SOME THEOREMS IN OPERATIONAL CALCULUS BY N. A. SHASTRI (King Edward College, Amraoti, Berar) Received in revised form July 12, 1944
WE know in operational calculus that if the 'image' f (p) of an ' original '
function h (x) can be broken up into `images' fl (p), f (p), ....fn (p), etc., of different ` original ' functions h l (x), h 2 (x), .... h (x) and if f(P) =
pn-i A
(P)fz(P)....fn(P)
then h (.x) can be related with h l (x), h 2 (x) .... h„ (x) by means of the relation Ii (x) ff .... J hi(ti)h2(t2)....hn_1(tn_1)h.[x tl t2.. tn _1]dtldt2....dt n _I —
--
—
where t 1 , t2 ,....t_ 1 are all > 0 and
t l +t2 + t3 + ....+ t l - 1
x] x _
f _ 1 g (y) p + ve
(ii) exp [— (p + y) x] x` 4- .° (y) R (s) > - 2 and p +ve. Hence it is sufficient for the inversion of the order of integration that g should be bounded and integrable in y > 0 and
f
Al
U
(v)
v
I g (y) I dy is convergent. 211
212
N. A. Shastri
The relation f(p) - h (x) means that the functions f( p) and h (x) are ` operationally ' related, i.e., 00
f(p) = p f e -Px h (x) dx.
R (p) > 0.
eJ
the integral being supposed convergent, and f(p) is called the 'image' of the 'original' function h (x). In what follows we will use two theorems of operational calculus and they will be denoted as F l and F., viz., If h (x) = f (p) then (i) h (ax) a constant > 0 (F l)
=f
(le a)
(ii) e h (x) = p + A' (P + A) (F2). 1. Now if f (p) = h (x) and h (p) 0 (p) = g (x), then by definition we get
.f (P) = P f e - h (s) ds -,
p> 0
0
and
00
h (s) 0 (s) = s f e -`t g (t) dt
s>0
0
both integrals being convergent. Hence CC
00
(1l) f(P) =P f b'(t) f f e -(p+t)s_.Ss) ds] dt Lo after substituting the value of h (s) and then changing the order of integration. If we assume in (1.1) (s) = s m we will have by using the integral 00
r J
e - cP+ t^ s s „: +1 ds = F (m + 2)) , R (rra > -- 2, (p + t)
0
THEOREM I.--
p + t being positive h (p) = g (x•). then
If f (p) - h (x) and
(1.2)
00
1(p) = T (m + 2) P
(p + t) + 2 dt R (m) > 0
p being positive and g (t) satisfying the conditions given in the introduction. As an example on this theorem, consider n
g (x) — x2 J. (a ,/x)
R (n) > — 1.
Some Theorems in Operational Calculus
213
By applying F l to the operational representation'-' 2
n —
x Jn (2 %/x) _ p
we get
e -'
R (n) > — 1
yt
x 2 Jn (a -,/ x) = g (x)
az
( a n —n )
2
e
p
( a)" + P 2
Z
^gip 'e
p
=(p) h(p). -
Again from the integral 3 CO
e -2u-4u a —s—i du= 21 ( 3 s+2) W —s Ks
(,V 2 w)
(1.3)
where Ks is the Bessel function of the second kind and R (u, 2 ) > 0, we get by the substitutions w 2 = 4p and x
?t = x,
ex = 2p
the operational representation
2 K s (2 1/1)
R (p) > 0
(1.4)
(at') R(p) >0
(15)
By applying F. to this we obtain _ a"S
s^1
_
e 4x '
Hence
h(x)=x
2 () p1 l2
w—n—i
K5
a2
e
—4
—n 2 (ay p
'
-
2
-n K(a1) =1(p).
Therefore by Theorem I proved just now we get after simplification (ay I (n — ,^) K. _.,r (a s /P) _ .P (µ -f- 11 2 2 p
0o
n
x .2 Jn (a/4 w^x 1 x)F (p
(l 6)
Changing x into x 2 and p into p 2 we get the well-known result*
(l
n
\2lup
=
n+1
µKW-n(aP) (
R (v)> — 1
2R (IL) + > R (n) > — (p2 +Jx21 ^x) ^ dx f R(P)>0
• Walson, Bessel Functions, p. 434 (2). Ala
N. A. Shastri
14
As a second example on the theorem consider the operational representation .xm T„,
(-r) = {1- p) - r(n + 1 m+„
(1.7)
)p
where n is an integer and R (in + n) > 1-. T'; (x) being Sonine's polynomial which is a generalization of the parabolic cylinder function. By applying F 2 to (1.7) we get g (x) = e-x x "t Tm (x)
T' (n + 1) (1 + p)m - pp-1 h(p)•
Hence (
n^ x*
—Y
h(xl =T(!T +1)
(1 +xl^`+n s
f (p) the ' image ' of h (x) is the value of the integral 00
f
( xe_^.^- ^Ix 1'(n+1)(1 } xlm^.n_ p>0 -
(-)n r(µ+n+ 1) e) P (n
+ 1)p}
(-
W -.4 -'4 (nt+µ+21,+1), 4 (µ-m) (P)
obtained by using the substitution
p>O, R ( µ -n±l)>
px = t
e kz z k Wk,on (Z) = r (1- k+ m) f e
and the integral'
tk a t -k-t nL (1 +Z)
b
0
/,t dt
(1.8)
\
R(k-2 - in) yc0 Therefore by applying the theorem investigated we get 00 e _t tm Tm (t) dt
f
0
(p -1- t}+^ +
( -)nr(w
+ n +1)eV
- 1' (+ 1 ) r (n + 1) p?tw- m+ 1) W-$ (m+ +2n+1), (w m) (p)
R(i+1)>0andR(y + n 4-1)>0. If we put p = a 2 and = x 2 we get
f 00e_x x 21'1 .^- 1
2
n Tm(x) 2
(a2 +x2lµ- 1 / 0\
n r ((t + n + 1) ela' 2 I i2n } 1). 1 (µ—m) Q ) ( P (ii } 1) r (fc { 1) Qw m+ l W —1 (m µ
__ (
- )
(19 )
R( ± 1) > 0, R (u +n + 1) > 0.
Some Theorems in Operational Calculus
215
The particular cases of (1.9) for (i) m = Z and µ = 0 and (ii) in = — ? and µ = 0 yield results already obtained 5 as both Sonine's polynomial T; (x) and Whittaker's function Wk,,, (x) are connected with parabolic cylinder functions by well-known relations.
p) 6 O
II. if f(p) = h (x) and h
-
g (x) instead of h (p) (fi (p) __
g (x) as assumed in the previous case, we get
f
00
J( p) = P
00
g (t)
0
If
-(Px+
e
t l xx
0
0 (x)
dx] dt
p> 0
This yields two interesting theorems if we take for ' (x) 1 (i) xor (ii) xs
In the first case we get by using the integral
fe
(
px f -
x
-
z x dx=
0
IT
(21j
/P e
f
the theorem: if f(p) = h (x) and
J
-2tP
h (p) = g (x), then
00
fV71p (P) ` (" e
g (t) dt
p> 0
0
provided g (t) satisfies the conditions already given and that the integral is convergent. If we change p into
4 and t into x and use the definition of 2
2
the operational representation we can restate the above as Theorem II: 2 If .f (P) (x) and 'P h (P) - g (x), then - f ( ) x g (x 2
=h
),
a theorem useful in getting new operational representations. Let us now take the second value of ¢ (x) and use (1.3) with the substitution px = 2u, we get Theorem III: If f(p)r=
h (x) and p, h (') = g (x), then .1(P) = 2p2
f t 2 g(t)KS( 2 /pt)dt G
provided R (p) > 0; g (t) satisfying the conditions already given, K s representing Bessel function of the second kind,
216
N. A. Shastri
We will now take some examples by way of application of the theorems investigated. n
(i) Let g (x) -= x2 Jn (2 V) —n —
i
!p e p R (n)> -1 = pI . p -tn+jl e
=Vph \ 1 = e - '` x
Therefore h (x)
n+}
T (n + $)
p n++l'
( 1 + p) = f(p)
by applying F 2 to x-
I'(np+ 1)•
Therefore from Theorem II we get the operational representation I' (n /-^ ) --p2-4 -- - x. xn Jn (2x) v n (l +p 2 /4) This changes by the application of F t into fl+ 1
r (^)
F2 + (1+p) n
R (n) > - 1
- xn { Jn (x) .... R (n) > - 1
(2.2)
This result can also be obtained by applying to the operational representation obtained by Po1 6 2n r In + P — K yl J (x) n
1/77 — (1 + p 2) n + f - . - -
the theorem of operational calculus p ( dP ) pp) = x h (x) if h (x) == f(p)• (ii) Consider' `
1
p
V [2 n— I) -n] Jn - 2 p^ == x -
Jm-1, n (3)
Some Theorems in Operational Calculus
217
and choose m = -f (2n + 1) we get n-1
g (x)=
x3
2 ( 3 \'x-) J?n—t n
n
VPP y ` Jn (-2 =Vnh(). n
Hence h (x)
= x2 Jn (— 2V3c) n
= ( — ) n x 2 J,, (21/x)
( — ) n P e P =f(p) where n is an integer. Therefore from Theorem lI we get x
r
I (2n+1)
_(— (2)2n J2u-1 (3x 2'3) - 1^? P e
(iii) Consider $(x) =e_
x x m Tm(.Y)= 1-
(2'3)
(—)YYpn_.JA
r(n +. 1)(1 + pr-n-^
P J h(-)
R(m + n)> — 1
from (1.7) with F9. (—)
^ w'
1
Hence h (x) = T (1 I n) (1 x)m+ n+i'
— µ + 2) I (n+µ) elrp W (—)'--I'F( (m 1 + n)P -+n—A h21n 12),.--I(,A+n_1)(P) =.f (p)R(m_ +2)>0. by following the same method for obtaining the 'image' as was done in the case of (1.9) before. Therefore by Theorem III we get
( —)n F (m -- u-}- 2 ) I (n--1) e p W P 2 I' (n -}- 1) —I(n—^+2m-{-2), —}(µ: L fl _ !) (P) 00
P mt (W-i) = /' -f
J
T (t)Kµ_](2-Nlt)dt
N. A. Shastri
218 Putting j. — 1 = 21 we have 00
f
e
-t m-1
t
Tn, (t) K 8, (2./) dt
— 2 r (n + 1) r(m— 21+ 1)P (iv)
4(n- 1 ) 1p
e W -*(n+2m-21+1) ,—j (21+n)(P) (2 4)
p>0 R(m+n)> —1 -}(m+n) Jm,n (3 ^V ) Let? e (x) = x
and R(m-21 +1) 0.
—J (n—m) J_ _2
n-m VP P
Pnh
GO
Therefore h (x) _ (—)n X
_ ( — )n: X
n-m 2
Jn_m (— 2 i/)
n-m 2
Jn-m ( 2 ',/x-)
(—)m p -(n-m) e- 7^
=1(P)
provided n and m are integers. Hence by Theorem III we get -n+1 r *(2'n m rn-n (_ ..) p e p= 2 p t
Jan, n (3 y t) K_ n (2
dt
0
Putting t = x 2 , changing p into p 2 and using k, (z) = k - , (z) we get after: simplification
IZ
1
2m-n-2 - $
00
n-2m { 5
p e p = f x T.. n ( 3 x 2) kn (2px 3 ) dv (2.5)
n and m being integers. This gives a relation between the Bessel function of the third order and the Bessel function of the second kind.
(v) Now consider o (x) = x n
cos (ti'2x)
(—) n Vw
(p
`
)
2-n y'P e-gyp Den (ç)
pn
219
Some 7'lieorems in Operational Calculus
an operational result obtained from Adamoff's integral" involving Weber's parabolic cylinder function. Therefore h (x) _ (— But 9
2—nDen ( Vx)
4J2n ! p p)". x D :zz 1^x +} (2 ) = ._2n . n i (I(+1 -p)n
Hence by F i and F. a
e
Dzn vz \/a 2n ! (— )n /2p'
and therefore
n! Hence from Theorem III we get
5
00
1 - 2
n-I
Z cos
22Kn(21%z)d'=izi
p
2 +_ ) ^ 2n: (
n_}
an integral more general than the one already given. (Watson, B. F., p. 388 (10)]•
Similarly, if we consider the operational relations zi
re —n—}
sin V/2z — (—) 2
ands M n Li (1^x)
— 1 1 ^7T e 4pPn D1 (-^p).
(1 — 4p)n V -r (2n + 1) ! 4p_ —
2n r2 .t
(1+4;)
and proceed on lines similar to the above we will get the integral CO
n; 2 1 _ !^ r^+ f z2 nsinlK1 (2 Vz ^ 4' _ i(2n n+t± 1) ._ (2.7)
2 n!
p
G +2pl
(vi) Take 1(P)=l+
Z
-sin x=h(x)
Therefore
h ( p ) — sin p _, bei (2 Vx) = g (x) Hence
00
1 +p2
= 2 f bei (2V) 1(0 (2
pt) cit
(2.8)
220
N. A. Shastri
Similarly if we start with
2
1 p-Pcos u
v we get
00
1 +pz = 2f her (2 't) K o (2 %/pt) a'
(2.9)
0
(vii) As the last example consider n
I' (1 -I- n) = h (x)
f(P) = ' Hence 1
1 h \P) I' (1 +n)ps+n
'
1
'
xr+n
-- -.. g (x) r^i +n) r(1 +n +s)
-
--
J
2
Therefore by applying the theorem and the substitution pt = 4 we get 00
22fl,. s
r (n + 1) r (n
+ s + 1) =
a result given in Watson's B.F., p. 388.
f x 2ft+ s ,.1 K, (x) dx
f o
(2.10)
III. We will conclude this paper by proving the following theorem. THEORFM IV.--If f(p) = h (x) ; h (p)p' - 't = g (x) and g (p) p1 " - j (x)
then f(4)= r(1+n)r(l+ m)p
x
f
m—n
x
O° 1 } (n—m--l) eW —+ (n+ n + 1), I (n—m) () J GO de ,
R (in +1)>0 and R(n+1)>0.
The proof of this theorem can be easily deduced. For this we have to reverse the order of integration twice. After the first inversion of the order of integration we use (1.2) to evaluate the integral occurring in the result. Then we proceed with the second inversion and use (1.8) and the substitution p = $. We then get the theorem after little simplification. The process of inversion is Justifiable if R (m)> — 1, /(x) is bounded and integrable in x> 0 and d
f' If (x) I dx is bounded in x> 0, and the last integral involving 0
Whittaker's function and j (x) is convergent. We will now consider two examples on this theorem.
Some Theorems in, Ofteralioltal Calculus s (i) Let
_x
10
] (x) = e
3
221
Nk. l (Y)
Pk- 1 +j
(1 4 p)k+ l-_ g (P) • P I_m
Hence
-^ n - l -k
g (x)
^p
P(k+m-1-f-+)e p
t m1^ -
- -
x
m(P) R(k-l+m +1)>0
W m
by using a process similar to one used in getting the operational representation in (1.9). Denoting the operational representation of g (x) by h (p) nl-'°, we have h (.x) = P (k + m - 1 + ) e
x
xl
j7
-
m2
1
m
m (.K)
-k-2 , -1+ 2 .
• 1"(k-{- m— l+ 2) P (— k— l+ 4) I'(/— k— m+ )
F(- - -- n + )
x P( 1— pl
--k-1—n-•+ 2F,
-f(p)
— k— 1+-},— k+ 1— m+ , .—k-1—n+, ( 1— P)]•
using the operational representation of Whittaker's function as given by Dha rll )P(-inf-k -I z) (.V) _ x1-1 e -IxP(in+k± W 1 (k-I+1) k,m xP(-p1-1
f(- )
2F,
[m+ k+ -1, k- m+Zi k-l+i ; -- p
with F. Hence we have by the application of Theorem IV the integral x
0
_
W+(n--m) (x)Nk,I ()^1a P
m— 1+4) ?'(— k— 1+ I) P(I — k— pz. m+ +) (In_„ I T(—k—1—n I^14)P(l+/ l(l+m) r(
x2F1 [—k—l+ , —k. +I—m±. —k-1—n+-;(1—p)) (3•1) R (m+ 1) > 0 R (n+ 1) > 0 anti R (k-- 1--- m+ 1) 0
222
N. A. Shastri (ii) Let .1(x) = x 2 JI (2 %/x) —1
1
.p aP m—I-1 —
1 1—m
e Pp
=P
= g (P) -P Hence g (x) = x
R(1)> -1
1—m
m-1-1 — 1
e
-1 m
2P n—
2
I-, (2'3P)
m-1
=2p
Km-., ( 2 1/P) P
1—n
1-n
=h(P)•p from (1 5), where K S denotes the Bessel function of the second kind. Again we have from the inteera1 12 00
emu ' k2, (2eu) u -2k du =
J
T ( ± m- k)]2(4----- m- k) etf ' Wkm (e2)
R(j ±m -k)>0
by using the substitutions e u = - and e _gip , p positive and the definition of the operational representation, 1 xk-# k-t 1 Wk 2x k k ?m (2 v/z) = P(+ m- k) V (-- in - k) e2P p
Hence h (x)
= 2x
,n, (P) (3•2)
KI (2 /v)
P(n+1)P(n-m+1+1)p x W_4(_1_1),
—n+1 (in--1±1)
1
e2 x
r 1 J (in-1)
\PJ = f(p) R(n+1)> 0 and R(n-m+1+1)>0 Therefore from Theorem IV we have
f
Co x
^ + (I+n—m-1) _
e x
W —3 (i! } ))!'7' 1) ,
I'(n- m + 1+ 1) *U-m) 2P -
1 (1 + m)
P
k (n-nr) (x) JI (2
e W
^ -^)
`P
dx
(
1
-n+4 (m-1-1), } (m -l) \ p) (3.3)
Some Theorems in Operational Calculus provided
223
R(m+1)>0, R(n+1)>0 and R(n—in+l+l)>0
The substitutions v,p=
s,n=— (in'+k'+ ),m=(m'— k' — )
change this result into the Hankel transform of the Whittaker's function obtained by Erdelyi. 13 If m = n = 1. Theorem IV reduces to a simpler form as: THEOREM V.-- •
.f (P)
=f e2 x
—
If f(p) h (x) ; h (p) g (x) and g (p) _ j (x) Mien
W- 4 , o (x)
.i (P) dx
0
As an application of this theorem consider the series of operational representations; f(p) = log p —y — log (x) =h(x) h (p) = — y — log (p) - log (x) _ g (x) and a (p) = log P — y — log (x) =j (x) Hence 00
— log p = f U
x
e2
x --I W-3, 0 (x)
[I
+ log
P] dx (34)
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Bromwich P. Humbert Watson Whittaker and Watson Gorakh Prasad Van der Pol P. Humbert
.. Theory
of Infinite Series, 1926, 503-04.
.. Le Cakul Symbolique, Paris, 27-30. .. Bessel Functions, 15, 183. .. Modern Analysis, Chap. 16. .. Proc. Benares Math. Soc., 1922, 4, 21-25. .. Phil. Mag., 1929, 8, 861-98. " Nouvelles Remarques sur les Fonctions de Besse! du Troisie'me Ordre," Att. Bella pont Accad. Scienze, Anno 87 Sess., IV del Marzo, 1934, p. 325. 8. Whittaker and Watson .. Modern Analysis, Chap. 16, p. 353. .. Phil. Mag., 1936, 22, (7), 30. 9. Varma .. Math. Zeit., 1936, 42. 10. A. Erdelyi .. Dhar Phil. Mag., 1936, 21, (7), 1087. 11. Meijer Koninklijke Akad. Wetenchappen Amsterdam, 1934, 37, 3. .. Proc. 12. C. S. .. Proc. Camb. Phil. Soc., 34, 29. 13. A. Erdelyi A2
..
