Appl. sci. Res.
THE
Section B, Vol. 6
FERMI-DIRAC
INTEGRALS
co
f ~(~)=(p!)-l f eP(e~-'7 + l ) - l d e 0
by R. B. DINGLE *) Department of Physics, Umve~sity of Western Austraha, Nedlands, Australia
Summary F o l l o w i n g a discussion of t h e r e l a t i o n s h i p of t h e F e r m l - D i r a c integrals to o t h e r functions, c o m p l e t e e x p a n s i o n s are d e v e l o p e d w h i c h e n a b l e t h e integrals of all orders to be c a l c u l a t e d w i t h o u t recourse to n u m e r i c a l i n t e g r a t i o n I n o r d e r to s u p p l e m e n t e x i s t i n g tables, values are given ior orders ---1 a n d 0 for p o s i t i v e a n d n e g a t i v e a r g u m e n t s , a n d for orders 1, 2, 3, 4 for p o s i t i v e arguments.
w 1. Introduction. The literature on Fermi-Dirac integrals can conveniently be divided into three main groups, (i) mathematical discussions of their properties, (ii) tabulations, and (iii) applications. Perhaps the most noteworthy papers in these groups are as follows: (i) S o m m e r f e l d 14) 1928, and N o r d h e i m 11) 1934. Derivation of an asymptotic series. G i l h a m 7) 1936. Discussion of the remainder term in the asymptotic series. M c D o u g a l l and S t o n e r lo) 1938. Analytic continuation of the integral, differentiation formula, and expression of the value for zero argument in terms of the Riemann zeta-function. R h o d e s 13) 1950. Relationship, for the integrals of integer order, between values for equal and opposite arguments. (ii) M c D o u g a l l and S t o n e r 1~ 1938. Tables for orders 1/2, 1/2 and 3/2. R h o d e s 13) 1950. Tables for orders 1, 2, 3 and 4 when the argument is negative. -
-
*) This work was commenced in 1954 while the author held a Post-Doctorate Fellowship at the National Research Council, Ottawa. The later work was supported by the Research Grants Committee of the University of Western Australia.
- -
Appl. sci. Res. B 6
2 2 5
- -
226
R.B. DINGLE
Beer, Chase, and C h o q u a r d 1) 1955. Tables for orders --1/2, 1/2, 3/2 (these from 10)), 5/2, 7/2, 9/2 and 11/2. (iii) The integrals arise in the great majority of problems in the theory of metals and semiconductors, and appropriate references appear in standard texts (e.g. W i l s o n 17) 1953). For energy bands of 'normal' form (energy proportional to square of wave-vector), only integer and half-integer orders are required. The major aim in the present paper is to obtain expansions for the Fermi-Dirac integrals sufficiently complete to render computation by laborious numerical integration unnecessary. The methods developed are of quite wide application, and might profitably be applied to other integrals. MATHEMATICAL
w 2. R e l a t i o n s h i p s
DISCUSSION
to other [ u n c t i o n s .
(i) To t h e 9~ a n d Ei i n t e g r a l s . Since ei~8 = - - 1 when s is an odd integer (including negative values), 1 + en-* vanishes when e ~ 7 - - i x s . Thus 1
as -
1 +e~
-~
8odd
e--
(,/--i~s)
'
where the coefficients as are given by as ~
=
Lt
,-.,-ins
1
(1 + e~ -*)
1 + ev-r
z
1.
e=n_in s
Thus co
oo
f ~ ( 7 ) = ~en _ . f e~Oe-*de _ e n X __[_1 f 1 + en-* 0
e,e-~de
sodd P ! d e - - (7 - - i:~s) 0
----en X s odd
91~(i~zs - - 7) = X e i'~* E i v + l ( i m
- - 7),
(1)
s odd
the integrals 9~ and Ei having been treated in previous papers 5)6)12). However, no tabulations exist for general complex argument. (ii) To t h e g e n e r a l i z e d R i e m a n n z e t a - f u n c t i o n . It maybe shown from relations due to H u r w i t z , L e r c h and J o n q u i ~ r e s), that f~(7)-=--cos ap f~(--7)--(2=)~+l(P !)-1R ett~,~+l,r
89189
(2)
where 7 is real, r being the generalized Riemann zeta-function 16) and R denoting the real part.
FERMI-DIRAC
(iii) T o t h e B o s e - E i n s t e i n
INTEGRALS
integral.
227
Let
oo
~(~)
=
p! J e*-~-
1 '
0
the principal value of the integral being understood for ~ > O. Then the observation t h a t 1 e x -~ 1
1 e x -- 1
2 e 2x -- 1
leads easily to the relation (e.g. 2)) ]'~(~/)
w 3 . Expansions x -- e-%
=
(3)
by the method o/ Mellin trans/orms. Writing
oo
f
:Bp(~) -- 2-PBp(2~]).
oo
xz-ldx xe* + 1
__
e - ~ z ( uz-ld u - - ( - - z ) l ( z - J u + I
o
1 ) ! e -*z,
o
~%, is
so t h a t the Mellin transform 15) of oo
M(z) = j ~ p
x z-1 dx = (-- z)! (z -- 1)! (p ! ) - l f e ~ e-~zde = o
o
= (-- z) ! (z -- l)
! / Z P +1 ~
~/ZP+I sin ~z.
(4)
~ e~z dz zp +1 sin ~z
(5)
B y the Mellin inversion formula, e+r
c+ioo
7~(~1-
1
#tt
2~i _J M(z)x -z dz e--ioo
--
1 f 2~i
c -- i co
where the line z = c separates the poles of (-- z)! from those of ( z - 1)!, i.e. 0 < c < l . In passing, it is noted t h a t (5) shows t h a t (e.g. 10))
7~(~) = f~-l(~).
(6)
E x p a n s i o n f o r n e g a t i v e 7. B y (5), the expansion in ascending powers of e~ is equal to minus the sum of residues of ~e~ffzP+1 sin nz at the poles of (-- z)! at z = r (r = l, 2 . . . . ), i.e. r
f p(~) = ~, (-- 1)r-ler~/rP +1, r=l
a well-known series convergent for ~?<0.
(7)
228
R. B. D I N G L E
Expansion
n e a r ~ = O. Combining the relation co
y
(0) =
p tJ
- - - - + 1
(1 - - 2-P) ~(p + 1 ) /
---In2 w h e n p = 0
1
(8)
]
(e.g. 9), p. 269) with the differentiation formula (6), it is seen t h a t f can be expanded as the Taylor series oo
oo
f'p(~) = Z
~vfP-v(()) -- Z ~v(1 -- 2v-~) ~(p + 1-- v)
,=o
v!
.=o
v!
,
(9)
tables of the ordinary R i e m a n n zeta-function being readily available (e.g. 9)). G i l h a m ' s 7) expansion in powers of ( 1 - e-r) is equivalent to this, b u t involves coefficients which are less easily evaluated, being combinations of the coefficients in (9). W h e n written in the form (9), the series is convergent only for [~] <~t. Expansions f o r p o s i t i v e ~. B y (5) the expansion in descending powers of eV is equal to the sum of residues of 7teVZ/zP+l sin ~z at the poles of (z -- 1) ! at z --~ -- r (r = 1, 2 . . . . ), and at the multiple pole at z = 0. I[ p is an integer, the contributions from the poles at z = - - r yield co
(-- A)P Z (-- 1)r-1 e-rv/r2~
~-- (-- 1)P f ' p ( - - ~/).
(10)
r=l
The contribution from the multiple pole at z = 0 is equal to the coefficient of zP in ~eVZ/sin ~tz. The most convenient m e t h o d of evaluation consists in noting t h a t oo
where to
1 and
~z/sin Jtz = 2 2~ t2, z 2~ ~=0
(11)
oo
t2v = Z (-- 1)~-1/# 2v = (1 -- 21-2v) ~(2v) ---/t=l
= 89
-
21-2 )Bv/(2
) !.
(12)
A v e r y comprehensive table of these coefficients t has been given b y D a v i s a), from which the values quoted here are t a k e n (table I).
FERMI-DIRAC
229
INTEGRALS
TABLE
I
J)
t2v
.50000000 .82246703 .94703283 .98555109 .99623300 .99903951 .99975769 .99993917
The required coefficient of z~ is thus equal to oo
2 Y~ t2~ • (coefficient of zP +1-2~ in enZ) v=O
[ 89
= 2
Y~ t2, r?+l-2v/(p + 1 -- 2v)!,
(13)
v=O
where [{(p + 1)] is the largest integer contained within 89 + 1). Combining (1 O) and (13), [89
1)]
f p ( ~ ) =- (--1)Pf~0(-- r/) q- 2 Y, t2~P+l-2v/( p + 1 -- 2v) l,
(14)
v=0
where p is an integer. This is equivalent to an expression found b y R h o d e s la). I / p is not an integer, it is convenient to change the sign of z in (5), which then becomes -c-ioo
1
f
~ e-n z
f~(~7) -- 2~i J (--
dz
1 f
7 ~ 7sin zz
2i
e-~zdz (-- z-)771sin :zz
--c+ioo
C
where C is a contour which starts from c~ q- i0, encircles the origin once counter-clockwise, and ends at oo -- i0. Taking - - z = ze -*~ on the first half of the contour and -- z = ze+i~ on the second half, with appropriate indents at the poles,
ffp(~) =l[e/~r~p+l) + e-i~(P +1)] {residues at poles of oze-~Z/z~+1 sin ~z atz=
1,2 . . . . }
oo
1 [ e'''p+l'
e-'~'~~
~_ e-~ z _d_z_
2i
.I z~+l sin ~z 0 oo
-- cos •ff ) r ( _
e - , / z dz 7) -}- sin =p f zp +1 sin ~zz '
0 where the principal value of the integral is to be understood.
(16)
230
R.B.
DINGLE
In the particularly i m p o r t a n t cases where p = integer + 89 the first term, of type e-n b y (7), vanishes identically. Since it will shortly be d e m o n s t r a t e d t h a t the second term is not of t y p e e-n for a n y order [cf. (21) and (23) to follow~, this disposes of the frequent erroneous implications appearing in the literature t h a t the c u s t o m a r y asymptotic expansion for these cases omits a contribution of t y p e e-~. If direct application is made of (11), the second term in (16) becomes oo
2 sin_ ~p y, t~
z2~-P -e e-~ z dz -- 2 sin z~p y~ t~ (2v -- p -- 2)! Y~ v=O ~2v-p-1 '
v=O 0
whence t2v ~ p + l - - 2 v
[ 89
= cos
+ 2 X
,=0
2 sin 4-
ap
pz
co
(p 4- 1 -- 2v)! t2~
X
[ 89
+
(2v -- p - - 2) T " ~2v-p-1
(17)
W r i t t e n in this form the expansion remains valid even when p is an integer. Apart from the first term, which does not appear to be generally known [except for integer 13) p], (17) is equivalent to the c u s t o m a r y asymptotic series first derived by S o m m e r f e l d 14). The terms of the last s u m m a t i o n begin to increase again for v > l(p 4- ~), and the expansion (17) is therefore severely restricted in accuracy. An expansion which is both exact and convenient can be constructed b y noting t h a t since b y (12) t2, ---- I -- 1/2 2~ 4- 1/3 2 ~ - . . . , z~z/sin az = 2
t~, z ~" + 2 Y, [z2~ -- (z/2)2~ 4- . . . v=O
v=n+l
= 2 X t,~ z 2~ + 2 1 -
z2
1 -
(z/2) 2 +
(18) .
.
.
.
B y (16), the contribution to )rp(~) from a term 2(--1)m-l(z/m)2n+2/[1 - - (z/m)2J in ~z/sin ~z is oo
2 (-- l) m-1 sin $em 2n+2
~p ~z2n-P e-~z dz
J
1 --
(z/m) 2
0
2 (-- 1)m-1 sin zp (2n -~t2n+2~]2n-p +1
p)!~2n_p(m~),
(19)
231
F E R M I - D I R A C INTEGRALS
where
~s(x) = 89 {9/s(x) -- ~Is(-- x)},
(20)
oo
the integral
9~s(x)= (s!)-lfese-*(e + x) -1 de
having been intro-
0
duced in an earlier paper 6). Hence
Y~('I)=COS=p y~(-- ~) + 2 +--2sin~p{
o~2n-~(~)
~2v ~]p+l--2v
Z
.=o
(p q- 1 -- 2v) l
t2,(2v--p--2)!+
~+ [89
+ ~2n-~+1
[89
3)]
~2v-:p--1
2~n+2 o~2n-p(2~)+ . . . .
(21)
Since by (20) As(x) ~ 1 for sufficiently large x, this expansion is convergent for n > -- 1. In actual practice, it is convenient to take n ~ 89 -i- 7) and regard the series in ~.~'s as an easily-calculated remainder. It will be shown in a forthcoming publication that the exact convergent series for o.~s(x) is *)
[
X2
~s(X)=
x2
x4
s(s--1) i l q (s--2)(s--3) Jr (s--2)(s--3)(s--4)(s--S) + "'} y~x8+l (eX--e-Xcos~zs). (22) 2(s l) sin ~zs
For large x and s this series is somewhat troublesome to compute, owing to large-scale cancellation between the two portions. Fortunately, in the ranges required o-~s(x) can alternatively be calculated from the expression X ( X X(2X--S) X(6X2--SXS~-S 2)
(x+s)2+ (x+s)~ x(24xa-58x2s+22xs~-sa) )
~s(x) - 2(x+s) .1 +
(x+s)8
-""
(x+s)6
x(1
+ 27s
+
1
1
)
- 12~s+ 2-88s~ . . . .
(x-s)/(2s)89 9
I(2s)89
f e"dt~ ( l + (x--s)a3s 2
(X--S)44S 3
+ ~(x--s)61 8s
" " ")
o
1 (
s
7(x--s)al2s 2 + (x--s)56s3
. . . . )1
(23)
*) This form is unsuitable for integer orders s, but these are not required since (17) p.
tcrminatesfor integer orders
232
R.B. DINGLE
Normally, only a few terms of (23) need be retained. For instance, when x ~-~ s, as it will indeed be for the first (dominant) term in the series if n is chosen ~ 89 + ~), the quantity in square brackets in (23) can be approximated b y [ ~ -"- (x - - s - - 8 9
- - s)E(x - - s ) ( x - - s + 1)--1]/3s.
(24)
For other ranges, f ~ ' e ~ ~ d# can readily be evaluated as a series in #', or reference made to tables 4) of this integral. As an example of the power of (21), consider f 89 The first term in (21) vanishes since cos 89 = 0. Taking n----3 so as to go as far as the least term, the four terms in the v-summation contribute 6.52135. Retaining only the major few terms in (23), o~11/2(4)= -- .432 and ~11/2(8) = 1.25, and these contribute --.00978 to f , yielding a final value f 89 This is correct to the s i x figures quoted, whereas the usual asymptotic series (17). gives f89 ---- 6.52 . . . . which is not quite accurate even to three figures. B y (23) and (24), r can be approximated as to order of magnitude by ~s(x) ~-~ 89 -- s + 1/6) when x ~-,s, giving the 'remainder term' in (21) as of the order sin~p . .
R .
(2n--p)! . ~]2n-~+l
(V + p -- 2n + 1/6)
when the series is broken off near the least term, n ~ 1(~] + p). In the same notation, G i l h a m ' s 7) result for the m a x i m u m b o u n d to the remainder would be essentially
Rs ~
sinsp y~
(2n--p)! ~]2n-p+l 4(p + l) (n + 1).
Hence Rc/R
~-, 4(p + I) (n + 1)/(~ + p -- 2n + 1/6),
(25)
from which it is clear that G i l h a m ' s stated upper bound fails to provide any realistic estimate of the remainder term. [Cf. 10), p. 82, for a numerical comparison of the actual error with R c ] . w 4. E x p a n s i o n s by e l e m e n t a r y methods. In view of the importance of the complete expansions, and of the desirability of similar complete expansions being obtained for other integrals at present
233
F E R M I - D I R A C INTEGRALS
computed b y numerical integration, it will now be shown how all the results of the last section could have been found b y elementary methods, though less generally, and much less expeditiously in the case of (21). Expansion
f o r n e g a t i v e ~. co
f f p(rl) = (p!)-l f ev(eS-V + 1)-1 de = 0 oo
co
=(P!)-l f ev de {e'l-* -- e2<~-*) + - - . ) = 2 (--1)r-lerv/r p+I, 0
(26)
r=l
in agreement with (7). E x p a n s i o n n e a r ~ = 0. Integrating b y parts,
~'v(~)
co
r p+l ]co+
_ __'
(p + 1)! keE-~ § 1 0
1
(27)
f ev+le -,7 de
(p + 1)!
(e*-n § 1) 2
Considering for simplicity only the cases for which p > -- 1, the integrated term vanishes, and the other term is seen to be equal to y~+l(~/). This establishes the differentiation formula (6), and the required expansion follows as in w3. E x p a n s i o n f o r p o s i t i v e ~. The surviving term in (27) can be written 0
(p§
1)!fv(~) =
--
co
§ --oo
(en-~ § 1)2 --oo
In thus extending the integration to negative values of e, it is necessary to adopt systematically some convention regarding the phase to be accorded to --1. The convention adopted here will be that the integrand is always real when ~ is real ; in other words, the mean will be taken throughout of results obtained b y writing -- 1 ------e • Then oo
oo
c o s a p (s'+lQ+~ds ~p+l Yv(~/) -- (p § 1)! d(eE+n + 1) z ~- (p § 1)! 0
f(l§
(28) (e-e § 1)z
'
--oo
on putting # = e -- ~ in the second integral. Comparison with (27) shows that the first term is equal to cos :zp t i p ( - - ~ ) , corresponding precisely to the first term of (21).
234
R.B.
DINGLE
Invoking the exact form of the terminated Taylor expansion (e.g. 16), p. 95), 1
(1 +~/~/)~+1
2.
(~/~/)t,
(~/~7)~'"+1 f
(1--t)zn
(29)
dt
(p + 1) ! -- ,~0 # !(p-t- 1--#) ! t- (2n) !(p-- 2n) ! (1 +t~/~) 2n-;" 0
By (28), the contribution of the summation in (29) to )z'p(~) is oo
~n E
( ~t, e-~ d~
~+i-~
.=0 ~!(p + 1 - ~ ) ! - J ( e - ( ~ 1)~ --r co
The integral obviously vanishes if # is odd. For even/,,
r
f = 2 f, --oo
0
and expansion of the integrand in rising powers of e-~ then yields 2(#l)tt, [cf. (12)1. Making use of the well-known formula (--z)! . ( z - 1 ) ! = ~/sin ~z (e.g. 9), p. 11), it is seen that the resultant contribution to y~(~) corresponds precisely to the two summations in (21). The integration over ~ in (28) for the remaining term in (29) can be achieved by means of the following artifice. Noting t h a t co
(I + t~/~l)-(2n-P I = [(2n -- p -- 1)!]-1/e-(l+t~/~)a a2n-p-1 de, 0
the required integral over ~ can be written as proportional to r
oo
( ~2n+l e-~ e-~t~'/n d~ (~:2n+le-~(e-~V,/n_e~t,/n)d~ J (e-~-T 1~2 = J (e-~ + l) 2 ' --co
0
since 2n + 1 is necessarily odd. Expanding (e-$ + 1)-2 in rising powers of e-S, and integrating term by term, this reduces to
(2n + 1)!
(1 + 22.+~
t~/~)2.+2
-
1 (I + t ~ / 2 ~ ) 2 . + 2
(1 -
t~/~) 2~§
-
-
(1-t~[2~)2.§
} 1 + ....
(3o)
It is now possible to integrate over t. For substituting 1 -- t = 1/r, 1
f
00
(1--t)2ndt
__f
0
dr
_
1
{(1 + alm~)r-- alm~l} 2n+2 -- (2n + 1) (1 + (~lrml)'
(1 + talm~l) 2n+2-1
FERMI-DIRAC
INTEGRALS
235
so that a typical pair of terms in (30) yields -
-
2 ( -- 1)m-1 (2n) !(o'/~)/m2n+2[ 1 -- (a/m~) 2].
The contribution of such a pair to the second term in (28) is therefore co -2(-- 1)m-1 f a2n-pe-*da (31) (p-- 2n) l (2n--p-- 1) ! m2n+2~ 2n-p+l 1-- (aims)2 ' 0
which is precisely equivalent to the corresponding result (19) of w3. Thus the complete expansion (21) has been obtained by elementary methods. TABULATIONS
The author prepared in 1954 a comprehensive table of the FermiDirac integrals _Yp(,/) for integer and half-integer orders. Consideration of the integral in this form, rather than the customary oo
Fp(~) = feP(e~-n + 1)-1 de = p! fp(,/), 0
possesses the following advantages: a) Unlike F, f exists even for negative integer orders. b) In the classical limit ~ 0, f~(~) is independent of the order p. c) Interpolation is possible between different orders p as well as between different arguments ~ (cf. 6), w 1) *). d) The relation between the function and its derivative is simpler (see (6)), thereby facilitating interpolation by Taylor series. In the meantime, however, an extensive table of F~(~) for halfinteger orders has been published by B e e r , C h a s e , and Choq u a r d 1) ~), and comparison between their F's and the author's f ' s shows excellent correspondence. In view of this, the tables below record only those orders (and arguments ) which do not appear in M c D o u g a l l and S t o n e r 10), R h o d e s is), or B e e r et alia 1). O7" *) F o r l a r g e p o s i t i v e a r g u m e n t s it is b e s t to i n t e r p o l a t e In f ~ between orders. t) M y t h a n k s are d u e to D r B e e r for his k i n d n e s s in s e n d i n g me a c o p y of t h i s w o r k p r i o r to p u b l i c a t i o n .
236
R.B.
DINGLE
NEGATIVE --7~
I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.O 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
*
Z--18
ARGUMENT
-TO t .6932 .6444 .5981 .5544 .5130 .4741 .4375 .4032 .3711 .3412 .3133 .2873 .2633 .2410 .2204 .2014 .1839 .1678 .1530 .1394 .1269
.5000 .4750 .4502 .4256 .4013 .3775 .3543 .3318 .3100 .2891 .2689 .2497 .2315 .2142 .1978 .1824 .I680 .1545 .1419 .1301 .1192
(n)=(e-,+l)-l.
I
Z-1
fo
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6! 1.7 1 1.8 1.9
.5000 .5250 .5498 .5744 .5987 .6225 .6457 .6682 .6900 .7110 .7311 .7503 .7685 .7858 .8022 .8176 .8320 .8455 .8582 .8699
.6932 .7444 .7981 .8544 .9130 .9741 1.0375 1.1032 1.1711 1.2412 1.3133 1.3873 1.4633 1.5410 1.6204 1.7014 1.7839 1.8678 1.9530 2.0394
f--l*
.TO t
.1091 .09975 .09112 .08317 .07586 .06914 .06297 .05732 .05215 .04743 .04311 .03917 .03557 .03230 .02931 .02660 .02413 .02188 .01984 .01799
.1155 .I051 .09555 .08684 .07889 .07164 .06504 .05904 .05356 .04858 .04407 .03995 .03625 .03283 .02975 .02696 .02442 .02213 .02003 .01815
t Yo(n)=ln (e,+l).
POSITIVE
n
--'Y] 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0
ARGUMENT
Z, .8225 .8943 .9715 1.0541 1.1424 1.2367 1.3373 1.4443 1.5580 1.6786 1.8063 1.9413 2.0838 2.2340 2.3921 2.5582 2.7324 2.9150 3.1060 3.3056
Z2
f3
Z4
.9015 .9873 1.0806 1.1818 1.2916 1.4105 1.5391 1.6782 1.8282 1,9900 2.1642 2.3515 2.5527 2.7685 2.9997 3.2472 3.5116 3.7939 4.0949 4.4154
.9470 1.0414 1.1448 1.2578 1.3814 1.5164 1.6638 1.8246 1.9998 2.1906 2.3982 2.6239 2.8690 3.1349 3.4232 3.7354 4.0732 4.4383 4.8326 5.2580
.9721 1.0715 1.I807 1.3008 1.4326 1.5774 1.7363 1.9106 2.1017 2.3111 2.5404 2.7913 3.0658 3.3658 3.6936 4.0513 4.4415 4.8668 5.3301 5.8344
FERMI-DIRAC
POSITIVE
,7
ARGUMENT
Z_,
Zo
71
.8808 .8909 .9002 .9089 .9168 .9241 .9309 .9370 .9427 ,9478 .9526 .9569 .9608 .9644 .9677 .9707 .9734 .9759 .9781 .9802 .9820 .9837 .9852 .9866 .9879 .9890 .99005 .99098 .99184 .99261 .99331 .99394 .99451 .99503 .99550 .99593 .99631 .99666 .99698 .99727 .99753 .99776 .99797 .99816 .99834 .99850 .99864 .99877
2.1269 2.2155 2,3051 2.3956 2.4868 2.5789 2.6716 2.7650 2.8890 2.9536 3.0486 3.1441 3.2400 3.3363 3.4328 3.5298 3.6270 3,7244 3.8221 3.9200 4.0181 4.1164 4.2149 4.3135 4.4122 4.5111 4.6100 4.7091 4.8082 4,9074 5.0067 5.1061 5.2055 5.3050 5.4045 5.5041 5.6037 5.7033 5.8030 5.9027 6.0025 6.1022 6.2020 6.3018 6.4017 6.5015 6.6014 6.7012
3.5139 3.7310 3.9571 4.1921 4.4362 4.6895 4.9520 5.2238 5.5050 5.7957 6.0958 6.4054 6.7246 7.0534 7.3918 7.7400 8.0978 8.4654 8.8427 9.2298 9.6267 10.0334 10.4500 10.8764 11.3127 11.7589 12.2149 12.6809 13.1567 13.6425 14. I382 14.6438 15.1594 15.6849 16.2204 16.7658 17.3212 17.8866 18.4619 19.0472 19.6425 20.2477 20.8629 21.4881 22.1233 22.7684 23.4236 24.0887
237
INTEGRALS
(continued)
f2 4.7563 5.1185 5.5028 5.9102 6.3416 6.7978 7.2798 7.7885 8.3249 8.8898 9.4843 10.1093 10.7657 11.4545 12.1767 12.9332 13.7250 14.5531 15.4184 16.3219 17.2647 18.2476 19.2717 20.3379 21.4473 22.6008 23.7994 25.0441 26.3359 27.6758 29.0647 30.5038 31.9939 33.5360 35.1312 36.7804 38.4847 40.2450 42.0623 43.9377 48.8721 47.8665 49.9220 52.0394 54.2199 56.4644 58.7739 61.1495
f4 5.7164 6.2099 6.7408 7.3113 7.9237 8.5804 9.2841 10.0373 10.8427 11.7032 12.6216 13.6011 14.6445 15.7553 16.9366 18.1918 19.5244 20.9380 22.4362 24,0229 25.7019 27.4772 29.3528 31.3329 33.4219 35.6239 37.9435 40.3853 42.9539 45.6541 48.4907 51.4687 54.5931 57,8692 61,3021 64.8972 68.6600 72.5960 76.7109 81.0104 85.5004 90.1868 95.0757 100.173 105.486 111.019 116.781 122.776
]
6.3828 6.9788 7.6261 8.3283 9.0897 9.9145 10.8074 11.7730 12.8166 13.9434 15.1591 16.4697 17.8815 19.4009 21,0349 22,7907 24.6758 26,6982 28,8662 31,1884 33,6739 36.3320 39.1727 42.2060 45.4429 48.8942 52.5716 56.4870 60.6528 65.0821 69.7882 74.7850 80.0868 85.7086 91.6659 97.9745 104.651 111.712 119.176 127.061 135.385 144.167 153.429 163.206 173.470 184.294 198.682 207.658
238
R . B . DINGLE P O S I T I V E A R G U M E N T (continued)
6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0
-Y'-I
Yo
fl
.99889 .99899 .99909 .99918 .99925 .99932 .99939 .99945 .99950 .99955 .99959 .99963 .99966 .99970 .99973 .99975 .99978 .99980 .99982 .99983 .99985 .99986 .99988 .99989 .99990 .99991 .99992 .99993 .99993 .99994 .99994 .99995 .99996
6.8011 6.9010 7.00091 7.10083 7.20075 7.30068 7.40061 7.80055 7.60050 7.70045 7.80041 7.90037 8.00034 8.10030 8.20028 8.30025 8.40023 8.50020 8.60018 8.70017 8.80015 8.90014 9.00012 9.10011 9.20010 9.30009 9.40008 9.50008 9.60007 9.70006 9.80006 9.90005 10.00005
24.7638 25.4489 26.1440 26.8491 27.5642 28.2893 29.0243 29.7694 30.5244 31.2895 32.0645 32.8496 33.6446 34.4496 35.2647 36.0897 36.9247 37.7697 38.6248 39.4898 40.3648 41.2498 42.1448 43.0498 43.9648 44.8898 45.8249 46.7699 47.7249 48.6899 49.6649 50.6499 51.6449
Received 28th August, 1956.
f3 63.5920 66.1026 68.6821 71.3317 74.0523 76.8449 79.7105 82.6501 85.6647 88.7553 91.9229 95. I685 98.4931 101.898 105.383 108.951 112.602 116.336 120.156 124.062 128.054 132.135 136.305 140.564 144.915 149.357 153.893 158.523 163.247 168.068 172.986 178.001 183.116
129.013 135.497 142.236 149.236 156.504 164.049 171.876 179.993 188.408 197.129 206.162 215.516 225.198 235.217 245.581 256.297 267.374 278.820 290.644 302.854 315.459 328.468 341.889 355.731 370.005 384.717 399.879 415.499 431.587 448.152 465.204 482.752 500.807
Y4 220.245 233.469 247.353 261.924 277.209 293.235 310.028 327.619 346.037 368.311 385.473 406.554 428.587 451.605 475.642 500.733 526.914 554.220 582.690 612.362 643.274 675.467 708.981 743.859 780.142 817.874 857.100 897.865 940.216 984.199 1029.862 1077.256 1126.430
239
F E R M I - D I R A C INTEGRALS
REFERENCES 1) Beer, A. C., M. N. C h a s e and P. F. C h o q u a r d , Helv. Phys. Acta 2B (1955) 529. 2) C l u n i e , J., Proc. Phys. Soc. A 67 (1954) 632. 3) D a v i s , H. T., Tables of the Higher Mathematical Functions, Vol. 2, Principia Press, Bloomington, Indiana, 1935, p. 247. 4) D a w s o n , H. G., Proc. Lond. Math. Soc. 29 (1897) 519. Stablein, F. a n d R . Schl~ifer, Z. angew. Math. Mech. 23(1943) 59. T e r r J l l , H. M. and L. S w e e n y , J. Franklin Inst. 237 (1944) 499, 238 (1944) 220. z z ~/ R o s s e r, J . B . , Theory and Apphcatlon of f e -z2 dx and f e-P2u 2 dy f e -x2 dx, 0
5) 6) 7) 8) 9) 10) 11) 12)
13) 14) 15)
16) 17)
0
0
Mapleton House, Brooklyn, N.Y., 1948. D i n g l e , R. B., Appl. Sci. Res. B 4 (1955) 401. D i n g l e , R. B., D. A r n d t , a n d S . K. Roy, Appl. Sci. Res. B 6 (1957) 144. G l l h a m , C. W., Proc. Leeds Phil. Soc. 3 (1936) 117. H u r w i t z , A., Z. Math. Phys. 27 (1882) 86; E. L e r c h , Acta Math. l l (1887) 19; J o n q u i ~ r e , Bull. Soc. Math. de France 17 (1889) 142. J a h n k e , E. and F. Erode, Tables of Functions, Dover, New York, 1945. M c D o u g a l l , J. and E. C. S t o n e r , Phil. Trans. Roy. Soc. A 237 (1938) 67. N o r d h e i m , L., Muller Pouillets Lehrbuch der Physik, 4, iv, 277, Braunschwelg, 1934. P l a c z e k , G., The Functions En(x), M T - - 1, National Research Council of Canada, Division of Atomic Energy, 1946. (Reprinted in Applied Math. Series no. 37, pp. 57-111, U.S. Gov. Printing Office, 1954). R h o d e s , P., Proc. Roy. Soc. A 204 (1950) 396. S o m m e r f e l d , A., Z. Phys. 47 (1928) 1. T i t c h m a r s h , E. C., Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937; G. D o c t s c h , Theorie und Anwendung der Laplace Transformation, Berlin, 1937; R. B. D i n g l e , Appl. Scl. Res. B 4 (1955) 401. W h i t t a k e r , E. T., and G. N. W a t s o n , A Course of Modern Analysis, 4th edition, Cambridge University Press, 1927. W i l s o n , A. H., The Theory of Metals, 2nd edition, Cambridge University Press, 1953.
