another convergence condition I x [ < 1: oo

X'(x)=~

res

(r[o~+s,--slx-*)~

(6.4)

k=0 s=-- ~--k

--~r(c~+k)x~+~(--l)k=F(a)x~(l+x)-% --

k[

k=0

]x[
"

The right-hand sides of (6.3) and (6.4) continue analytically each other in the analyticity sector [ arg x [ < (A +

B - C - D)rr/2 = Ir, which exists in view of the fact that A + B - C - D = 2 > 0. Therefore, the condition on ] x [ can be considered as a derivation condition and the value of the integral I 1 can be written in the f o r m (6.1). The convergence condition 91cx > 0 follows f r o m the inequalities (6.2), while condition ~1(1 + 1/x) > 0 ensures the exponential decrease at c~ of the integrand and it is obtained as the analytic continuation of the condition x > 0, under which the convolution (2.9) is considered usually. Besides, this condition can be relaxed to ~(1 + I/x) = 0, $(1 + l / x ) ~ O, but in this case one has to adjoin the restriction 91a < 1, in order that the integral (6.1) should converge at least conditionally at oo. Example 2. We consider the integral oo

I2 =

~sin bt

I tcz-I ( t 2 - tZ2)la-1(COS bl~}dt.

(6.5)

a

Here it is convenient if after the substitution

t=a]/u

we introduce the notations

I~ = 89 a~+et~-eg~' (x), 0~0,

Ydl ('r

v ~' z(v __ 1)~-I,

fsin 2rl -~/2 )

2

(6.6)

Vx From formulas 8.4.2.4, 8.4.1.5, 8.4.5.4 of [164] there follows that the Mellin transform of the functions 3%(q) have the f o r m

Yg'~(S)--F(~)F(1--c~/2--f~--S]'I --r Re(c~/2+~+s)<

~(s)=V~r[(6

+6 / 12)-/s2 + s J]'

.7/'1 (~) and

Re ~ > O,

I;

--1/2~Res~8/2,

(6.7)

(6.8)

where 6 = 1 if one considers the formula with sin bt and 5 = 0 if one considers the formula with cos bt.

1257

The product of the formulas (6.7), (6.8) gives the following expression for the MelIin transform of the convolution /~ (x) : [

6/2--s,

~*(s)---K~r(~)r L(6+l ) / 2 + s , --1/2 < Re s < 6/2,

Re[~>0,

1--~/2--[~--s]

1--a/2--s

' (6.9)

1--Re(a/2+[~).

In this case A = 0, B = 2, C = D = 1, i.e., A + D - B - C < 0 and the corresponding Mellin--Barnes integral (4.39) of Yg'*(s) has to be evaluated by the sum of the residues at the right poles s = 6/2 + k, s = 1 - a/2 -/~ + g, k, g = 0, 1, 2 ..... We assume that these poles are not superposed, i.e., (~ + 6)/2 + 13 ~ 0, _+1, _+2..... Then, by Theorem 5.1 or by analogy with Example 1 (see also (4.47)), we obtain

--(a+6)/2,

X~F~ [ (c,-+-6)/2; \(cr

1/2+6J X

--x< ~ ~,2-o ~1- (cr ] ' 1/2+6}-+- x , ~-,'- t' [[3, ( 3 + ~ - - a ) / e - - ~ J X

{ 1--[~; X a,= 2~2_(a+6)/2_~,

--A;

(6.10)

-1

(3+~_~)/2_) /

This result can be simplified somewhat by taking into account that the parameter 6 satisfies

[ c+6/2,

F [(1 += 6F)c/~2 - ~- c/ 2]

(1--6)/2+c] [(1 + 6 ) / 2 - - c, (1 - - 6)/2-t-cj =

l

='-~-

2~c-~lAY [.cos c

"

Performing in (6.10) the inverse substitution of (6.6) and making use of the fact that the pair 6, 1 - 6 assumes the values 0, 1 in some order, we obtain finally the value of the integral

I2= ~-~a"+2~+~-~b~B(1~,I--I~----~)• XIF2\

_9

2

~--~-6;-----

~cos(=/2+[~).~v~ '-I~; 2 - ~-13,

(a+2[~--2)X

(6.11)

2

[b, Re[~>O; Re(a-}-2[3)<3]. The restriction 9~(c~+ 2fl) < 3, following from the inequalities (6.9), ensures the conditional convergence of this integral at infinity. If in the equalities (6.5), (6.11) we set 13 = 1, then we obtain the value of the following indefinite integral:

i t~-i Lcosbt fsinbt} dr--

a~+~ b6 F2 ( a + 6 . a + 6 a~462) _~T_.ff 2 ' 2 ~ - 1 , ~1+ 6 ; -+

a

. , (sin (~a/2))

+O-~T ~/Z)'l.cos(~za/2)~ [O>0, Re~< 1]. This integral can be evaluated also in a straightforward manner by turning to formula 8.4.2.2 of [ 164], rather than 8.4.2.4, and performing the substitution Yg'l('0 =x~/2H(~--l), where H(O) is the Heaviside function (see (3.3)). Example 3. For the evaluation of the integral e~

Ia ~ S t~-le-ma sin bt dt 0

1258

(6.12)

we perform tile substitution pt 3 = r 3 and we write it in the form

la~p-=/~

sin 2

~

(1:3)ct/ae-'~'d'r -~-.

0

Setting then sin 2 ] f x ~ Yg'1 (x)

2~Up 3

x ~ ' e -~ ~ - ~ 2

(x),

--

x ~--T--,

lx=~/3,

I3 .~ p-~la fft'o (X),

(6.13)

we obtain the modified convolution co

I gL/'l / x -2 0

In accordance with formulas 8.4.1.6, 8.4.5.1, 8.4.1.5, 8.4.3.1 of [164] and the multiplication formula (3.16) from [116] for the Gamma function we have

63~o (6S) ~- l / ~ r[ 1 / 2 -

+3s2S+~]----

2~-' t2 ~ 3-61~ r[s--}-lx/2' s-+-(Ix+ 1)/2, 1/6--S, 1/2--S, 5 / 6 - - s ] - - V 3 - " "- " s+l/3, s+2/3, s+l

(6.14)

For the function r[::] in the right-hand side of (6.14) we have obviously A = 2, B = 3, C---3, D = 0. Therefore, making use of Theorem 5.1 under the condition -9t#/2 < ~ls < I/6, i.e., for 9t(z > -1, with the aid of formula (4.51) it is easy to obtain the corresponding value of the original P.B(1/z) = Q(z), where B = 3. Performing then the inverse substitution (6.13), we obtain the desired value of the integral (6.12):

This value contains a combination of three series 2Fs. Example 4. We evaluate the integral co

14 ~ S t"-le-Pt sin bt dr.

(6.15)

9

Here one can proceed in two ways: 1) Performing, as above, the substitutions

bi=2V~,

Y{'l (x) =x~'n sin 2 ] f x , ,7{2(x)=e -2~-112, I4=2~'-Ib-~ a~ (x),

b/p=]/-x,

we obtain the convolution (2.9) and then, with the aid of the formulas 8.4.5.1, 8.4.3.2, 8.4.1.5-6 of [164], its image

1259

(A = D = 1, B -- 2, C -- 0) and, finally, the original

(x)=xO+~,)/zp [(1-}-c~)/2, 1+~/2] 3/2

rc~+l ~ 3 2F~ ~ , ~ + I ; T ;

--x.

)

2) A different f o r m of the result in terms of an arbitrary parameter c can be obtained after a substitution of the form

t=c~, Oc=sin~,

p c = = a + c o s ~ , x-----lla,

I4 =c~x~;~ (x), ~ (t)-----e-t~~ ~

sin (t sin I~),

(rl) = e-l/nT1 - ~ .

(6.16)

Then the Mellin transform corresponding to the convolution (2.9), by virtue o f formulas 8.4.5.15, 8.4.3.2, 8.4.1.5 f r o m [164], will be

IS,

0~-- S

]

&**(s)--~r[13s ' l_lgs],

--l
[~1<112.

(6.17)

This expression differs f r o m the analogous representations (6.2), (6.9), (6.14) by the presence in f r o n t o f s o f the coefficients f l r 1. Here the analogues o f A + D - B - C and A + B - C - D are 1 + fl - 1 - fl = 0 and 1 + 1 - fl - fl > 0, formed by the coefficients o f s. Therefore, just as in the second case o f T h e o r e m 5.1, the f u n c t i o n ~ ( x ) has here various representations, continuing analytically each other through the c i r c u m f e r e n c e I x ] = 1 into the sector I arg x I < (A + B - C - D)~r/2 -- (1 - fl)Tr. These representations are obtained by formulas similar to (4.46), (4.47), (4.48), (4.49), by introducing the values o f the poles s --- - k and s = c~ + k into ~ * (s)x-q the replacement o f l"(-k) by ( - l ) k / k ! , and summation with respect to all k:

o~(x)=r(~)2sin(l~kz)(~)~ ~,

Ixl
#=0

:e{x)=r(~)x-~ si~tl~(~+k)=l{~)o{~-~~-~, l~l>t. The obtained series converge in the indicated domains and, after the elimination o f x and fl f r o m them by means o f the formulas

x=(pc--l/--f--s following parameter Condition 2 ~.

~3z~=arcsinbc,

c
f r o m (6.16), they represent and equivalent f o r m for the integral 14=c~x~(x), containing the arbitrary c. Obviously, the integral (6.15) converges for 91a > -1 and 9tp > 0 (if b > O) or ~tp > ] ~b [ (if ~b r 0). ~lc~ > -1 follows f r o m the inequalities (6.17).

Logarithmic Cases. Example 5. We prove that

I x=

t ~-1 In n

dt := ( - - 1)'~rt! ~

a

t~T ~z=O

b>a>O,

b

~,,+1 ,

"

(6.18)

~z4= O.

This equality can be verified also in a straightforward m a n n e r by differentiating both sides with respect to b; however, we make use o f the general method. P e r f o r m i n g in the integral (6.18) the substitution t = ar and extending the definition of the integrands with the aid of the Heaviside f u n c t i o n by the formulas

Sr (x) = x ~ l n ~ H (x - - 1), 15 ~ a ~ 1260

(x),

~2(n)=HCn-- 1), b tz

(6.19)

we obtain the convolution (2.9). By virtue of the formulas 8.4.6.4 and 8.4.2.2 of [164], we have

~(s) The desired value o f

(--1)"-'n! (s + cz)~+~ ' R e s < - - R e c z ;

Yd(x)

Sr 2 (s) - - - - s , R e s < 0 .

(6.20)

can be represented by a Mellin--Barnes integral o f the p r o d u c t o f the functions

(6.20): y+ioo

9g' (x) =

(--l)nn[

I

2,i

x-~

s (s + c~)~+, :ds,

~, ~- Re s < 0,

- - Re oz.

(6.21)

y--ioo

Each of factors (s + a) -~ can be written in the f o r m

(s-f-~)-~ = - - r r( (l - - a - - s )

9 This means that to the f u n c t i o n s - l ( s +

cz)-n-lthere correspond the conditions A = C = 0, B = D = n + 2. Thus, here, just as in Example 4, the second case of T h e o r e m 5.1 holds. T h e r e f o r e , the integral (6.21) for b > a, i.e., for x > 1, has to be evaluated by the sum o f the residues at the right poles s = 0 and s = - a , taken with the minus sign. At the simple pole s = 0 the residue a - n - l i n accordance with (4.48) is obtained by replacing s -- 0 in the coefficient of s-l: res Is -I (s + o,,)-"-zx -'] = c~-"-1.

(6.22)

s=O

For the determination o f the residue at the (n + l ) - f o l d pole s = - a , first one has to p e r f o r m the substitution s = - a + and to expand the integrand in powers of e up to e-l:

xae-elnx

X--.x

s (s + cz)TM

- - ~ (1 --sic9

xCL ~.~ (--~ ln x)k ~ s"+' --

--c~8"§

k=0

1

s]

~-~

~- =

j=0

x" ' ~ (--In x)e

Obviously, the desired residue is contained in the r o u n d parentheses o f the last expression: T/

res [s-l (s + oO-'~-lx -s] s~--~x

x ~1 ~ c*~§

(--~,ln /el x)k ""

(6.23)

k~O

Now, in order to obtain the value o f the integral (6.18), in accordance with (6.19), (6.21) we multiply the sum o f the residues (6.22), (6.23) by (-1)nnla a, we introduce b / a instead of x, and we change the sign o f the obtained expression since the poles s = 0, - ~ are situated to the right rather than the left of the integration contour (~t - ioo, 7 + ioo). Equality (6.23) can be obtained also on the basis of the formulas (4.54), (4.55), writing first the integrand f r o m (6.21) in the f o r m

s (s + cqn*, - - ( - - 1 ) nF

sz, s l - - c z . . . . . s l - - c z s l + l, s l + l - - o c . . . . . s l - q - l - - ~ 9 "

sl = - s,

(1/x)-""

n+l'~a3

[Re s~ > O, Re cq.

Introducing then into the indicated formulas

N = 2 , r = 1, qi =n.-~l, a~z=a~l . . . . =an+l, z = - - c ~ , a t 2 = 0 , c t = l , c z = c a = . . . = c a + 2 = 1--cz

1261

and taking into account that in the given case we have the equalities

G,,-- A(2) ,n,

l=0,

T~--Ck,

/~n+l,I,0 --

X~ (~

it is easy to obtain f o r m u l a (6.23). E x a m p l e 6. We evaluate the following integral of the product of two exponential integral functions:

(6.24) 0

After the substitution ct = r and the introduction of the notations

I8=c-~3~(x), x = b c , ,~1 (T) = ~ E i ( - - " 0 , Yd=(n ) = E i (--rl)

(6.25)

this integral will have the f o r m of the Mellin convolution (2.9). The Mellin t r a n s f o r m of the functions Ydj ('c) are given in 8.4.11.1, 8.4.1.5 f r o m [164]. Multiplying them, we obtain the value of Yd(x) in the f o r m of the Mellin--Barnes integral 1 ~' oo

.7{' (x) = ~

Yd* (s) x - s d s ,

.Td* (s) =

r (s) P (a + s) s (~ + s)

(6.26)

y--ioo

"f> 0 , - - R e c~. We consider the four cases of the disposition of the poles of the integrand of (6.26), taking into account that all of them are situated to the left of the integration contour. 1. Let c~ r _+n, n = 0, 1, 2 ..... T h e n the points s = 0 and s = - a are double poles, while s = - k - 1, s -- - a - k - 1, k = 0, 1, 2 ..... are simple poles. T h e residues at the simple poles are obtained by the f o r m u l a (4.48) by substituting the corresponding values of s into the f u n c t i o n 3r (s)x -~ with the subsequent r e p l a c e m e n t of l"(-k - 1) by ( - l ) k + l / ( k + 1)!: res

s~-k-1

res

(Yg'*(s)x-0=

r ( a - - k -- 1) (--k

(k+l)!

(--~--k

' "

P (--o~--k--

(~*(s)x-O

(--x)k+,

1)(~--k--1) --

1) x a

1)(--~--1)

(--x)/~+ t

(k+l)!"

(6.27)

s=--ct--k--I

At the double poles the residues are obtained by the method described in E x a m p l e 5, with the aid of the formulas (2.15), (2.16):

yd*(S) X - S N S s--~O

S [ I + ~ p ( 1 ) S ] - if- 1 - -

X[1--slnx]--~s

X'*(s)x-~=x

~a)

l+,(1)

~ r(--~+8)r(8)

r(a)[l+q~(a)s]X

s---d-+q~(a)s--slnx x-~--x~

r(--~______~)•

F r o m these expansions we obtain the following values for the residues:

--

1262

-s,

r (~) I , ( 1 ) _ 1 + ,

(cz)_ in x ] '

,

r (--~) x ~

res (Yd* (s) x -*)

q) (1)-[-- -~- + q~ ( - - o~)-- In x ].

(6.28)

S~--G6

S u m m i n g all the o b t a i n e d r e s i d u e s , we o b t a i n the value o f the integral (6.24): co

Z i., (c,__ k__ 1) (__x)~ X' (x) = x k=o (k + 1) (co--k-- l)(k + I ) ! oo

~Xa+l~O

P(--~--'k--l)(--x)k

= (~ZU(Tg~(~u

-§

-1(6.29)

_+x, +_2

. . . .

2) L e t c~ = n, n = 1, 2, 3 . . . . . T h e n the points s = 0 and s = - n - k - 1, k = 0, I, 2 ..... are d o u b l e poles, s -- - k 1, k = 0, 1,..., n - 2, are s i m p l e poles, while at the p o i n t s = - n we have a triple pole. T h e residues at the points s -- - k 1, k = 0, 1..... n - 2, a n d s = 0 are o b t a i n e d b y the f o r m u l a s (6.27), (6.28) b y i n t r o d u c i n g the v a l u e ~ = n. T h e residues at the points s = - n - k - 1 are o b t a i n e d f r o m the e x p a n s i o n o f the f o r m

fit'* ( s) x-s = x n§ s=-n-k-I +e

r(--n--k--I

+s)r(--k--1 +s) x_~

(--n--le-- 1 + e) ( ~ k - - I + s)

(n+k+l)(k+l)

1-{n+~+l ( ~ 1)k+~

X [l + ~p(n + k + 2 ) ~ ] ~ [ l

~

(n+k+l)!s

~z~

+ ~ ( k + 2) 8l [ 1 - - e l n x],

w h i c h follows f r o m f o r m u l a s (2.11), (2.12). Thus, we have

xn+k+~(-- 1)r~ res (9~* (s) x-O - = - ( n + k + l ) ( k + l ) ( n + k + l ) l ( k + l ) [ • s=-n-k-I

+[,,+I k + l

1 +

+k-'-~

(6,30)

~p(n+k+2)+~(k+2)--lnx].

In o r d e r to o b t a i n the r e s i d u e at the t r i p l e pole s -- - n , we have to c o n s i d e r a g a i n the c o r r e s p o n d i n g expansion in p o w e r s o f s + n = e, b u t this time w i t h the a c c u r a c y o f terms up to s 2 in the n u m e r a t o r . M a k i n g use again o f the f o r m u l a s (2.15), ( 2 . t 6 ) , we o b t a i n as a result the f o l l o w i n g e x p a n s i o n

~=_n+~

•

(--n+~)~

~0--ns

-h-+~

•

8']

(--1)= I1 .~ ~-+*(n+l)~+(~+3r

X

-~mx-t-~j,

•

~' (1)=~. Thus,

res (Jd*(s)x-~)-- x~(--1)~-'n!n Iln~ ----s

x (~(1)+~(n+ 1 ) + 1 ) +

n2 + 2 ~ ( l ) + 2~t(n+ l)--2~" (n+ +V(1),(n+

1 ) + --~-+*(1) l,(n+

~_1._~_

(6.31)

1)].

1263

Summing the residues (6.27), (6.28), (6.30), (6.31), we obtain the value of

-

=

Yg' (x):

[, ( , I - ) + ,

n--2

P ( n - - k - - l ) (-- x)k+,

--k~=O (k+l)(n--k--1)(k+l)!

@

co

&-k

+ ( - - 1)~x~+l ,~=o(n + k + l)(k + I) ( n + k + 1)! (/~ + 1)! >< X [n+}+t

(6.32)

+k-@i+*(n+k+2)+*(k+2)--lnxl+ q-

res

(if/'* (s) x-O.

3) Let c~ = 0. T h e n the point s = 0 is a quadruple pole, while the points s = - k - 1, k = 0, I .... , are double poles with residues obtained by the f o r m u l a (6.30) for n = 0. In order to obtain the residue at the point s = 0 we consider the expansion o f the f u n c t i o n ,7{*(s)x -~ in powers of s = e with the a c c u r a c y o f terms up to e z in the numerator:

ffl,,(s)x_ ~ r~(~) _~ a=0

~ e----T- X

+2"(1)+~'~*(I)+2'"(I)12

I 1 [1+,(1)8_1_6,~(1)+.~ ~

a~o e'-' e 2

"

12

e2"-~-

~ 3 2 [ ] . - - ~ l n x - ~ ~'ln2x2, ~ln~x]3' "

The coefficient o f e-1 gives the required residue: res (YE* (s) x - O = n*@(1)q'-4@a(1)+~"(1) s=ct=0

3

(6.33)

1 [12~ 2 ( 1 ) + zt2] In X + ~ (1) In 2 x--Ing-Yx

6

6

Summing the residues (6.30) f o r n = 0 and (6.33), we obtain the following value of the f u n c t i o n

Yd (x):

co

9Y'Y(x),=x = ( k + l ) . ( k ! ) 2 ln3-~x+ ,

2~(k+2)--lnx

--

(1) In2 x - - ~ -I [12q~2(1)+r~ z] in x +

(6.34)

_~ : ~ * ( ] ) + 4 . ~ ( I ) + ~ " 0 ) 3 The last value can be obtained also with the aid of the formulas (4.54), (4.55) if we set there r = N = 1, qt = 4 , /=0, [ a ] = 0, 0, 0, 0, ( c ) = 1,1 and we take into account the equalities V~=Fk, Th=Ck, R410 ~ i 4) Let c~ = -n, n = 1, 2, 3 ..... This case is similar to case 2 since the integral (6.26), after replacing s by s - a reduces to the p r o d u c t o f x a by the corresponding integral of (6.26), in which instead o f c~ one must have -~. Therefore, in this case the value of ~ ( x ) can be obtained by multiplying the r i g h t - h a n d side o f (6.32) by x -n. P e r f o r m i n g now in the formulas (6.29), (6.32), (6.34) the inverse substitution (6.25), we obtain the expression for the integral (6.24) f o r any values o f c~. This integral converges for 9~b, 9~c > 0 since f o r these values of the parameters the integrand decreases exponentially at 0 and at c~. In conclusion we note that the sums in the f o r m u l a (6.29) can be written in terms o f generalized h y p e r g e o m e t r i c functions. For example,

r (a--k--1)(--x) k k=0 (k + l ) ( ~ k ~ l ) ( k + l ) !

F(r a--1

1264

3F4(I I, I--~; 2, 2, 2--r

2--r

x).

Example 7. We evaluate the integrals I ~ = ~3 t e-I (t - - a)~-' t In (ct + a) l dt.

a

(6.35)

(ln l c t - - d l ]

For this we can use two methods: 1) Obviously,

I+ - -

dpa A (p) [o=0,

A (P)

=i t~-1(et(t-a)~-1+ el)9

dt.

tTt

For the d e t e r m i n a t i o n of A(a) we a p p l y the technique presented in Examples 1-3, which, in the most reduced f o r m leads to the following Mellin c o r r e s p o n d e n c e Yd(x)-,~-,Td* (s):

1

~ct a+~- Id-~

for a, c, d, 9t/3 > 0, ~l(c~ + 13 - p) < 1. D i f f e r e n t i a t i n g n o w A(p) at p = 0, we can obtain the desired value of 17+. 2) H o w e v e r , the value of 17+, as well as that of I7-, can be obtained in a simple w a y by a n o t h e r m e t h o d , namely by the s t r a i g h t f o r w a r d application to 17-+ of the above presented c o m p u t a t i o n scheme. Making use of the previous substitution t ----ar and of the f o r m u l a s 8.4.2.4, 8.4.1.5, 8.4.6.5, 8.4.6.7 f r o m [164], we obtain

l~-----a"+~-I

~ - 1 ( ~ - - 1)~-1 j'In ('c/x + 1) ][ l n ] f i x - - I [f d~/x=(ac)-,a +

(6.36)

Since 0 < 91s < 1, 1 - 9t(cz +/3), it follows that the double pole s = 0 is situated to the left of the contour ~Rs = 7. We introduce the notations

In accordance with the signs f o r s, the function [164]); h e r e A =--D= 1, B = 2 ,

C=0,

3~{'~(s), can be r e f e r r e d to the type

( "+- -_- - - )

i.e., for it we h a v e A + D - B - C = 0 a n d A + B - C - D = 2 .

by virtue of the equality cos s~r = 7r/I'[1/2 + s, 1/2 - s], belongs to the type

(~ /

(see 8.4.52 f r o m Function

-- --)

YdL(s)

, for w h i c h ,4 + D - S - C /

A + B - C - D :-- 0. T h e r e f o r e , the original 3~'~-(s) , corresponding to Yl'+ (x) will be an analytic f u n c t i o n in sector [ arg x [ < 2.r/2 (see (5.9)), while ,7/'_ (x) will be a piecewise analytic function with a possible discontinuity on the

1265

c i r c u m f e r e n c e I x I = 1. Function o~d'+(x) can be evaluated by any of the following methods: for I x [ < 1 it can be represented in terms of the function P~A+(x), which is the sum of the residues at the left simple poles s = - k - 1, k = 0, 1, 2 ..... and at the double pole s = 0; for I x I > 1 it can be represented in terms of the function XlB+(1/x), equal to the sum of the residues at the right poles s = k + 1 and s = 1 - ~ - fl + k, k = 0, 1, 2 ..... which m a y be s u p e r i m p o s e d on each other under the condition cz + fl = 0, -1, -2 ..... If the obtained expressions for ,7/'+ (x) are expressed in terms of the function pFq(X), then f r o m the two functions EA + and EB+(1/x) one can retain that which has a simpler f o r m , discarding the restrictions on x. T h e other function is an analytic continuation of the first one. In the case of the f u n c t i o n ,Td_(x) both sums I2A-(x)and X2B-(I/x), for any way of writing their r i g h t - h a n d sides, have to be retained since they do not continue analytically each other. Thus, replacing in if/'; (s) x -~ the value of s by - k - 1 or k + 1, or even 1 - cz -/~ + k, and then replacing r ( - k - 1) by _+(-1)k+l/(k + 1)! (in accordance with (4.48), we obtain the following formulas for the residues at the simple poles: res

(Y{'~ (s) x - ' ) =

s=--/~--I

=F[k+1 89

s

1

(k + 1) res (3~d~ (s) x - ' ) =

(6.38)

a} (--x),+, (e + 1)! '

s=k+l

--a--/~ res

(6.39)

(~ + 1)! '

( ~ : (s) x-~)----- - - r [1 - - a - - ~ l + ; ' _ _ ~ + [ ~ - -

1 - - k ] )<

s=l--~--~+k

X(cos(1--c~--~+k)

. -~.~

~+[~#o, -L

(6.40)

,

-2 ....

(the minus signs in front of r[: :] in the last equalities appear because of the minuses in f r o m of s in the arguments - s and l - e - fl - s, which are related with the corresponding right poles). The residues at the double poles s = 0 and s = n + k + l, k = 0, l, 2 .... (for c~ + fl = -n, n = 0, 1, 2 .... ) are c o m p u t e d as in E x a m p l e 6, f r o m the following expansions in the powers of e: 9 1 [1 + r Ed++_(s)x-'N W S~8--~O

9=.+k+~+,

1 [ 1 - - ~ ( 1 ) ~ ] r (1--a--I~) _----7r(1--a) X[1--~(1--~z--~)el[l+~p

[~--/~-- e

cos(n+k+

(1--~)~][1--elnx],

1+e)=

.~0 r n

1

a+f~=-n

X [1-4:-, ( n + k + 1) el [1 + 4 (13-- k) ~1•

(--l)"+k+l --(,,+ k + ~)t [ 1 - - , ( n + k + 2 )

a] (-1)* [ 1 - - , ( / ~ + 1) , l x

X { ( _ 1;.+k+l} x-~-k-~ [1 - - e In x ] . The coefficients of e -1 give the required values of the residues: res s=O

(ark

x-,)=

[ 1 - -t--c~ ~--[3] [, (l_~)_V(l_ = --r L res s=n+k+l ~+~=-n

X [,(n+k

( *,1~ ~,y-t'=,s,x-,,=rL + 1)+,

[ n + k + 1] (-- 1)n+~x-n-k-~ ( +_1)~+k+l

([~-- k ) - - ,

[3--k j

(6.41)

~ _ ~ ) _ l n x],

(n+k+l)!kl

(n+k ~2)--,(k

X

(6.42)

+ I) - - In x].

S u m m i n g the corresponding residues (6.38)-(6.42), we obtain the following expressions of the original of the function (6.37): cc

,9g'+(x)=~,

res

k=O s ~ - - ~ - - I

1266

* -8 (,3f*+_(s)x-*)-kres(Yg+(s)x), s=O

Ix[<1,

(6.43)

~"~"~ s ~ - ~ + l

=

]x[>l,

s=I-c~-13+~

~z+[~@0, - - 1, - - 2 . . . . .

n--I

(6.44)

oo

res (~;(s)x-~)--~_~

,7/'• ( x ) = -- x k~0

s=k+l

k=0

[xl>l,

a+l~=--n,

res s~n+k+l

(i~l'*~(s)x-'),

(6.45)

n=0,1,2 ....

(the minus signs in the last formulas are due to the fact that one sums the residues at the right poles). Introducing now the values (6.38)-(6.45) into the formula (6.36) instead of the function (6.37), replacing x by (ac)-~d, carrying out transformations, and expressing some sums in terms of the functions vFq, we obtain finally the following expressions for the integrals (6.35): d

I ~ = + e (i3, 2 - - ~ - - fi) 7 a,~+o -2 •

•

1, 2 - ~ - ~ ;

2, ~-~: ~

d ~-)+

+ B ( ~ , l - - a - - ~ ) a ~ + t 3 -1 [~ ( 1 - - a) - - ~b( 1 - - a - - [ l ) + l n ( a c ) ] ,

[dl<[acl, arZ+~-2d

B(~,--~-~)3e~(l, ~, ~+~; ~ + ~ + ~ , 2; A-

~)•

__.

~' 1--~'---~

[ ctg (a + [I) r~ J )'( (6.46)

-~-B(~, 1--a--~)~.a+~-ltrid,

[dl>lact,

o~+~--/=0, - - 1 , - - 2 . . . . .

r/--i

[~=-+a~+~-2d X (a+l)k (7) ------~--B ( 1 ~ , - a - - [~) (k + 1 ) ( a + p + 1)~ -~

k_

k=0 "co

[ [dl>lac[,

a-b~--n,

n=O, 1,2 . . . . .

Obviously, the integrals (6.35) converge under the corresponding conditions a, R e i g > 0 ;

Re(a+[~)<(1,

f[arg(cx+d)l<~ I c, d > 0

for

a~
which in the cases ac + d = 1 can be relaxed somewhat. The expression for I7 +, by virtue of the analytic dependence on (ac)-ld, is valid also for [ d [ > [ac [. 3 ~. Functions of Exponential Growth. Example 8. We evaluate the integral oo

I8= ~ t"-Ie-atoF2 (b, c;

-- zt) dt.

In order to transform it to a convolution form, we perform the substitution

~,(T)=*%F2(b, c; --*),

(x) = z~Is,

(6.47)

zt = r and

X'2(n)=e-'/~,

z

a

1267

On the basis of the formulas 8.4.51.1, 8.4.1.5, 8.4.3.2 from [164], we have the relations

~ (~) ,~--~ ~e~'(s) = r [b, c] r[b[ ~ -+ s~ - s, c - a - s l ' s=--a--kE~0

~(s)=r

+,

k=0,1,2 .....

(-s),

~,=Res <0.

The first of them means that the Mellin--Barnes integral (4.39) of Yd~*(s) converges only along the left loop L_~, containing all the left poles s = -e, - k, situated in the domain ~ + to the left of the contour L_oo. This loop cannot be straightened into the contour ( ' / - ioo, 7 + i ~ ) since for Yd~*(s) we have the inequality A + B - C - D -- -1 < 0 and corresponding function Yd~0;) increases exponentially at 0% having a principal term of order O(~ca~-b-~+~e30/3/2). Therefore, the Mellin transform of Yd~('c) does not exist. Making use of Parseval's equality (2.12) and of Theorem 5.1. for A + D = 3 > B + C = 1, it is easy to obtain the value of the integral (6.47) in the form Z

I8=a-~F(~)~F2(~; b, c; ---d )' Recx>0. This integral remains convergent for 9ta > 0 and for arbitrary arg z since the function e - ~ at oo decreases exponentially faster than oF2(b, c; -zt) increases exponentially. Example 9. We consider the integral co

i9 ~ j" t ok-1e-~tI~ (bt) dt.

(6.48)

0

Turning to formulas 8.4.3.2, 8.4.1.6, 8.4.22.1 from [164], here we could apply the arguments from Example 8 since for x --* + ~ the function Iu(x) increases exponentially. Instead of this we shall make use of formula 8.4.22.3 from [164]. For the functions e-X/2I~,(x/2)we introduce the notations

*=2bt,

2b x-- a--b '

~1

(rt)=e-1/nq c*, (6.49)

Yg'2(x)=e-~/2I~(~),

Yd(x)=x-~(2O)~I 9.

Then the integral (6.48) turns into the convolution integral (2.9). Multiplying the images convoluted functions, we obtain Yd*(s)=--~F[ Computing the original integral (6.48):

3~(x)

sq-v'lq-v-sl/2-s' ~ - - s ] , - - R e v K R e s < l / 2 ,

3tdj*(s), ] = 1 , 2

, o f the

Rea.

by Theorem 5.1, after the inverse substitution of (6.49) we obtain the value of the

I

(2b)~,a 2v+l;

b~_~_vp[~-~l/2, a-j-v]2 F ( ~ + 1 , 2b ~. ~),

Re(a+v)~0,

Rea~]Reb]~0

(z+u; (6.50)

(condition ~ v > - 1 / 2 can be omitted here, being a derivation condition). Example 10. We consider the integral oo

Iao=S 0

1268

t~.--I

K~(at)I~(bt)dt.

(6.51)

Making use of formula l~(z) = (-i)~Jv(iz), we transform the function I u into Ju, for which the Mellin transform exists. Then the integral (6.5 I) assumes the form co

I10-- ( - - i) v j' t~-lK~ (at) Yv (ibt) dr. 0

From here, after the substitution

t~

bi '

x ~ -- ~a ,

X'~(~I)=K~

V-

~

'

we obtain equality (2.9). The images of the corresponding functions can be found by making use of formulas 8.4.23.2, 8.4.t.5, 8.4.19.1 f r o m [164]. Multiplying them, we obtain

3r

1

(s)=-~ F

[(aq-~)/2--s,

(a--~)/2--s, lq-v/2--s

--Rev/2
v/2-t-s]

'

Re (o~ + ~)/2.

Now, with the aid of Theorem 5.1 it is easy to find the original ff/'(x), and then the value of the integral (6.51) itself:

X~F~ c,+ + v , Re ( ~ + v +

2

'

(6.52)

a~'

ix), Re(a__+ b ) > 0 .

Setting here # = 1/2, replacing c~ by c~ + 1/2 and making use of the formula (6.52) we can obtain the equalities (6.48), (6.50). 4 ~. Some General Integrals. Exarnple 11. We calculate the integral

Kt/2(x) =l/zt/(2z)e -z,

from (6.51),

(90

lu(x)=Sta-le-x/t-trdt,

r>0,

x@0.

(6.53)

0

Setting

f f d 1 ( x ) = e -x,

3f2(x)=e-xrx%

and taking into account formulas 8.4.3.1, 8.4.1.5 - 6 from [164] and (2.5), (2.9), (2.11), we obtain the equality

(6.54) Res >0,

- - R e oz.

The function (6.54) differs f r o m the Mellin transforms Y{~*(s) of the previous examples (except Example 4) by the presence in front of s of the coefficients 1 and r - I , where, in general, r ~ I. The analogues of the dimensions A, B, C, D for it are the quantities 1 + r -1, 0, 0, 0. Since A + D - B - C = I + r -a > 0, it follows that the original I n ( x ) for the image (6.54) must be evaluated in terms of the residues in the two series of left poles s = - m and s = -c~ - rk, m, k = 0, 1, 2 ..... which may be partially superposed if the equation

m--rte=~z,

m, k = 0 , t, 2 . . . .

(6.55)

is solvable. 1269

!) Assume that equation (6.55) is not solvable in integers. Then all the poles of the function (6.54) are simple and, moreover,

s=--m res

s---a-lz r

m!

{F[s, L-~]x-~}

(--lpr i,(_cz_kr) le I

x~+kk.

For the derivation of these formulas we have made use of the equality (4.48), but in the second case the coefficient r -1 of s has led to an additional factor r in the right-hand side of the formula. These residues can be easily summed, leading to the value of the integral (6.53):

o~

oo

IlI (de) = m~=0-~-'~r Ia ('~--~)(--'"xT)m-~'X~ k--0 ~ I~ (--~--kr)/~!" ( - - Xr)k'

(6.56)

Since the integral (6.53) converges for all r > 0 and a, not only for x > 0 but also for 91x > 0 (or ~Rx = 0, x ~ 0 and 91a > -1), it follows that the equality between (6.53) and (6.56) is preserved for any r > 0 and ~Rx > 0 (or 9tx = 0, x 4,0 and ~Rc~> -1).If a ---, m - k r , m, k = 0, 1, 2 ..... then the left-hand side of (6.56) is continuous; therefore, also the right-hand side of (6.56) is continuous, where for a = m - k r one has an indeterminacy of the oo -- oo type. This indeterminacy can be solved by continuity or by using the residues at the corresponding double poles s = - m = - a - r k . 2) We assume that the equation (6.55) is solvable for some values of m and k. In order to determine the residues at such double poles s = - m = - a - r k , we expand the functions (6.54) and x -s in powers of ~ = s + m = s + a + r k , taking into account in the numerators of the terms containing ~1 (see Examples 5-7): I ~ , ( s ) x -~

1 (_~)~ [1+8,(.z+~)]

~0 r

m!8

r(--1)ky k~ --

X [1 q - -sV , (k q- 1) ] x'~ [1-- ~ In x]. Multiplying these expansions and separating the coefficient of e- i , we obtain

res

(1~1 (s) x -~) =

1)m+kxm [~l~(rrt+ 1) + ~I- * ( k + "=: ( - - m.k'

1)-- In x ].

Thus, in the general case we have 1 In(x)=

X

~2)

[ap(l+m)+~-,(1-4-k)--ln

X]

(--1)rn+kxm

re!e!

m=~+kr

m,k

where the symbol

2)

Z( m=~x+kr

denotes summation with respect to all the double poles, i.e., with respect to all m, k = O, 1,

2 ..... connected by the equality (6.55), while

~ ( 1 ) denotes summation with respect to the remaining indices m, k,

tt~k

which correspond to the simple poles. The empty sums are replaced by zero. 3) Assume now that r is a rational number: r = p / q , where p and q are relatively prime natural numbers. We show that in this case the function (6.56) can be written in terms of generalized hypergeometric functions. For this we transform r ( - c t - k r ) according to the formula F(z)r(1 - z) = r / s i n Zr and then according to the multiplication formula: ~p--Ct--kolq--1/2

I' ( - - ~ - - k r ) =

~ n (cz + k p / q ) z P i l + ~ + k p / q ) ~--- - -

O--1 l=0

1270

(2Z~)(p--I)f2

Let k = nq + j, where j = 0, 1, 2 ..... q - 1, while n = 0, I, 2 ..... Then, with the aid of the formulas of Appendix 11.3 in [164], we obtain the representation (-- l)r

r(-oc-kr)=

12-rip (2g) (o-1)12

sin (]r -}-a) z~~

P n .q-

l=0

( __ p ) _rtp

sin(jr+cQM'(jr+l+c~)

off(-~--4 l + a +pl )n l=0

(--- p)-nPr ( - - s - - j r )

II'[J l=0

l+ct+t

i-g+

)n

p

If we perform similar operations also with c r ( - ~ - ~ ) , ml, kl, then equality (6.56)can be written in the form

Iu (x)=

o-1 rilH(i'q-lq-l~

~=0 n=0

q-t II/i--r

/=0 q--1

q_x ~ ' ~

=

l-t-l\

ol,--?--+-7-).

eo

~'~

_ q - ~ q ( - - p ) ~ (--cr

J l X) nI t( = J +o \l +ql

j

= (J~-7+ l + ~ + l ) n O

p--I

~--i (-7-)

5' (--W

(1; A(p,i+l), a(q,

t=O

(6.57)

q--I

i=0

X1Fp, o(1; A(q, j - t - l ) , A(p, 14-, co-t-jr); z),

Z ~- ( -- 1)P+qxPp-Pq-q, where A (p, ~) - ~-, ~ p+ I " ' "

.~ + pp- - 1

We note that, since the vector A(p, i + 1) contains a component equal to 1, the function aFp+q from (6.57) reduces to oFp+q_l .

4) If in the relations (6.53), (6.57) we set r = 1, then we obtain the known formula

~t~-ae-*/t-tdt=F (C~)oFa(1 --r

x)q-x~F (--cr

oF, (1-1- c~; x ) =

0

2x~/2K~(21/r-.~)[Rex>O]

or

[Rex=O,

x@O,

Rea>--l].

Example 12. We evaluate the integral o~ l Lct_le_p t

I12~-- ~

dr, y, r>O.

(6.58)

0

A f t e r the s u b s t i t u t i o n t = ry a n d

Y/'~(rl)=e -~/n, X =(pL])-I

..~2('t')=cr__l,

1 1 2 = y a - - r ~ (X)

(6.59)

1271

we obtain the convolution (2.9). The formulas 8.4.3.2, 8.4.2.6, 8.4.1.5 -6 f r o m [164] lead to the relations

Y,f~*(s)=F(--s), Res
O
r ctg--/--n,

Thus, by virtue of f o r m u l a (2.12) we have

:~ (x)

n

=

- - -2•ir

v co

[(a-k-s)/r,

1--(a-q-s)/r,

--s

v-i~ r [1/2q-(a-~s)/r, 1/2--(~+s)/rJ

x>O,

] x_~ds, (6.60)

r--Rea.

--Rea<~=Res<0,

Here, the analogue of the expression A + D - (B + C) o f T h e o r e m 5.1 is the quantity r -1 + r -1 - (r -1 + 1 + r -1) < 0. Therefore, the integral (6.60) has to be evaluated by the sum o f the residues at the poles s -- k, s -- r + r e - a, k, e -- 0, 1, 2 ..... We assume that all of them are simple; for this it is sufficient that the condition of non-solvability be satisfied for the following Diophantine equation:

rl--k=o:--r,

k,l=O, 1, 2 . . . .

Then, by the same m e t h o d as the one used at the derivation of f o r m u l a (6.56), we obtain

.71' (x) = - - 7-

~

ctg-7-

a+

k=O

i:(-') " ' r rL 3 / 2 + l , r,=o---F--

-1/2--1J

lx ' .... ,/l"

Performing the inverse t r a n s f o r m and making use of a f o r m u l a for the G a m m a f u n c t i o n f r o m A p p e n d i x II.3 o f [164], we finally obtain the following value o f the integral (6.58):

112=

F-

~

ctg-7--- ~ +

k=O co

pr

(py)", r, g, R e ~ , R e p > O . I,-O

Conditions for its c o n v e r g e n c e follow f r o m the conditions in (6.60) and f r o m the requirement for the convergence of the integral (6.58) for complex p. We mention that in [618] a detailed investigation was p e r f o r m e d of an integral o f general form, containing a product of an exponential, sine or cosine, and generalized Laguerre polynomials. 5 ~ Integrals of Several Functions of a Special Form. Example 13. We consider two integrals o f the f o r m

L

oo

r

0

9=

(6.61) "

L

]=0

9@--1, --2, --3 .....

1272

]t

I13=

~--'

e- p ~ -

t

(--tr~2)J

][

(c~'~-~ k r (

2K,~(c+)

j)

j=O

]=0 v = O, + l ,

+2 .....

where g and m are integers such that in the first case [ r n - k + ~ l ( c ~ - p ) [ < l, in the second case [ m - k + 9 1 v [ < 1, and in both cases -2 < g + k + ~tp < 0. Performing, respectively, the following changes of variables and functions

t=pr, ~,(n)=r(o)

x =pz ,

~z=p+v,

[0+n)-~176162 j=o

1,3---- P-~ n-o

jt

(x),

,

(0)j (--~)-J ] j!

j=O

(6.63)

I

2 V T ~- cr,

C2

x=Tfi,

~o/~J ' ~z=29-I-v, llz=2V-'c-vp-OK(x), j=o

J!

'

k

(0 = 2t"K

(2 V r ) -

r

(6.64) Jl

( --

t)i

--

1=0 P

j=0

(--v--j) jt (--04

in both cases we obtain the Mellin convolution formula (2.9). In order' to find the Mellin transform of functions o~, 0 l ) , Jg2(t)we make use of formulas 8.4.2.5, 8.4.3.1 2, 8.4.23.1 from [164], expressing the Mellin transforms of the functions ( l + x ) - L e-*, e -~/~, K~(2t/x). It is easy to see that a~, (TI) differ from the enumerated functions by the absence (or presence) of segments of Taylor series in the neighborhoods of the points 0 or oo. By virtue of formula 8.4.3.3 from [164], we have 17l

e - ~ - ~.~ ~ , - + F j=O

(s)

[ - - m - - 1 < Re s < - - m].

(6.65)

Taking into account the values of the residues from the equalities (6.3), (6.4) and assuming that the line (7 - ioo, 7 + ioo) intercepts k + 1 left and g + 1 right poles, from formula 8.4.2.5 of [164] we derive the correspondence

r

(v) i (1-1- x) - ~

k (P)i (--x)J j=O j[

~,t (o)j(--x)-J ]

--X-O~= 0

j]

]~"

(6.66)

,~r [s, o - s ] , [Re [~+l, - - k - - 1 < Re s < Re p - t - l + 1, - - k ] (here the functions EA+(X) and EB_(X)from Theorem 5.1 and Remark 5.3 assume the values

,

x-O ~ (P)j(~ x .j =0

)_g_)

j ~_~ (P)J (--x) i!

and

j=k+,

. Obviously,

in this case p, e, and k must be connected by the condition -2 < l + k + ~lp < O.

Otherwise, the mentioned line does not exist. If in formula (6.65) we replace s by s + v, and also s by p - s and m by e, then we obtain the correspondences

[

~, (_x)l ..

L j=~ j! J I - - m - - R e v - - 1 ~ R e s ~ - - m - - R e ~],

(6.67)

1273

(--

x-o e-~/~--

~I' (0-- s)

[Re O + l < Re s < Re 9 + l +

(6.68)

1].

In a similar manner, from formula 8.4.23.1 of [164] one derives the relation

i!

1=0

( - x F - x~

2

It

j=O

( - x)J (6.69)

~ r [s, s + "~l I--k--l, --m--Rev--1
--k].

It should be mentioned that the correspondences (6.66) and (6.69) hold only under the conditions p ~ -1, -2, -3, ... and v = 0, _+1, _+2. . . . . Otherwise, their right-hand sides have multiple poles and instead of the indicated correspondences we have r

IS 1

- - -e

~] (~-V-4-i7 ( - - X ) T - - I 2-1 ~ (1, 1; r + 2 ; - x ) + ( oj~

r--/~--I

1)r-lxk+l •

xY

j~__o(k + ] + 1)1 (m--k

[--9=r-~k+

1)I [q~(k + j q- 2) -- ~ ( r - - k -- ] ) - - lnx[

1, k + 2 . . . . ; - - k - - 1 < R e s <

--k];

C~

F [s, n-t-s],-,(-- 1)"xk+a~

#

1=o (k+ j ' t ' l ) t ( k ' t - j m n ' l ' l ) l

X

[~p(k + j + 2) .-t-~ (k + j -- n-t- 2)-- In x] [ ~ = n - ~ 0 , 1, 2 . . . . . k; - - k - - 1 < R e s < - - k ] . Multiplying the obtained Mellin transforms of the functions ou{'~(x), a~2(x), i.e., the right-hand sides of the formulas (6.66), (6.67) and (6.68), (6.69), by the convolution theorem (2.3) we obtain the following formula for the Mellin transforms of both integrals:

~ * ( s ) = r [ s , v+s, o--s] [ - - k - - l , - - m - - R e v - - l , Re p + l < R e s < R e p-t=l+ 1, - - m - - R e v; --k].

(6.70)

Making use again of formula 8.4.46.1 from [164], we obtain the following correspondence:

r [s, v + s , p - s l ~ r [o, p + v l ~(p, 1 -,~; x ) -

~ r [~--J,p+J] ]I

m

(__x)/_xv x

] =o

r [--~--1,j!p + v + j ]

(--xy--

j =o

(6.71)

l

x-~

P [P+J'

P+~+]]/

fi

,--

x)-]

.

[9#--I,

+__1, +_2. . . . ; - - k - - l , - - m - - R e v - - 1 , /+l,

--m--Rev,

--2, --3,

....

" v4:0,

Rep+l
For v = n _< k this correspondence turns into

r[s, n + s , o - s l u r n--1 j =0

1274

[p, p + n ) ~F(p, l - a ;

X)--

l

v [ n - j , p + j l ( _ _ x ) J _ x _ o ~ rio+J, ~+n+ Jl ( - x ) - J - j! jl j=O

k--n

(_x)~ ~ r(p+n+ l) xj i=o (n+j)!j! [q~(n+j+l)+,(j+l)-~ ( p + n + j)--lnx] [p@ --1, - - 2 , - - 3 . . . . ; v-~n~O, 1,2 . . . . . k; --k--l, Rep+l
(6.72)

while f o r u = n _ _ , k a n d p = - r < - k i n t o

F [s,n+s, --r--s]~, r--k--1

~-* (--1)k+r+n xk+l X (--x)] '" 2 (k+j+l)l(k~-j--n+l) (r--k--j--1)t X j=o {ln~ x - - 2 [~ (k + j + 2 ) + a p ( k + j - - n + 2 ) - - ~ z 2 + , ~ (k + j 4 2 ) + ,

(r - - k -- j)] In x +

2 (k + j - - n + 2 ) + , z

( r - - k -- ] ) - -

, ' (k + j + 2 ) - , , (k + j - n + 2 ) - , ' (r - k - ] ) + 2~ (k + j + 2) ~ (le + j --n + 2)-- 2o2(k + j + 2) ~p(r-- k - - j ) - -

- - 2 , ( k + j - - n + 2) , ( r - - k - - j)]+ ( - - 1)n xr~:I

(6.73)

llxJ

j=o (r+j+l)t(rq-j--n+ 1)1 X

[,(r + j + 2 ) + , ( r + j - - n + 2)~,a;(j + l)--lnx] [ n = O , 1,2 . . . . . k; r = k + l , k + 2 . . . . ; - - k - - l < R e s < : - - k ] .

We denote by fir(x) the functions given in the right-hand sides of the formulas (6.71)-(6.73). Their form depends on the conditions on the parameters v, p, but in all cases the right-hand sides of these formulas express the desired original Yf(x) , whose image has the form (6.70). If in the obtained expression for ,7f(x) we perform the inverse substitution of (6.63), (6.64), then we obtain the corresponding values Iz3 and I13" of the integrals (6.61) and (6.62). If - m - 1 < ~ s < -m, then it is convenient to transform the right-hand side of (7.65) into the form

r (s)'=(-1)m+'r [ s + m +l'- s - m - s ] "

(6.74)

To formula (6.7'4) one can apply directly Theorem 5.1 since gl(s + m + 1), 91(-m - s) > 0. Making use of Theorem 5.1 in the case A = B = D = I, C = 0, from the right-hand side of (6.74) we obtain easily the correspondence F (s) ~ (--x)m+' r (m + 2) 1F1 (1;

m+2; --x),

which, together with (6.65), leads to the equality _

O; ,,, +

-

r

_

+

In the same way, with the aid of formula (6.74), the function (6.70) can be transformed into

~ * (s)~-(-- 1)l+~+k-lr I S + k + 1, - - k - - S , S+ ~ + m + 1, [ 1--s, 1 --v--S, --~--m--s, s--p--l, l+p+l--s l s--p+l

1275

We mention that in this case the formal application of Theorem 5.1 would have led to an equivalent, but significantly more complicated finite expression. Now, if in the inequalities from (6.70) we perform the substitution u = a - p (for the integral (6.61)) and p = (c~ u)/2 (for (6.62)) and then we add the restrictions ] arg z I < 7r, 91p > 0 for the integral (6.61) and 91c, ~lp > 0 for (6.62), then we determine all the convergence conditions for the integral (6.61) in the form IRe (c~--p) + m - - k I < I, - - 2 < R e o~qzmq--l 0 , and for the integral (6.62) in the form

--2O. The conditions 91p, 91c > 0 mean the exponential decrease at c~ of the functions e-p ~, e-p ~' and K~,(cr), the condition I arg z I < r means that z ~ (-c~, 0), while the remaining conditions are a direct consequence of the existence of ~Ils, satisfying the system of inequalities from (6.70). 6 ~ Integrals Reducing to Multiple Mellin Convolutions. Now we consider two examples of integrals that are reduced to the multiple Mellin convolution (2.10) for n = 2 with three functions ~ j (t). In the first example we make use of the series decomposition of one of the functions o~5 ; in the second example we apply formally the computation scheme from Section 3. Example 14. We evaluate the integral cx~

N~ (b, p ) : S ~-~e-~'~'-Pr err (c'0 dr.

(6.75)

0

Turning to 8.4.3, 8.4.14 from [164], we note that in the table there are given functions which allow us to evaluate this integral only in particular cases, when p = 0 or b = 0. We assume that the value of Na(b, 0) has been found. Then it is easy to express also the value of the integral Na(b, p) if we take into account that the function e-P'can be expanded into a series of powers of - p r and if we assume the possibility of the t e r m - b y - t e r m integration of this series: oe

~ (_p)k ~v~+~ ~r (b, 0). N= (b, p ) ~ ,z~"---s

(6.76)

/~0

In order to determine (6.75) we perform, in the integral N~(b, 0), the substitution

b~=V-t,

cl/x=b,

X l ( ~ ) = e r f ( l @ ~ ), (6.77)

,7l'2 ('0 ~e-~'cr

,7t' (x)= 2b =N~ (b, 0),

after which we obtain the one-dimensional Mellin convolution (2.9). Turning to the formulas 8.4.14.4, 8.4.3.1, 8.4.1.5 from [1641, we find the product of Mellin transform 9g'* (s) =,Jr (s)Jf2* (s) in the form ,Y/'* ( s ) = I--~--F I s ~' l / _

sSq-C~/2'q1 1/2--s],

0,--Re~/2
The preimage ,7/'(x) of this expression is evaluated with the aid of Theorem 5.1. After the inverse substitution of (6.77) we obtain

Na(b, O ) ~

-I.-~-1

/2b

~ ' a q - 1 ) . ~ (1 aq-1 3 zt'l 2 ' 2 ; 2;

cl ( ~

Recz> --1.

1276

c2 )

bz '

Replacing in this equality a by c~ + k, we form and evaluate the series (6.76), dividing it into two series with even and odd indices k:

b-c'-~c V~

N ~ (b, p)

~_ ( k,

p~kr((a+k+l)/2+m)(I/2),~ /

o - - ~-]

kl (3/2)raml

c~

c'~"

k--~'}

21~

(6.78) ]/-~"~+I F

a~J'l

7;" ' 2' 2' 2;

b~'

3

V~-0~+2 cp r ( 2 ) l+t S1l ( , ~ _ . A y l , ~ ;. 1 ~3 _~;

4b 2

cz , _ ~ p') " b2

For the derivation of the last part of this equality we have used the duplication formula for the Gamma function (see Appendix II.3 in [164]) and the notation of the degenerate hypergeometric function of two variables ~l(a, b; c, c'; w, z). Taking into account the asymptotic properties of the function erf(x), it is easy to establish that for the convergence of the integral (6.75) at oo it is sufficient if one of the following groups of conditions holds: a) ~ > - I , 9~(b9) > max(0, -~(c2)); b) ~ > -1, ~(b z) = max(0, -~(cZ)), ~ p > 0; c) ~l(b2), = max(0, -~(c2)), ~ p = 0 and ~l(c ~) < 0, 1~1~ I < 1 or ~ ( c 2) > 0, -1 < ~la < 0. Under these conditions, the term by term integration of the series in the interval (0, oo), carried out above, is legitimate and the integral (6.75) takes the value (6.78). Example 15. We evaluate the integral

i t~-l (tr--ar) ~-lsin(btu)[cosctjdt, /si n ct~

11.5=

r, tt>O.

(6.79)

a

After the substitution t =

ar and ,7/'1 (x) ='c~ (a : r - 1)~+-',

,Td2(X)= sin x-u,

9 , (sin~-'] t,~ 3 ["C)=~COS ~.-1~, I15=a~+~-~YC(xl, x2),

a-'b - 1 1 ~ xl,

(6.80)

(ac)-l= x2

we obtain the equality (2.10), where n = 2. By virtue of the formulas 8.4.2.4, 8.4.1.4-6, 8.4.5.4 from [164] and (2.77), (2.10), (2.13), the Mellin transforms of the convoluted functions have the form

Yt; (sl q- s2)= rr@ v [1--~--(~+ s, + s2)/r] ' ~[ 1--(~-k-s,+s2)/r J' Re l]> O, 9g'~(s,)

=

Re(s,+s2)
V-~- 2-~,/u-,F [1/2--s,/(2u)] / l + s , / ( 2 u ) J'

ff/,~(s~)=if~-2-~r~Fr[(1(6-sz);2 ] + 6 + s2)/2J'

[R~s,l
--1 < R e s 2 < 6 ,

where 6 = 1 if from the two functions Y/'a(x) one takes the function sin r -1, and 6 = 0 if one takes the function r -1. Thus, by virtue of the equalities (2.8), (2.13), the problem of the evaluation of the integral (6.79) has been reduced to the determination of the value of the Mellin--Barnes double integral

~(xl, 9"~2! ~ = 4 nr(13) v'ii~176 v'i~~176 [(6--s2)/2, ~ ) 2 (2x2)-~'ds2 r 1(1 + 6 + s2)12, y~--ieo

y1--ioo

1277

1/2--sl/(2tt), 1--~--(~+ sl+ s2)/r] 1--(~+sl+s2)/r (21/~x,) -~' dsl.

l~Sl/(2tt),

(6.81)

We evaluate this integral with the aid of the residue theory, without dwelling on the question of the justification of the intermediate operations. If we fix the value of s~., then in general the poles of the integrand in the plane sl are the points s 1 = u + 2uk, s 1 = (1 -/~ + k)r - a - s 2, k = 0, 1, 2 ..... Introducing these values into the integrand and replacing r ( - k ) by 21ri(-1)k/ (k!~o), in accordance with (4.48), (4.49) we obtain fig'(X,, X2)=

~r(ts)~ ~(--,?{(2!mxl)-~-2~k2ttX

4ur2~i

k=0

W~~~176[(6--S2)/2, 1--~--(c~+tt+2ttk+S2)/r F [(1 + 6 + s 2 ) / 2 , 3 / 2 + k , (1 --(~+tt+2uk d

yz--too

] +s2)/rj X

)< (2x~)-*~dsz+ (2t/Ux~) ~-(1-~+n)~ r X w+~i~176 F [(~-- s2)/2, 1/2--((.1 --~+ k) r - - ~ - - s~)/(2u), [(1+6+s2)/2, l + ( ( 1 - - ~ + k ) r - - o ~ - - s 2 ) / ( 2 t ~ ) , d

]

~--kJ X

~ls - - i oo

ds 2j.

-)

s2=6+2l, s~= (l---~.+l)r--~z--u--2uk, s2=6--}-2l, s2---=(l--13+k) Taking into account that, in general, the points r--a--u--2ul, l = O, 1, 2 . . . . are simple poles of the last integrands, using again the theory of residues, but in the sz-plane, we obtain 4ur

k!l!

(2'/UX,)-u-2uk4tt (2X~) -~-2t X

k,l~O

[1 - - f i - - ( a + u + 2ttk + 6 + 21)/r F [ 6 - + - / + 1 / 2 , 3/2-+-k, 1 - - ( a + u + 2 t t k + 6 + -{- (21/"x0 -"-2"~ 2ttr (2x2) ~+"+2~-~ [(6--(1 --~+l) r + = + u)/2 + uk FL(l+6+(1--~+l)r--~--u)/2--ule, H (1 - - u) X,, ([ xl [--] x2 l) +

2l)/r] + X

3/2+k, ~--

l] X J

(6.82)

(2'/~xa)~-O-~+k)~2r ~(2'-'/~X~x-----7---/]-~-et X

F [(tt + o~--1-6-1- 2 / - - ( 1 - - {~-1-k) r)/(2u) [1/2+6+l, l+((1--~+k)r--o~--6--2l)l(2u),

~--k] X

H (u-- 1) x. (I x~i --Ix, I)}, where Xu(t) = 1 for u ~ 1 and Xl(t), while H(t) is a step function. The factors of the type H(u - 1)Xu(t) characterize the presence of the second or the third terms: if u < 1, then the first and the second terms are present and there are no restrictions on xl, x2, while for u = 1 the first and the second (for I x 1 [ > Ix2 I) or the first and the third (for Ix1 I < Ix2 I) terms are present. If, however, u = 1 and [ x l I = I x21, then one has to take only one of the last two terms. After the substitution (6.80), from (6.82) we obtain the desired value of the integral (6.79). The conditions for its convergence have the form b, c > 0, 9 ~ > 0 and ~ ( a + r/~) < m a x ( l , u) + r (if a ~ b or u r 1) or ~(c~ + r/3) < r (if a = b and u = I).

Formula (6.82) can be obtained from the double integral (6.81) also directly in the following manner. The integrand has poles at the points

s1=u+2uk, s2=6+2l,

k~O, 1,2 . . . . . l-=0, t,2 .....

sl+s~=(1--~+m)r--o~,

1278

m~-O, 1,2 . . . . .

We introduce the values s 1 = u + 2uk, s 2 + 6 + 2e into the integrand of (6.81) and we replace I'[-k, -el by 4u(27ri)L (-1)k+l/kff! in accordance with (4.48), (4.49); then we obtain the first term f r o m (6.82). We introduce the values s 1 = u + 2uk, s 1 + s 2 = (1 - fl + m ) r - ~ and we replace I'[-k, - m ] by 2ur(27ri)2(-1)k+m/k!m! and m by e; then we obtain the second term and then, similarly, also the third one. The advisability of the use o f the functions H(t) and Xu(t) follows f r o m the following arguments. We f o r m the matrix f r o m the elements o f the integral (6.81), containing s 1 and s2: - - s2/2, - - st/(2u), s2/2, sl/(2u),

- - (s~ + s2)/r~ - - (sl 4- s2)/rJ"

(6.83)

First we consider the term corresponding to the residues at ~the poles s 1 = u + 2uk, s 2 = 6 + 2L Fixing sl, we separate from the matrix (6.83) a submatrix with respect to the variable s2:

--s2/2, - s 2 / r 1 s2/2, - s2/r}' and, similarly, fixing s 2, we separate a submatrix with respect to sl:

-st/r/

/(2u), --st~r/" N o w we; f o r m the analogues of the expression A + D - B - C f r o m T h e o r e m 5.1 for the last matrices: O+l/r--(1/2+i/r)--1/2,

O+l/r--(1/(2u)+l/r)--l/(2u).

Obviously, they are always less than zero (since u > 0) and, therefore, by analogy with the case A + D - B - C < 0 of the indicated theorem, there arise no restrictions on x 1 and x 2 in the first term o f (6.82). The second term is the sum o f the residues at the poles s 1 = u + 2uk, s 1 + s 2 = (1 -/~ + m ) r - a. We fix s 1 + s 2 = s and we separate f r o m the matrix (6.83) the corresponding submatrices with respect to the variables s 1 and sz: --(S--St)/2, (S--St)~2,

--Sl/(2tt)' 1 St/(2tt)]'

{--S2/2, ~ S2/2,

--(S--S2)/(2U) I (S--S2)/(2U)]"

The analogues o f the expression A + D - B - C for them are equal to 1/2-FI/2--(2u)-J--(2u)

-t, ( 2 u ) - ~ + ( 2 u ) - t - - 1 / 2 - - 1 / 2

respectively, and they change sign when passing through the point u = 1. Therefore, the second term f r o m (6.82) must be present only for u < l or for u = 1 and Ix1 I > I x2 [. In a similar m a n n e r one establishes the conditions for the presence of the third term f r o m (6.82). At the evaluation o f (6.81) one has assumed that the lines ~ls 1 = 71 and 9~s2 = 7),2 separate the poles in such a manner that the functions u%/'~,Yd2, Y~s occur entirely (by complete series) in the integral (6.79), i.e., the cases of the type considered in Example 13 are excluded here. 7. INTEGRALS OF GENERAL FORM

1~ T h e F u n d a m e n t a l T h e o r e m . Let f and f be the functions introduced by the formulas (4.41)-(4.44). We consider the integral oo

.i'x=-lf(cfxOf(o3x)dx~-.~(cr,

o),

r•0,

or, o ~ 0 .

(7.1)

0

Obviously, this integral is invariant under transformations of the following three forms: 1) the interchange o f the parameters a and a, b and b, c and c, d and d, R and R, w and a, N and At, and the replacement o f r by r - l , a by o~/r with the subsequent division of the integral by r; such a t r a n s f o r m a t i o n interchanges the functions f ( x ) and f ( x ) . 1279

2) the interchange of a and b, c and d, t~ and b, ~c and d, R and R - I , / ~ and ~ - 1 , e and a -~, w and w-~, c~ and - a , which leads to the r e p l a c e m e n t of f(x) by f ( l / x ) and f(x) by f(l/x). _ 3) the interchange of a and b, b and a, c and d, d and c, R and R - I , a and w-x, N and N, r by r -~, and o~ by -c~/r, with the subsequent division of the integral by r. This change is the composition of the first two changes and leads to the transformation of f(x) into f ( l / x ) . It is easy to see that the restriction r > 0 does not lower the generality since if r < 0, then, a f t e r the interchange oc of a and b, c and d,/~ and R - I , a and a - x , a and - a , we obtain the integral ~ x~-~f (exit,) f (ox) dx f o r which [r [ = 0

--r> 0, T H E O R E M 7.1 1118]. Assume that there is given a contour La such that qs + c~r-1 ~ L and -ps 9 L for s 9 ~ , while the functions f(axr), f(wx) are defined by the formulas (4.41)-(4.44) and satisfy the conditions

Re(a-i-rai+"@,)>O, Re(--~+rbk-{-~E)>0,

j-=l, k=l,

a~+b~,

2..... 2.....

A, B,

j~l,

2.....

k=l,

2.....

A, B,

(7.2)

ay+b~--/=O, --1, --2 . . . . .

as well as some group of restrictions f r o m the list of the convergence conditions in part 2 ~ of this section. Then the integral (7.1) converges and has the value q j - , , /r

o~,=wlf*(qs+~)f*(--ps)~-~'~"ds=

(q,

a-~/'.Y-d~ (z),

(7.3)

z =~-~t%,

where Yt'l (z) has the f o r m (4.45) and, m o r e o v e r , for 6 = A r - - A < 0 or for 6 = 0, ] ~r [ < 1, as La one has to take the left loop ~e_oo, while for 6 > 0 or for 6 = 0, [ ~ [ > 1 one has to take the right loop La_+oo.U n d e r the additional conditions

2~o = E + Er > 0, [ arg(aw) -r) [ < ~o~ror 2~ _ 0, [ arg(aw)-q [ _<_~ , 9iB > ,/(qA - pA), these loops m a y be unrolled into some contour ~ioo = (ff - ioo, '7 + ioo). The function Yd~ (z) d e p e n d s analytically on z in the cases when 6 # 0 (for any arg z) or f o r 6 = 0, ~o > 0 (for I arg z I < ~o~r/r). For 6 = 0, ~o = 0, 9ifl > -1 the function ffda(z) is piecewise analytic. In the general case the value of 3~g'1(z) can be represented by the residue sums E,__(~), E_,(1/a) (see (4.56)(4.62)) using the f o r m u l a s

~1 (z)= E~__(~) for

0
!

if

E~(1/~ A--Ar,

if

A
if

iE__,(1/~) for l ~ l > 0 ,

~(z)=E+_t~)=E~(1/~), if

~Kr,

Re~>O,

for

I'~]>l, (7.4)

!ol=l,

A--C>~(D--B)r.

2 ~ Convergence Conditions. In the list of the convergence conditions for the integral (7.1) we use the notations (4.10)-(4.33), ( 4 . 3 7 ) a n d , m o r e o v e r , we assume that all the points -(a), (b) and -(fi), (b) are irreducible poles of the functions f*(s) and f*(s). 1. The cases w h e n regular singularities: la) ABAB # 0 , E > lb) AB # 0, E > 0, lc) AB = 0, E > 0, ld) E = A = E = A

(~/R) 1/r ).

at the points x = 0, x = oo (and, possibly, x -- R, x = J~) both functions f(x) and f(x) have 0, E > 0 ((4.29), (4.31)); E = A = 0, 9iv < 0 ((4.30), (4.31)); ~ = A = 0, 9ib < 0 ((4.29), (4.32)); = 0, (4.30), (4.32) and 9iv < 0, 9ib < 0 (for to=/=~? (cr/R) 1/r

)orgt(v+v)<-I

(for ~ o = R

2. T h e cases when one of the functions f(x) or f(x) has at 0 or at oo a p o w e r - s i n u s o i d a l singularity of the type O(xnsin ax), while at the remaining singular points both functions are regular:

1280

2a) 2b) 2c) 2d)

A/~ ~ 0, E _> 0, _E > 0 ((4.30), (4.31)) and one of the conditions (4.25) or (4.26); AB 4=O, E >(3, E >_ 0 ((4.29), (4.32)) and one o f the conditions (4.23) or (4.24); E = A = 0, E >_ 0, ~tu < 0 ((4.30), (4.32)) and one o f the conditions (4.23) or (4.24)); E -- A -- 0, E _> 0, 91u < 0 ((4.30), (4.32)) and one o f the conditions (4.25) or (4.26).

3. The cases when both functions f(x) and f(x) have at 0 or at oo a p o w e r - s i n u s o i d a l order and, moreover, at distinct points: 3a) E ___ 0, ~ _> 0, A > 0, ~ < 0 ((4.23), (4.26), (4.30), (4.32)); 3b) E >__ 0, E _> 0, A < 0, A > 0 ((4.24), (4.25), (4.30), (4.32)). 4. The cases w h e n both functions f(x) and f(x) have at 0 or at oo a p o w e r - s i n u s o i d a l order and, moreover, at identical points: 4a) E >_ 0, ~ >__ 0, A > 0, ~ > 0 ((4.24), (4.26), (4.28), (4.30), (4.32), (4.33)); 4b) E >_ 0, E >_ 0, A < 0, A < 0 ((4.23), (4.25), (4.27), (4_.30), (4.32), (4.33)). 5. The cases when at least one of the functions f(x) or f(x) tends to zero exponentially at 0 or at o~ in such a manner that the integrand >:~-~f((~x~)y(o~x) tends to zero exponentially: 5a) AB = 0, E > 0 (4.31) and one o f the following conditions: 6 < 0 (for A > 0) or 6 > 0 ( f o r / } > 0) and, moreover, arg ~ is arbitrary: E > 0, (4.29); Zk = E = 0z ~ u < 0 (4.30); E ___ 0, A < 0, A > 0 ((4.25), (4.30)); E _> 0, A > 0, B > 0 ((4.26), (4.30)); 5b) AB = 0, E > 0 (4.29) and one of the following conditions: 6 > 0 (for A > 0) or 6 < 0 (for B > 0) and, moreover, arg co is arbitrary: /~ > 0 (4.31); = E = 0, ~Rb < 0 (4.32); ~_> 0, 4 < 0, A > 0 ((4.23), (4.32)); E_> 0, A > 0, e > 0 ((4.24), (4.32)). 6. The cases when both functions f(x) and )(x) have at 0 or at oo an exponential order and, moreover, one of them increases but the f u n c t i o n x = - ~ f (ax r ) . f ( o x ) is integrable: 6a) AB = 0, E > 0, 6 = 0 ((4.31), (4.33)), one o f the three conditions: Ar > 0 or A~ = 0, ~Rfl > -1 (for A8 ~ 0) or Ae = 0, ~ > 0 (for A~ = 0), and also one o f the following conditions: E<0,

A>C,

larg~J<(A--C+l)~

E<0,

B>D,

[argtrl<(B--D+l)z~

(for

A>0);

(for / ~ > 0 ) ;

E>O,

A>O,

z~E/2
(for A>O);

E>0,

A<0,

zE/2
(for /~>0);

6b) AB = 0, E > 0, 8 = 0 ((4.29), (4.33)), one of the three conditions: Ae > 0 or )'e = 0, ~Rfl > -1 (for ,X8 4=0) or ,Xe -0, 91fl > 0 (for A8 = 0), and also one of the following conditions: E<0,

A>C,

Iargol<(.4--C+l)#

E<0,

B>19,

largo~[<(B--D-kl)#

(for A>0); (for .'B>0);

/~>0,

z~?~>0, u E ? / 2 < t a r g ~ o l < ( _ A - - C - k ~ ' ) z t ( f o r A > 0 ) ;

E>O,

A
~ E / 2 C ] a r g ~ o [ < ( B - - D q - e ) ~ x (for B > O ) ;

Remark 7.1. If points s ~ -- a]-- r - ~ , bk-- r-~a , r-la~~, r - ~ ~ are not poles o f the f u n c t i o n r-- f * (s + t'-la) f * (--rs), then the corresponding convergence conditions can be relaxed somewhat. 3 ~. Proof of the Fundamental Theorem. Assume that certain conditions hold that ensure the absolute convergence of the integral (7.1) and o f the Mellin transform [ 121 ] f*(qs 1 + ct/r) o f the f u n c t i o n f(x) at all points qs 1 + a/r f o r which s~6.o~ , while qs 1 + a/r ~ L and -ps z ~ L. In (7.1) instead o f f(x) we introduce its expression (4.44) and then we inter-

1281

change the order of integration and we introduce new variables s = -ps 1, ax r = t. assuming that So = 0. Under the mentioned conditions, these operations are legitimate and lead to the equalities

0

l

L

= 2~i i

7" (S) o~-*as i

xc~-~-~f (ox r) d x

=--

o

=

(7.5)

( - - PSi) cr-q"co~

~

~a

o

f 7" ( - - p s O

=Td

t(~+~

(t) dt

=

r)g-qs~o~pS'dsl.

The contour ~e, obtained from/~ by the substitution s ; -ps 1, is considered at integration in the same direction as/~ and, moreover, ~e will separate the left poles of the functions f*(-psl), f*(qs 1 + a/r) f r o m the right ones. The conditions for the absolute convergence of the integrals (7.5) can be relaxed somewhat to conditions that ensure conditional convergence, since all the functions, occurring in (7.5), are analytic with respect to the parameters and, therefore, they can be continued analytically to a larger domain. Thus, the integral (7.1) can be represented as the Mellin--Barnes integral (4.45), (4.41), (4.42), (7.3). However, the latter, under appropriate conditions is computed in terms of the sums of the left or right residues of the integrand, Formulas (4.56), (4.57) for these sums are obtained by analogy with (4.46), (4.47), (4.48), (4.49) by introducing the values s = - - ( h - J - ~ / r - l - a j ) q-r , s = --(h-t-b])p-1 and s =(bi4-. h--c~/r ) q-l, s ~ ( a i + h ) p-1 into the function q f * ( - - p s ) f * (qs-}-~/r) z p', the replacement of r ( - h ) by (-1)h/h!q (or (-l)h/h!p in the second and the fourth cases), and by the subsequent summation with respect to the indices of all the poles. This is how one has to proceed in the regular (not logarithmic) cases, when the enumerated poles are not multiple. I f r --- p/q is rational, while p and q are relatively prime integers, then the integral (7.3) can be transformed into the form (4.39), while the sums (4.56), (4.57) can be written in terms of generalized hypergeometric functions in the form (4.59), (4.60). For this one has to make use of the multiplication theorem for the G a m m a function. Indeed, introducing the notation (7.3), (4.45), we obtain the representation

a~, (z)=-~T~ f rq [(a)-i-~/r-I-qs' [(c)@cz/r @qs, •

I_(d) +

(b)--~zlr--qs] (d)--o~/r--qsJ X

(7.6))

'

ps, (c)-- psJ

where

bl ~ q N N R ~'/" (1/-2-~)0-q)e+Cl-p)~'qv+'~/r-e/~p'~-~/2,

r,l(a)l=r [ (a) q ' (a)+l ~ '

....

(7.7)

(a)+q--1 ]' q

while a is connected with z by the formulas (4.16). We compare the integral (7.6) with (4.39). To the A-dimensional vector (a) + s f r o m (4.39) there corresponds the Aq + Bp-dimensional vector

=

A(q, ( a ) + c ~ t r + q s ) , A(p, ( b ~ ) + p s ) = (a),@ w . ( a ) + q ~ l jr_ c~ q

W +s .....

&-} + s , .. p

"'

q

- T + s,

(b3+P-~ +s; p

similar relations hold among the vectors involving b, c, d. To the variable Z from (4.39) there corresponds ~, while to the quantities A + D - B - C, A + B - C - D the expressions ZXq - LXp, Eq + Ep. Therefore, on the basis of Theorem 5.2 1282

one can conclude that the integral (7.6) f o r qA > ZXp (qA < Z~p) or f o r qA = Ap, but < 1 ( > l ) has to be c o m p u t e d by means of the sum of the residues at the left (right) poles, taking as La the left (right) loop La_oo (La+oo). U n d e r the additional conditions 2q0 ~ E + / ~ r ~ 0, I arg (aco-r) ] < q~n or :- q~>~ 0, I arg (ao~-') [ ~< qD~, ~ (qA - - pA) < Re [~ (see (4.15)), t]hese loops can be unrolled into some contour Laioo = (7 - ioo, 7 + ic~). T h e f u n c t i o n Yg~ (z) in the cases 6 = rA - A r 0 (for any arg z) and 6 = 0, ~o > 0 (for [ arg z [ < ~o~r/r) is an analytic function, while f o r 6 = 0, ~o = 0 and 9119 > -1 the f u n c t i o n YE~ (z) is piecewise analytic. F r o m here we obtain the equalities (7.4), (7.3), (4.16). It is sufficient to establish only the first of the formulas (4.59), (4.60) since the second one is obtained f r o m the first one by the t r a n s f o r m a t i o n 2) of 1% By the method of the comparison of (7.6) with (4.39) and of E,_(~) with (4.50), it is easy to find the p a r a m e t e r s occurring in the h y p e r g e o m e t r i c function n+iFmf r o m (4.59). Indeed, to the vector (b) + aj f r o m (4.50) there correspond two vectors: the vector

(a)+=+p--lq a~+h p -'-'~

'

p

,

(b)q-ai-bh-Fq--I ,

('~)-bo:~_ai-Fh

q q P q where s u m m a t i o n has to be carried out with respect to j (j = 1 2 ..... A) and with respect to h

(h = 0, 1 ..... q - 1), and also the vector

(a)-b biq-hh-p--I

(b)+a/+n

(b)q _] ~'j + h--~p ,

...,

(b)+q--lq

[_

-#j+~--CCp, , (a~)+b~+hp ,

...,

where s u m m a t i o n has to b e c a r r i e d o u t with respect to j (j = 1, 2 ..... /~)and with respect to h (h =

0, 1..... p - 1). In a similar m a n n e r one finds also the other p a r a m e t e r s of the f u n c t i o n n+lFm and the quantities m, n. T h e coefficients o f n+iFmin f o r m u l a (4.59) are equal to the r i g h t - h a n d sides of (4.56) if the s u m m a t i o n with respect to h is carried out f r o m 0 to q - 1 (or to p - 1). This is established in the following manner. The first sum f r o m the r i g h t - h a n d side of f o r m u l a (4.59) can be derived f r o m the first sum of the r i g h t - h a n d side of (4.56) if we represent the index h (h = 0, 1, 2,...) in the f o r m h = kq + h 1, where k = 0, 1, 2 ..... while h I = 0, I ..... q - 1. For k = 0 the s u m m a t i o n in both sums is carried out only with respect to h = h 1 f r o m 0 to q - 1. M o r e o v e r , the first term of the generalized h y p e r g e o m e t r i c series for the function n+lFm(tO which there corresponds the index k -- 0) is equal to 1; this leads to the m e n t i o n e d equality between the coefficients of (4.56) and (4.59). Introducing into (4.56) the value h = kq + h 1 and representing the first sum

2

in the f o r m

h=0

2'2

, we obtain

h~ =0 k=0

finally the first sum f r o m (4.59) if instead of the sum with respect to k we use the notation n+iFm f o r the series. Its parameters have been established above by means of comparison. Thus, to the group of terms f r o m h -- qk to h = q(k + 1 ) - 1 in the first sum (4.56) there corresponds the group of terms f r o m the first sum (4.59), obtained by the multiplication of the kth term of the series for n+lF m by the coefficient given there. It is easy to see that by changing a and b, c and d, N to N, R to R -a, r to r -1, ~ to -a/r, a to a l/r, r to wr in the first sums f r o m (4.56), (4.59) and by their multiplication by z-"r-l,we obtain the second sums. This corresponds to the invariance of the integral YC~(z) under the t r a n s f o r m a t i o n 3) from 1~ We show how one can establish f o r m u l a (4.62), expressing the value ~,._(d) in the logarithmic case, when a 2 = A

ax + n, b2 =/~1 + m, m, n = 0, 1, 2 ..... while equation (4.61) has nonnegative integer solutions k, h. T h e sums B

"V

n--1

Q~' ~ ]=4

/~=0

m--I

S~n, 2 h=0

2

PJ,

j~4

T~n' 2

T3h, 2

h(gk )

S3h, generated by the residues of the function qf*(qs + a/r)f*(-ps)zP" at the

k(~k)

simple poles, are obtained without d i f f i c u l t y by the simple introduction of the values s = -(aj + h)/q - c~/p (for Sjh) or s = -(bj + h)/p (for Tjh) into this function with the subsequent replacement of P(-h) by (-l)h/h!q or (-l)h/h!p (for Tjh ). The term, containing the sum of

S~h , can be also obtained by introducing the value

s=

a~+,~ q

-~ p

into the

mentioned function, by r e p l a c e m e n t o f F[-n - h, - h i by (-1)a/(n + h)!h!q2, and by multiplication by (In [R-qRpz-p]-- - q ~ * ['" ] -F p ~ [ :: 1). The last bracket is f o r m e d by analogy with the curly bracket f r o m (4.52). The reason of its appearance is elucidated at the analysis of the examples f r o m 2 ~ in Section 6. In a similar m a n n e r one finds the terms containing T'zb , T3h. N o w we shall dwell on the derivation of the convergence conditions of the integral (7.1). In accordance with the formulas (5), (7) f r o m [20], the integral (4.43), in the presence of both left and right poles at the points -(a), (b), in the composition of the principal terms contains terms of order x~a/ (for x ---, 0) and x -r~ (for x ~ ~ ) and does not contain exponentially increasing terms ff E _> 0, while [ arg a I < Ez/2. T h e r e f o r e , for the c o n v e r g e n c e o f the integral (7.1) at 0 and at ~ , one has to assume that conditions (7.2) are satisfied, which ensure the c o n v e r g e n c e of the

1283

"equivalent" integrals

Jx

~ J

dx

0

for small ~0 and large E 0. If some points - - a 1, bk,

and Eo

- - a I, b~

are not poles of f*(s), f*(s), then conditions (7.2) can be relaxed somewhat (see R e m a r k 5.1.). In the case A - C = 0, - 2 , - 4 .... and A = 0 the f u n c t i o n f(x) has as singular point also the point x = R (Theorem 4 o f [20]), at which, in general, f(x) = O[(R - x ) A - c - " - 1 ] + O(ln ]R - x ] ). We show that A - C cannot be negative. Indeed, if A - C < 0 and A = 0, then B - D < 0 and, therefore, E < 0, which makes impossible the selection for L of the contour Lioo. We select the left (right) loop L_oo (L+oo): then the function f(x) will be d e f i n e d for 0 < x < R (x > R), while f r o m the condition -ps E L~oo o f the fundamental theorem it will follow that s ~ LP_+oo.Thus, the f o r m of the contour LQ+ooand the value o f the integral (7.3) will depend on the sign o f x - R; however, this integral does not depend on x. Therefore, in the case A = 0 one has to take A = C and then E = 0. In order that the f u n c t i o n f(x) be_integrable at the singular point x = R, one has to require that condition ~ u < 0 be satisfied. If, however, E = A = E = A = 0, then the integral (7.1) has two singular points: x = (R/o)1/r and x = R/w. Therefore, there arises the additional condition ~R~, < 0 for the case w ~ R(a/R)l/r,.when these points do not coincide, and the condition 9t(u + b) < -1 in the case when the singular points coincide. If E > 0, then the function f(wa) is analytic with respect to w in the sector I arg w ] < Er/2, where the integral (4.44) is trivially convergent (see T h e o r e m 5.2), which leads us to the analyticity o f the integral (7. I) with respect to w in this sector. Thus we obtain the conditions l a, lb, ld f r o m 2 ~ Condition lc can be written by s y m m e t r y f r o m lb; it is established by the replacement of A by B, etc., indicated in the group 3) f r o m I ~ We consider the situation 2 f r o m 2 ~ If E _> 0, A ~ C, ] arg a ] = ETr/2, then the f u n c t i o n f(ax r) at 0 (for A < 0) or at eo (A > 0) has a power-sinusoidal order, i.e., the principal terms o f the asymptotics contain an additional term, including a sine. F r o m T h e o r e m 9 (part 3a) o f [123] there follows that this term generates additional terms in the asymptotics o f f u n c t i o n x~-lf(ax~)f(wx), which in case A > 0 at oo have orders O I x ~~ ik 1 sin ( A I x ' ~r 11a'17 ~o~

pA=v~

1--E 2

~-

)j,

Now, in order to obtain the additional restriction o f that case 2a f r o m 2 ~ where A > 0, it remains to oo

make use o f condition 91X < 7, guaranteeing convergence o f "equivalent" integral

j x z- ~cos (x v) d x , ~ > 0, at oo. 1

Interchanging a and b, c and d, b and a, a and - a , we obtain the additional constraint o f the case 2a for A < 0. If, however, E = A -- 0, arg oJ -- 0, then the function f(x) has a third singular point, which, as already mentioned, leads to the necessity o f a condition o f the f o r m ~ < 0. This aspect is reflected in case 2d. On the basis of the obtained conditions o f the cases 2a, 2d, and of the above mentioned T h e o r e m 9 of [123], it is easy to obtain by s y m m e t r y the conditions of the cases 2b, 2c and to f o r m the conditions for 3a, 3b, 5a, 5b, and part of the conditions f r o m 4a, 4b. The remaining conditions f r o m 4a, 4b are obtained in the following manner. If in the expansions at oo both functions x a - l f ( a x r) and f(wx) have power-sinusoidal terms, then by their multiplication there appears an additional term o f another type. Its principal term contains f o u r terms o f the order

Taking into account that the integral oz

j'xx c o s x ~ ' c o s ( a x n) dx,

a ~ O, ~ > 0,

~ > 0,

1

converges for ~lX < max(A, 7) (if), ~ "r or a :g 1) or even for 9~X < 0 (if)~ = 7 and a = 1), we obtain easily the condition

2 R e ( A ~ c z ~ r v A q - v A ) < t A ~ r ~ A I q - A E ( i f A ~ A r o r Atcr/RI llA ~ A l ~ / ~ t l / ~ ) , w h i c h f o r A = A r a n d r ~ R r ! ~ t = R t c o } r has to be strengthened to 2Re (~cz q- v @ Q < E - ] - / ~ - - 2, i.e., to ~fl > 0 (see (4.15)), since in the last case one loses the possibility o f the conditional c o n v e r g e n c e o f the integral. If A = Ar (which means the rationality of r and, moreover, then A = kp and lx = kq, where k is an integer), then the function Kz(z) has a third singular point z ~ z 0 = ~?R-t/rrXe-~ ~o~ (4.14), in which b = exp{[(C - A)p + (D - B)q]Tri}. In this case there arises another condition ] arg[l - (zo/z)P ] ] < r, denoting the necessity of a cut. In accordance with Theorem 4 f r o m [123], the order g o f the power singularity at this point is equal to t3 (4.15). Therefore, as mentioned in the conditions (4.33), for ~Rfl > 0 the value z = Zo. is assumed~ In the case 5a f r o m 2 ~ one has the condition AB = 0. Let B -- 0,/~ > 0; then, obviously, ~ > 0. If we assume also A > ~ r , then also A > 0. Since by T h e o r e m 9 (part 5) f r o m [123], in this case the f u n c t i o n f(wx) has at oo an exponential

1284

decrease of order O (e-cxl/A), c > 0 , while, in general, f(crxr) at ~ may increase exponentially, but not faster than 0 (ea*rtA), d > 0 , it follows that their product decreases exponentially for A > ZXr for any arg a and [ arg w I < Er/_2, while for A = Ar the product has a power-sinusoidal order. The case A > Ar is reflected in 5a, while the case A = Ar is singled out in 6a, 6b and requires a special consideration, which will be given below. Let B = 0 , k 2 > 0 , A - ~ r , l a r g o I < k ' ~ / 2 , A>C, h > 0 , max(O, E r ~ 1 2 ) < l a r g a l < ( A - - C - b g ) ~ .Then, in accordance with Theorems 6, 7, and formula (15) f r o m [123], the function xa-lf(axr)f(wx) for arg a.arg w 4=0 has at c~ order O {exp [ - - (~,~q- i ~ ) x~/a] x ~+~o+y-1} , where A~, As are given in the equalities (4.19), (4.20). Obviously, for Ar > 0 the exponential decrease is preserved, while for ,~ 0 the convergence may be conditional (when A8 ~ 0) or absolute (for),~ = 0). In these cases there arise the restrictions ~lfl > -1 or ~lfl > 0. However~ if, for example, arg a = 0, arg w 0, then instead of the condition As ~ 0 (or As = 0) one has to impose the restriction ,~+),~-~ 0 (or)~8+A~- =_0), where ),~-+ have the f o r m (4.21). This follows f r o m the relations (35), (38) of [123]. Setting, by definition, As = As+)~8- for arg a -0, we preserve in this way the previous f o r m of the conditions: A, ~ 0 (or As -- 0). Similar modifications are carried out also in the cases when arg a = arg w = 0 or arg w = 0, arg a , 0. The analyzed cases give us the possibility to formulate all the convergence conditions f r o m 2 ~. 4 ~ Applications of the Fundamental Theorem. For the evaluation of a concrete integral of the f o r m (7.1) of the product of any two functions f and f of the hypergeometric type, in particular, of functions contained in Table 8.4.2.51 of [164], one has to: I) represent these functions in the form of Mellin--Barnes integrals (4.43)-(4.44);_ 2) express the collections of concrete parameters (a), (b), (c), (d), N, R, (t]) ..... R, r, p, q, a, a, w, form the fundamental characteristics A, A, 6 of (4.12), (4.14), allowing to determine which of the sums E,__(a) or I3__,(1/~) of the form (4.56)-(4.62) expresses the given integral (7.1), and elucidate whether the logarithmic case holds or not and which formula has to be used to evaluate the sum E=; 3) after the composition of the necessary functions B= and argument (4.16), express the value of the given integral by the formulas (7.3), (7.4) and compute the fundamental convergence characteristics of this integral E, E, u, ~ (4.10)(4.13); 4) from the list of the convergence conditions of the integral (7.1), given in 2 ~ and taking into account the restrictions (7.2), establish the corresponding convergence conditions of the given integral; 5) compute, if necessary, the auxiliary characteristics ~o, p, z o f r o m (4.14)-(4.17) and f o r m the additional convergence conditions (4.23)-(4.28), (4.33) for the given integral; 6) in certain cases the obtained expressions of the integral can be simplified with the aid of the tables for the special cases of the generalized hypergeometrie functions f r o m the sections 7.3-7.18 of [164]. Remark 7.2. In complicated cases, when one considers an integral of the product of several functions of type f of (4.43) or when there appear logarithmic eases with a high multiplicity of the poles, as well as in the very simple cases, it is advisable to carry out the constructive method described in Sections 3-6. Remark 7.3. As it follows from (4.40), the functions f and f, defined by the formulas (4.43), (4.44), represent modified Meijer G-functions (2.2). Therefore, all the results presented in Section 7, in the case of a rational r, can be reformulated in terms of G-functions. In particular, the integral (7.1), (4.43), (4.44) reduces to the integral from 2.24.1.1 of [164], while its convergence conditions, after substitutions and rearrangements, obtain the f o r m of the conditions 1)-35) from 2.24.1.1. For such a transformation, in the integral (7.1) one has to change a to aR, A to m, B t o n , C to p n, D tq q - m, (a) tO (bin), (b) to 1 - (an), (c) to an+ 1..... ap, (d) to 1 - bin+l,., u 1 - bq, r = p/q to g/k, w to wR, A to S, B to T, C to U - T, D to V - S, (t~) to (dS), (b) to 1 - (CT), (c) to CT+x..... Cv, (d) to 1 - d s , 1..... 1 - d r , and also N = N I; then the fundamental characteristics are replaced in the following manner: A by q - p, A by V - U, 6 by -~o, E by 2e*, E by2b*,uby/~+e*1,~,byp+b*1. Remark 7.4. The integra_ls of the f o r m (7.1), (4.43), (4.44) express the values of the integral transforms with definite kernels x '~-1, f(ax r) or f(wx), if the remaining functions are assumed to be arbitrary. To the kernel x a-x there corresponds the Mellin transform, while in the remaining cases one can obtain in particular the Lap_lace, Fourier, Hankel, Stieltjes, Hilbert, Riemann--Liouville, Weyl, Meijer transforms, the G-transform, etc., if for f or f one takes appropriate concrete functions. 8. EXAMPLES

Example 16. Setting in the formulas (4.41)-(4.44), (4.10)-(4.17)

1285

N-----r-' (0,), N = v-' (g~), we obtain

f*(s)-----NI'[s,b,~s], E=/~2,

,~-~bt, ~=b~,

q~=r+l,

zr=~-~/~o,

J3=l--bz--b~,

A=:~=6=0, "~=(~qO3-P,

zo~exp[--(r-~ @ l)~i]

(we mention that the parameters 13, ~o, z o are not required here). Making use of the indication 8.4.52 from [164] and of formula 8.4.2.5 from [164], we find that

y (x)=(1 +x)-~,

f (x)=(l+x)-~.

Making use of Theorem 7.1 and of the enumeration of the convergence conditions in the case I a, we can write the value of the corresponding integral in the form ~o

S -~'-' 0 + ~')-~, 0 + ~)-~,a~ =/o-':'E~- 8)- o <2~1< ~,

(8.~)

[ct-"/rZ~.(l/cO, I~l> 1,

.o

where, by virtue of (4.56)-(4.60), in the regular cases we have

N

q--1

~lt~

p--1

}

~,--/zr--~] ~/,I •T - ~.Ca ( - l )hi~ z - ~ - h r [ - O ~ , b,+O, ' O,+hlx~ h=O

'

q~l ,o--1 } -t- r-1 ~ ~iI)n z"l~ [~1, b, -- ~1, b, -t-/l] X4 , h~O 0 1 = ( b l q - h - - ~ ) r -1, rh~--(b~+k)r--~, ~ = ( / z + ~ ) r - L Here p -- q - ~ , Xk = 1, k = 1, 2, 3, 4, if r > 0 is arbitrary; if, however, r = p/q is rational and p and q are relatively prime integers, then the functions Xk can be represented also in the following form (see (4.59), (4.60)):

X,=~r

(I, a(q, bl+k), A(p, kr+~); (--1)P+ooqoJ-,~, a(q, 1 .+h), A(p, 1--b,q-hr-q-~,)

(8.2)

)

( | , A (q, b t + 01), A (]7, bl -[- h); ( - - l)P+q(Iq(o-P~,.

~2 = l~.q+l F ll.+q \

a(q, l+oi), t~(p, 1q-h) (1, A (q, bl + h ) , A (p,

\

(8.3)

/

b,+111); (-- 1)P+q(J-qOJP~,

A (q, 1 + k), A (p, 1 + n,)

X4=p+q+lFo,q(1, A(q, ~,), A(p, ~ + k ) ; (--1)o*r A(q, 1-o,+~,), A(p, l + k )

]

/"

The functions E~(8) and ~3__.(1/a) are analytic continuations of each other through the circumference [ b[ = 1 and, therefore, the function ~__,(1/6) need not be written in the case of the use of the symbols (8.2), (8.3). The integral (8.1) converges if the conditions Reck>0,

Re(--~+rb~+b~l)>0, [argol<~,

b~, b~4=0, - 1 ,

--2 .....

Iargco [ <-~

are satisfied (see (7.2) and conditions la from 2 ~ of Section 7; moreover, the restrictions b 1, bl § 0, -1, -2 .... can be removed since ~R(rb1 + ha) > ~Rc~> 0).

1286

In particular, for r = 1 one has to set p = q = 1, while the function XL__(~)can be transformed by the formula 7.2.1(1) from [164] to the form

~_~)],=l=z_aF [o~, bt+~_~] F (~, bl; 1-Ot+t~l

bl+bl oa

]2 ~

-

1).

From here, after replacing e by z -1, w by y-l,/~l by A, and b 1 = p, we obtain the known integral 2.2.6.24 from [162]. Example 17. Setting

A=C-~R=r=I,

B-~D-~aI~-O, N=r(cO,

by the formulas 8.4.52, 8.4.2.3 from [164] we obtain f(x) = (1 x)el-1 for 0 < x < 1 and f(x) = 0 for x > 1. Selecting the function i f ( x ) = (1 + x) -~' , with the corresponding parameters, indicated in the previous example, we make use of Theorem 7.1 and of the enumerated convergence conditions in the case 1b. Then we obtain the value of the following integral: 0)-1.

9

L01, cl+~lJ

o XaFg_ c l + g l , Recz>0,

A(1, 1) '

~ 1 7 6 1 7 6 gl-~cz'

Rect>0,

argr

~-------cl,

largol<#.

By virtue of the analyticity of this integral with respect to the variable ~, we can restrict ourselves to the corresponding value of ~ ( l / c ; ) from formula (4.60). After replacing a by a -I, w by z -1, and c 1 = fl, b 1 = p, we obtain finally formula 2.2.6.15 of [162]. Example 18. Setting f(x)=(1

--

~'~ C':t -I

-w+

,

f ( x ) = (1-- x)~r -I,

r=l

(the corresponding parameters are indicated in Example 17), we form the integral mIn(aTx, o-x~

J'

x ~-1 (1 - - ax) *'-1 (1 - -

0

This integral is not an analytic function of a and w since here the formulas

X~F~(1, 1 - - c 1, a; ~) 9 1, c~Tl-c* '

ox)'L-ldx ~ Jt.

(8.4)

E = A = / ~ = A-~ 0. Its value Jx has to be computed from

a~cr~

(8.5) z=~176

la[
The value of J1 for I ~ [ > 1 is obtained from (8.5) if we interchange a and w, c 1 and cl. From the formulas (7.2) and the convergence conditions in the case ld we obtain the following convergence conditions for the integral (8.4):

Re~:>O,

o>0,

o>Oand

Recl>O, if

Recl>O

Re(cl+~i)>l

(for a ~ o ) (for or=o).

1287

Example 19. We consider the integral x "-I (1 q- o~x)- ~ sin V"o-~x'dx-~ J2.

(8.6)

0

We introduce the notations

y ( x ) = ( l + x ) -~,

/(x)=sinl#~,

reducing the integral J2 to the form (7.1). From the formulas 8.4.2.5, 8.4.5.1, 8.4.1.4 of [164] there follows that to the functions f, f there correspond the representations (4.43)-(4.44) with the parameters

A=D-~4=B=R=dI=I, B=C=C=D=a1=O, / ? = 4 , ai=l/2, ~N=I/'~-, .,V=I'-!(b~). In this case the parameters (4.10)-(4.17) assume the values ~,-------1/2, '~-b~, A = 2 , A = 0 , E = 0 , E = 2 ,

6=--2,~0,

~ - r , ~=4-qq-2actqo3-~p, z=~-~1%) (here 13 and z o are missing). Making use of Theorem 7.1, we can obtain the corresponding value of the integral N

j~ = o-~lr N,_ (~) in terms of an analytic function, which is not given here. The convergence conditions of the integral (8.6), in accordance with the case 2a from the enumeration of the convergence conditions in part 2 ~ of Section 7, have the "form Re(~q-r/2)>0,

argo=0,

largo~l
-' 2Re (2~-- 261 -- r/2) < r, and, moreover, the last restriction, reducing to the inequality ~(c~ - / h ) < r/2, guarantees the conditional convergence of the integral at ~ . Example 20. We consider the integral

eo

I x =-1 sin lf~-x sin -U-~ 1 cl x = J3. 0

Here f(x)=sinV'x,

j'7(x)=slnx-'/2,

r=l,

and, in accordance with Example 19,

A=D=[3=C=dl=~-~I, R=4, Y?~4 -~, B=C=A-~_D=O, a,=b,=l/2, N=N-~-I/'-~. The value of the given integral can be easily written according to the formula J3 = a - ~ ( ~ ) , where I3,__(~) is obtained from (4.56). The integral is an analytic function of the argument b = a/16o~. The conditions for its convergence 1288

can be obtained f r o m 2 ~ (Section 7), f r o m the convergence conditions in the case 3a since here E = E -- 0, A = 2 > 0 and A _- - 2 < 0, while conditions (7.2) are satisfied. The integral J3 converges if a r g o = a r g o = 0 , 2Re ( 2 ~ - - 2 / 2 - - 1 ] 2 ) < 1, 2t~e (--2o~--2/2--1/2) < 1, i.e., for

]Reoq
~>0, re>O,

Example 21. We consider the integral

~

x ~-1 sin V - ~ sin lf-~--xdx~-J4.

(8.7)

0

Here f ( x ) = f ( x ) = s i n V ' x , r = 1 , while the other parameters are obtained f r o m Example 19. The integral (8.7) is not an analytic f u n c t i o n but a piecewise analytic one. In the given case, E = E --- 0, A = ZX = 2 > 0. Making use o f the formulas (7.2) and the case 4a f r o m the c o n v e r g e n c e conditions o f 2 ~ (Section7), we find the conditions for its convergence: a r g ~ = a r g o = O, - - 2 R e ( 4 ~ - - 2 / 2 - - 2 / 2 ) > O, Re ( ~ + 1 ) > O, and we also express condition (4.33). A f t e r simplification, all these conditions reduce to the following: --l
m>~r>0

(or

o>io~>0),

and, moreover, for 91# = 91(1/2 + 1/2 - 2c~ - 1) > 0 the value a -- w is assumed. The last restriction is necessary since for o = w the conditional c o n v e r g e n c e o f the integral at oo is lost and the inequality 91c~ < 1/2 has to be strengthened to 91c~ < 0. Example 22. One can consider examples of integrals, whose integrands have exponential order at infinity: o~

0

0

fo - 1

(8.8)

oo

x "-I (1 --~x)~,-le-~dx, 0

x ~-1 s i n . . ~ e - , , ~ d x . 0

For these integrals we have r = A = R = A/= 1,/~ = C = / ) = a 1 = 0, while the parameters o f the remaining functions f(x) are such that the c o n v e r g e n c e conditions o f the integrals are obtained f r o m the formulas (7.2) and the corresponding conditions 5a f r o m the c o n v e r g e n c e conditions in 2 ~ (Section 7). A f t e r a change o f variables, the integrals (8.8) reduce to the integrals 2.5.36.2, 2.3.6.1, 2.3.6.10, 2.5.38.1 f r o m [162]. The values given in these formulas and the convergence conditions can be obtained with the aid of the fundamental theorem. Example 23. The most complex and varied situation o f the convergence conditions is contained in 6a and 6b in 2 ~ of Section 7 We consider the integral oo

x=_~e_~e_O~dx = o

r (~) (~ + ~)~' "

(8.9)

For it

r:= A----~4=R=f~= N = N=- I, B=C=D:]3=C=l):al:al=O, [3= - ~ ,

z=~-ko,

~=lc~lcoslarg~l+[colcoslargo[,

1289

the conditions (7.2) assume the form ~

> 0, while the conditions 6a-6b reduce to the following:

[argol <~/2, [arg(1-q-o/o~[ 0 (a•a ~ = 0 ,

(8.10) Re cz< 1 npn ~,~=/=0), ~ / 2 < ]arg g] <3z~/2. Obviously, )~e = ~R(er+ w) while condition h e > 0 leads to the exponential decrease at oo of the integrand (although e - ~ increases) and, therefore, also to the convergence of this integral at oo. If )~e = 0, then we have the equality e-l,r+~o)x ~ e~r ~ cos Cx-q- i sin oq~x,

where ~b = i(a + w) is a real quantity and, moreover, ~b ~ 0 if As ~ 0. For the convergence at infinity of integrals x - ~cos

dx

, obtained from (8.9), it is necessary that the additional condition ~ a < t, reflected in (8.10), be

0

satisfied. In conclusion we note that other examples for the use of Theorem 7.1 can be found in the sections "Integrals of general form" in [162-164]. LITERATURE CITED 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

1290

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1316

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1143. F. M. Ragab, "The inverse Laplace transform of the modified Bessel function l(,~(al/2mp~/:,~) ," J. London Math. Sot., 37, No. 4, 391-402 (1962). 1144. F. M. Ragab, "The Laplace transform of the modified Bessel function K,(t+mx) , where rn = 1, 2 ..... n," Proc. Edinburgh Math. Soc., 13, No. 4, 325-329 (1963). 1145. F. M. Ragab, "Integrals involving products of modified Bessel functions of the second kind," Proc. Glasgow Math. Assoc., 6, No. 2, 70-74 (1963). and I~(xt• where n = 1, 2, 3 ..... " 1146. F. M. Ragab, "The Laplace transform of the Bessel functions 1Q(xt• Ann. Mat. Pura Appl., 61, 317-335 (1963). 1147. F. M. Ragab, "The inverse Laplace transform of the product of two modified Bessel functions Kn~al/2:apl/Zn]Knu[al/2npl/2n] where n = 1, 2, 3 ..... " Ann. Polon. Math., 14, 77-83 (1963). 1148. F. M. Ragab, "The inverse Laplace transform of the product exp (--al/2p~/2/2)K~,.~ - (a~/'~pt/"), n=l, 2, 3~ ... ," Boll. Un. Mat:. Ital., 19, No. I, 26-30 (1964). 1149. F. M. Ragab, "Integrals involving Whittaker functions," Ann. Mat. Pura Appl., 65, 49-79 (1964). 1150. F. M. Ragab, "Multiple integrals involving product of modified Bessel functions of the second kind," Rend. Circ. Mat. Palermo, 14, No. 3, 367-381 (1965(1966)). 1151. F. M. Ragab, "E-transforms. I," J. Res. Natl. Bur. Standards, 71B, No. 1, 23-37 (1967). 1152. F. M. Ragab, "E-transforms. II," J. Res. Natl. Bur. Standards, 71B, No. 2-3, 77-89 (1967). 1153. F. M. Ragab, "Integration of Kamp6 de F6riet's hypergeometric functions with regard to their parameters," Rend. Circ. Mat. Palermo, 19, No. 3,225-242 (1970(1971)). 1154. F. M. Ragab and A. M. Hamza, "Integrals involving E-functions and Kamp6 de F~riet's function of higher order," Ann. Mat. Pura Appl., 87, 11-24 (1970). 1155. F. M. Ragab and M. A. Simary, "Integrals involving products of generalized Whittaker functions," Proc. Cambridge Philos. Soc., 61, No. 2, 429-432 (1965). 1156. F. M. Ragab and M. A. Simary, "Integrals involving E-functions," Proc. Glasgow Math. Assoc., 7, No. 4, 174-177 (1966). 1157. F. M. Ragab and M. A. Simary, "Integration of E-functions and related series," Monatsh. Math., 70, No. 1, 58-63 (1966). 1158. F. M. Ragab and M. A. Simary, "Definite integrals involving Whittaker functions," Proc. Cambridge Philos. Soc., 64, No. 4, 1033-1039 (1968). 1159. F. M. Ragab and M. A. Simary, "Definite integrals involving modified Bessel functions of the second kind," Ann. Univ. Bucuresti, Mat., 26, 91-101 (1977). 1160. R. K. Raina, "On Laplace transformations of H-function of two variables," Proc. Natl. Acad. Sci. India, A45, No. 2, 133-142 (1977). 1161. R. K. Raina and C. L. Koul, "On certain integral relations involving the H-function of two variables," Proc. Natl. Acad. Sci. India, A47, No. 3, 169-176 (1977). 1162. E. D. Rainville, Special Functions, Macmillan, New York (1960). 1163. S. L. Rakesh, "Infinite integrals involving generalized hypergeometric functions," Math. Ed., AS, 96-98 (1971). 1164. S. L. Rakesh, "Integrals involving generalized functions," Vijnana Parishad Anusandhan Patrika, 16, 27-30 (1973). 1165. S. L. Rakesh, "Integrals involving products of generalized hypergeometric function and generalized H-function of two variables. I," Rev. Univ. Nac. Tucumfin, A23, 281-288 (1973(1974)). 1166. S. L. Rakesh, "Integrals involving the H-function of two variables," Indian J. Math., 21, No. 1, 63-67 (1979). 1167. S. Ramanujan, "Some definite integrals, " Messenger Math., 44, 10-18 (1915). 1168. S. Ramanujan, "Some definite integrals," Proc. London Math. Soc. 17, 17-18 (1918). 1169. S. Ramanujan, "A class of definite integrals," Quart. J. Math., 48, 294-310 (1920). 1170, M. A. Rashid, "Lattice Green's functions for cubic lattices," J. Math. Phys., 21, No. 10, 2549-2552 (1980). 1171. A. K. Rathie, "An integral involving H-function. II," Vijnana Parishad Anusandhan Patrika, 22, No. 3, 235-238 (1979). 1172. C. B. Rathie, "Some infinite integrals involving E-functions," J. Indian Math. Sot., 17, No. 4, 167-175 (1953). 1173. C. B. Rathie, "A theorem in operational calculus," Ganita, 4, No. 2, 135-137 (1953). 1174. C. B. Rathie, "Some infinite integrals involving Bessel functions," Proc. Natl. Inst. Sci. India, 20, No. 1, 62-69 (1954). 1175. C. B. Rathie, "Some results involving hypergeometric and E-functions," Proc. Glasgow Math. Assoc., 2, No. 3, 132- 138 (1955). 1176. C. B. Rathie, "A few infinite integrals involving E-functions," Proc. Glasgow Math. Assoc., 2, No. 4, 170-172 (1956).

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1279. M. Shah, "Some infinite integrals involving Whittaker functions and generalized hypergeometric polynomials, with their applications," Proc. Cambridge Philos. Sot., 65, No. 2, 483-488 (1969). 1280. M. Shah, "Some integrals involving H-functions. I," Math. Ed., A3, 82-88 (1969). 1281. M. Shah, "Some results on the H-function involving the generalized Laguerre polynomial," Proc. Cambridge Philos. Soc., 65, No. 3, 713-720 (1969). 1282. M. Shah,, "Certain relations involving generalized hypergeometric and Tchebichef polynomials of the first kind," Comment. Math. Univ. St. Paul., 17, No. 2, 73-81 (1969). 1283. M. Shah, "Some Fourier series for generalized hypergeometric polynomials," Portugal. Math., 28, No. 1-2, 7-19 (1969). 1284. M. Shah, "Some results involving generalized hypergeometric polynomials and generalized Meijer function of two variables," Comment. Math. Univ. St. Paul., 18, No. 2, 95-110 (1970). 1285. M. Shah, "Fourier series for generalized Meijer functions," An. Stiint. Univ. Iasi, Sec. la, 16, No. 2, 293-313 (1970). 1286. M. Shah, "Some results of generalized Meijer functions associated with Tchebichef polynomials of the second kind," An. Univ. Timisoara Ser. Sti. Mat., 8, No. I, 101-113 (1970). 1287. M. Shah, "On some results involving generalized hypergeometric polynomials and associated Legendre functions," J. Indian Math. Sot., 34, No. 1-2, 89-97 (1970(71)). 1288. M. Shah, "Certain relations involving generalized hypergeometric polynomials," J. Sci. Phys. Sec., 1, No. 1, 7180 (1971). 1289. M. Shah, "Some results involving generalized Meijer functions associated with Gegenbauer (ultraspherical) polynomials," Indian J. Pure Appl. Math., 2, No. 3, 387- 400 (1971). 1290. M. Shah, "On Fourier series for generalized Meijer functions of two variables and their applications," Indian J. Pure Appl. Math., 2, No. 3, 464-478 (1971). 1291. M. Shah, "A result on generalized hypergeometric function and generalized Meijer function of two variables," An. Stiint. Univ. Iasi, Sec. la, 17, No. 2, 331-338 (1971). 1292. M. Shah, "Gegenbauer (ultraspherical) polynomial and heat production in a cylinder," Studia Univ. Babes-Bolyai Ser. Math.-Mech., 16, No. 2, 83-90 (1971). 1293. M. Shah, "On generalized Meijer function of two variables and some applications," Comment. Math. Univ. St. Paul., 19, No. 2, 93-122 (1971). 1294. M. Shah, "Some results involving a generalized Meijer function," Mat. Vesnik, 8, No. 1, 3-16 (1971). 1295. M. Shah, "Certain results involving Kamp6 de F6riet's functions," An. Sti. Univ. "A1. I. Cuza" Iasi, Sect. I a Mat., 18, No. I, 93-99 (1972), 1296. M. Shah, "Results on generalized hypergeometric and Jacobi polynomials," An. Sti. Univ. "A1. I. Cuza" Iasi, Sect. I a Mat., lg, No. I, 101-108 (1972). 1297. M. Shah, "On generalized Meijer's and associated generalized Legendre's functions," Portugal. Math., 31, No. 1-2, 57-76 (1972). 1298. M. Shah, "On some problems leading to certain results involving generalized Meijer functions of two variables and associated Legendre functions," Math. Student, 40A, 124-133 (1972). 1299. M. Shah, "On generalized Meijer's G-functions of two variables and generalized Legendre's associated functions," Math. Student, 40A, 157-168 (1972). 1300. M. Shah, "Several properties of generalized Fox's H-functions and their applications," Portugal. Math., 32, No. 3-4, 179-199 (1973). 1301. M. Shah, "On some applications related to Fox's H-functions of two variables," Publ. Inst. Math. (Beograd), 16, 123-133 (1973). 1302. M. Shah, "On generalizations of some results and their applications," Collect. Math., 24, No. 3,249-266 (1973). 1303. M. Shah, "Some results associated with extended Fox's H-functions and their applications," Vijnana Parishad Anusandhan Patrika, 16, No. 1, 47-66 (1973). 1304. M. Shah, "A new formula on generalized functions," Indian. J. Pure Appl. Math., 4, No. 11 - 12,889-897 (1973). 1305. M. Shah, "A note on Mellin inversion formula," Indian J. Pure Appl. Math., 5, No. 5, 464-469 (1974). 1306. M. Shah,, "An extension of Rice's result on an integral equation," Publ. Inst. Math. (Beograd), 18, 173- 179 (1975). 1307. M. Shah, "On applications of H-functions," C. R. Acad. Bulgare Sci., 28, No. 11, 1459-1462 (1975). 1308. M. Shah, "A property of generalized integral transform of two variables," Math. Balkanica, 6, 186-193 (1976). 1309. H. Shanker, "On certain integrals and expansions involving Weber's parabolic cylinder functions," J. Indian Math. Soc. 4, 158-166 (1940).

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1344. K. C. Sharma, "Infinite integrals involving E-function and Bessel functions," Proc. Natl. Inst. Sci. India, A25, No. 6, 337-339 (1959). 1345. K. C. Sharma, "An integral involving G-function," Proc. Natl. Inst. Sci. India, A30, No. 5, 597-601 (1964). 1346. K. C. Sharma, "Integrals involving products of G- functions and Gauss's hypergeometric function," Proc. Cambridge Philos. Soc., 60, No. 3, 539-542 (1964). 1347. O. P. Sharma, "Some finite and infinite integrals involving H-function and Gauss's hypergeometric functions," Collect. Math., 17, No. 3, 197-209 (1965). 1348. O. P. Sharma, "Certain infinite and finite integrals involving H-function and confluent hypergeometric functions," Proc. Natl. Acad. Sci. India, A36, No. 4, 1023-1032 (1966). 1349. O. P. Sharma, "Relation between Whittaker and generalized Hankel transforms," Proc. Natl. Acad. Sci. India, A37, No. 1, 97-108 (1967). 1350. O. P. Sharma, "On generalized Hankel and K-transforms," Math. Student, 37, No. 1-4, 109-116 (1969). 1351. O. P. Sharma, "Certain infinite integrals involving H-function and MacRobert's E-function," Labdev J. Sci. Tech., A10, No. 1, 9-13 (1972). 1352. S. D. Sharma and R. K. Gupta, "On multiple integrals involving generalised H-function and spherical functions," Ganita, 30, No. 1-2, 47-58 (1979). 1353. N. A. Shastri, "An infinite integral involving Bessel functions, parabolic cylinder functions, and the confluent hypergeometric functions," Math. Z., 44, 789-793 (1939). 1354. N. A. Shastri, "Some results involving Bateman's polynomials," Bull. Calcutta Math. Sot., 32, 89-94 (1940). 1355. N. A. Shastri, "Some theorems in operational calculus," Proc. Indian Acad. Sci., A20,211-223 (1944). 1356. N. A. Shastri, "Some theorems in operational calculus," Proc. Benares Math. Soc., 7, No. 1, 3-9 (1945). t357. B. M. Shrivastava, "Integration of certain products involving Kamp6 de F6riet's function and the generalized H-function," Jnanabha, A3, 85-93 (1973). 1358. J. Siekmann, "Note on a Riegels-type integral," Z. Angew. Math. Phys., 15, No. 1, 79-83 (1964). 1359. A. Siddiqui, "Integral involving the H-function of several variables and an integral function of two complex variables;," Bull. Inst. Math. Acad. Sinica, 7, No. 3, 329-332 (1979). t360. M. A. Simary, "Integral representations of the modified Bessel function of the second kind," J. Nat. Sci. Math., 8, 133-139 (1968). 1361. M. A. Simary, "On hypergeometric functions of matrix argument," Bull. Math. Soc. Sci. Math. R. S. Roumanie, 16, No. 1, 113-118 (1972(1973)). 1362. M. A. Simary, "Definite integrals involving generalized E-functions," Math. Student, A40, 119-123 (1972). 1363. M. A. Simary, "Integrals involving generalized hypergeometric functions," Rev. Roumaine Math. Pures Appl., 17, No. 2, 281-286 (1972). 1364. M. A. Simary, "Definite integrals involving Whittaker functions," C. R. Acad. Bulgare Sci., 29, No. 7, 931-934 (1976). I365. A. K. Singh and A. Siddiqui, "Integral involving the H-function of several variables and an integral function of two complex variables," Bull. Inst. Math. Acad. Sinica, 11, No. 1, 31-35 (1983). 1366. B. Singh, "On certain expansions and integrals involving wmu(x)," Proc. Rajasthan Acad. Sci., 9, No. 2, 9-22 (1962). 1367. B. Singh., "On certain integrals involving %,,,(x)," Mitt. Verein. Schweiz. Versicherungsmath., 65, No. 1, 55-62 (1965). 1368. R. C. Singh Chandel, "Fractional integration and integral representations of certain generalized hypergeometric functions of several variables," Jnanabha, A1, No. 1, 45-56 (1971). 1369. F. Singh, "An integral involving product of G-functi0n, generalized hypergeometric function and Jacobi polynomial," J. Sci. Eng. Res., 12, No. l, 155-160 (1968). 1370. F. Singh, "Application of the E-operator to evaluate an infinite integral," Proc. Cambridge Philos. Soc., 65, No. 3, 725-730 (1969). 1371. F. Singh, "Integration of certain products involving H-function and double hypergeometric function. II," Math. Student, A40, 47-55 (1972). 1372. F. Singh, "Application of E-operator in evaluating certain finite integrals," Def. Sci. J., 22, 105-112 (1972). 1373. F. Singh, "On some results associated with a generalized Meijer function," Math. Student, A40,291-296 (1972). 1374. F. Singh and B. M. Shrivastava, "On some integral relations of Fox's function with applications," Ranchi Univ. Math. J., 6, 48-53 (1975). 1375. F. Singh and N. P. Singh, "Evaluation of an integral involving generalised Fox's H-function and generalized Legendre associated function," Vijnana Parishad Anusandhan Patrika, 17, 71-80 (1974).

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(1 + t~)a+f~-I

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0

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," Indian J. Pure Appl. Math.,

5, No. 9, 825=831 (1974). 1430. G. P. Srivastava and S. Saran, "Integrals involving Kamp6 de F6riet function," Math. Z., 98, No. 2, I19-125 (1967). 1431. G. P. Srivastava and S. Saran, "A theorem on Kamp6 de F~riet function," Proc. Cambridge Philos. Soc., 64, No. 2, 435-437 (1968). 1432. G. P. Srivastava and S. Saran, "A general theorem on a Kamp6 de Feriet function," Math. Notae, 21, No. 3-4, 105-111 (1968/1969). 1433. H. M. Srilvastava, "On some sequences of Laplace transforms," Ann. Soc. Sci. Bruxelles, S6r. I, 67, No. 3,218-228 (1953). 1434. H. M. Srivastava, "On integrals associated with certain hypergeometric functions of three variables," Proc. Natl. Acad. Sci. India, A34, 309-316 (1964). 1435. H. M. Srivastava, "Hypergeometric functions of three variables," Ganita, 15, 97-108 (1964). 1436. H. M. Sr~.vastava, "Some integrals involving products of Bessel and Legendre functions," Rend. Sere. Mat. Univ. Padova, 35, No. 2, 418-423 (1965). 1437. H. M. Srivastava, "The integration of generalized hypergeometric functions," Proc. Cambridge Philos. Soc., 62, No. 4, 761-764 (1966). 1438. H. M. Srivastava, "Some integrals representing triple hypergeometric functions," Rend. Circ. Mat. Palermo, 16, No. 1, 99-115 (1967).

1335

1439. H. M. Srivastava, "Some integrals involving products of Bessel and Legendre functions. II," Rend. Sem. Mat. Univ. Padova, 37, 1-10 (1967). 1440. H. M. Srivastava, "A note on a generalized of Sonine's first finite integral," Matematiche (Catania), 23, No. 1, 1-6 (1968). 1441. H. M. Srivastava, "Integration of certain products containing Jacobi polynomials," Collect. Math., 19, No. 1, 3-9 (1968). 1442. H. M. Srivastava, "A note on the evaluation of a definite integral," J. Korean Math. Soc., 7, 29-32 (1970). 1443. H.M.Srivastava,"Certaindoubleintegrals involving hypergeometric functions," Jnanabha, A1, No. 1, 1- 10 (1971 ). 1444. H. M. Srivastava, "A contour integral involving Fox's H-function," Indian J. Math., 14, No. 1, 1-6 (1972). 1445. H. M. Srivastava, "Remark on some integrals involving products of Whittaker functions," Proc. Am. Math. Soc., 31, No. 1, 133-134 (1972). 1446. H. M. Srivastava, "On reciprocal functions," Indian J. Pure Appl. Math., 3, No. 5, 704-713 (1972). 1447. H.M.Srivastava, "Certain double integrals involving hypergeometric functions," Jnanabha, A1, No. I, 1- I 0 (1971 ). Collect. Math., 24, No. 2, 117-121 (1973). 1448. H. M. Srivastava, "The Laplace transform of the modified Bessel function of the second kind," Publ. Inst. Math. (Beograd), 26, 273-282 (1979). 1449. H. M. Srivastava and M. C. Daoust, "On Eulerian integrals associated with Kamp6 de Feriet's function," Publ. Inst. Math. (Beograd), 9, 199-202 (1969). 1450. H. M. Srivastava and H. Exton, "A generalization of the Weber-Schafheitlin integral," J. Reine Angew. Math., 309, I-6 (1979). 1451. H. M. Srivastava, S. P. Goyal, and R. K. Agrawal, "Some multiple integral relations for the H-function of several variables," Bull. Inst. Math. Acad. Sinica, 9, No. 2, 261-277 (1981). 1452. H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions of One and Two Variables, South Asian Publishers, New Delhi (1982). 1453. H. M. Srivastava, K. C. Gupta, and S. Handa, "A certain double integral transformation," Nederl. Akad. Wetensch. Proc. Ser. A, 78, No. 5, 402-406 (1975). 1454. H. M. Srivastava and Co M. Joshi, "Certain double Whittaker transforms of generalized hypergeometric functions," Yokohama Math. J., 15 No. 1,, 17-32 (1967). 1455. H. M. Srivastava and C. M. Joshi, "Certain integrals involving a generalized Meijer function," Glas. Mat. 3, No. 2, 183-191 (1968). 1456. H. M. Srivastava and C. M. Joshi, "Integration of certain products associated with a generalized Meijer function," Proc. Cambridge Philos. Sot., 65, No. 2, 471- 477 (1969). 1457. H. M. Srivastava and C. M. Joshi, "Integral representation for the product of a class of generalized hypergeometric polynomials," Acad. Roy, Belg. Bull. CI. Sci., 60, 919-926 (1974). 1458. H. M. Srivastava and R. Panda, "Some operational techniques in the theory of special functions," Nederl. Akad. Wetensch. Proc. Ser. A, 76, No. 4, 308-319 (1973). 1459. H. M. Srivastava and R. Panda, "An integral representation for the product of two Jacobi polynomials," J. London Math. Soc., 12, No. 4, 419-425 (1975). 1460. H. M. Srivastava and R. Panda, "Some multiple integral transformations involving the H-function of several variables," Nederl. Akad. Wetensch. Proc. Ser. A, A82, No. 3, 353-362 (1979). 1461. H. M. Srivastava and N. P. Singh, "The integration of certain products of the multivariable H-function with a general class of polynomials," Rend. Circ. Mat. Palermo, 32, No. 2, 157-187 (1983). 1462. H. M. Srivastava and J. P. Singhal, "Contour integrals associated with certain generalized hypergeometric functions," Proc. Natl. Acad. Sci. India, A36, No. 4, 824- 840 (1966). 1463. H. M. Srivastava and J. P. Singhal, "A note on certain hypergeometric functions of two and three variables," Ganita, 17, No. 2, 99-108 (1966). 1464. H. M. Srivastava and J. P. Singhal, "Double Meijer transformations of certain hypergeometric functions," Proc. Cambridge Philos. Soc., 64, No. 2, 425-430 (1968). 1465. H. M. Srivastava and J. P. Singhal, "Certain integrals involving Meijer's G-function of two variables," Proc. Natl. Inst. Sci. India, A35, No. 1, 64-69 (1969). 1466. H. S. P. Srivastava, "Integrals involving generalized Legendre functions and a generalized function of two variables," Vijnana Parishad Anusandhan Patrika, 21, No. 1, 27-33 (1978). 1467. K. J. Srivastava, "Certain integral representations of MacRobert's E-function," Ganita, 8, 51-60 (1957). 1468. K. N. Srivastava, "On some integrals involving Jacobi's polynomials," Math. Z., 82, No. 4, 299-302 (1963). 1469. K. N. Srivastava, "On some integrals involving Gegenbauer polynomial and Chebychev polynomial of first kind," Ricerca (Napoli), 14, genn.-apr., 10-16 (1963).

1336

1470. K. N. Srivastava, "On some integrals involving Jacobi's polynomials. II," Math. Z., 85, No. 3, 257-259 (1964). 1471. R. Srivastava, "Definite integrals associated with the H-function of several variables," Comment. Math. Univ. St. Paul., 30, No. 2, 125-129 (1981). 1472. S. C. Srivastava, "On certain integrals," Math. Balkanica, 8, 187-192 (1978). 1473. S. K. Srivastava, "Theorems on a generalised Laplace transform," Acta Mexicana Cienc. Teen., 6, 59-63 (1972). 1474. S. K. Sr~Lvastava,"A generalized Laplace transform," Istanbul lSIniv. Fen. Fak. Mecm., A37, 93-100 (1974). 1475.T.N. Srivastava, "A note on an integral transform," Univ. Nac. Tucumgn Rev., A21, No. 1-2, 85-94 (1971). 1476. T.N. Srivastava, "Some integrals involving generalized H-function of Fox," Acta Mexieana Cienc. Teen., 11/12, No. 31-34, 42-52 (1977/1978). 1477. R. Straubel, "Unbestimmte Integrale mit Produkten yon Zylinderfunktionen. II," Ing.-Areh., 13, 14-20 (1942). 1478. S. N. Stuart, "Non-classical integrals of Bessel functions," J. Austral. Math. Sot., B22, No. 3, 368-378 (1981). 1479. K. Sugahara, "Some applications of complex integral," Repts. Himeji. Inst. Technol., No. 25a, 11-13 (1972). 1480. P.K. Sundararajan, "Some integrals involving E- function of MacRobert and G-function of Meijer, n Proc. Natl. Acad. Sci. India, A34, No. 1, 97-104 (1964). 1481. P.K. Sundararajan, "Some finite and infinite integrals involving G-function," Proc. Natl. Acad. Sci. India, A36, No. 2, 435-440 (1966). 1482. C. D. Sutherland, "Footnote to the evaluation of certain complex elliptic integrals," Math. Comp., 19, No. 89, 132-133 (1965). 1483. R. Swaroop, "A study of Varma transform," Collect. Math., 16, No. 1, 15-32 (1964). o~

1484. M. Talfit--Erben, "Evaluation of the integral

I (n, a. b) = l

6

rne - a t

X Ei(br) ar ," Istanbul Tek. ldniv. Bill., 29, No.

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1337

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(x--e)~+, 1/ax2 + bx + e '" Edinburgh Math. Notes, 32, 13-14 (1941).

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1339

1568. E. M. Wright, "The asymptotic expansion of the generalized hypergeometric function," J. London Math. Soc., 10, No. 4, 286-293 (t935). 1569. E. M. Wright, "The asymptotic expansion of the generalized hypergeometric function," Proc. London Math. Soc. 46, No. 5, 389-408 (1940). 1570. C. R. Wylie Jr., "New forms of certain integrals," Am. Math. Monthly, 49, No. 7, 457-461 (1942). 1571. J. M. S. Yadav and Y. N. Prasad, "Some finite integrals involving generalized function of r variables," Vijnana Parishad Anusandhan Patrika, 20, No. 3, 211-217 (1977). 1572. J. M. S. Yadav and Y. N. Prasad, "Some infinite integrals involving generalized functions of r variables," Vijnana Parishad Anusandhan Patrika, 21, No. 3, 213-220 (1978). 1573. S. R. Yadava, "On certain integrals and a theorem in generalised Hankel transform," Proc. Natl. Acad. Sci. India, A42, No. 4, 277-286 (1972). 1574. S. R. Yadava, "Certain theorems in two dimensional Laplace transform," Indian J. Pure Appl. Math., 6, No. 10, 1167-1172 (1975). 1575. C. Yeh, "A note on integrals involving parabolic cylinder functions," J. Math. and Phys., 45, No. 2, 231-232 (1966). 1576. M. Yoshida, "Euler integral transformations of hypergeometric functions of two variables," Hiroshima Math. J., 10, No. 2, 329-335 (1980). 1577. J. van Yzeren, "Moivre's and Fresnel's integrals by simple integration," Am. Math. Monthly, 86, No. 8, 691693 (1979). 1578. R. Zanovello, "Integrali di funzioni di Anger, Weber ed Airy--Hardy," Rend. Sem. Mat. Univ. Padova, 58,275285 (1977). 1579. A. Zygmund, "On certain integrals," Trans. Am. Math. Soc., 55, 170-204 (1944). SUPPLEMENTARY LITERATURE 1.

2. 3. 4.

.

6. 7. 8.

.

10.

11. 12. 13. 14.

1340

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