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Theory of Fluctuations in Superconductors$

Anatoly Larkin (late) and Andrei Varlamov

Print publication date: 2005

Print ISBN-13: 9780198528159

Published to British Academy Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198528159.001.0001

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(p.372) APPENDIX B PROPERTIES OF THE EULER GAMMA FUNCTION AND ITS LOGARITHMIC DERIVATIVES

(p.372) APPENDIX B PROPERTIES OF THE EULER GAMMA FUNCTION AND ITS LOGARITHMIC DERIVATIVES

Source:
Theory of Fluctuations in Superconductors
Publisher:
Oxford University Press

B.1 Euler gamma function

The gamma function Γ(z) was defined by Euler in the form

(B.1)
Γ ( z ) = 0 t z 1 e t d t ,
where Re z > 0 is supposed. Its other, useful for our purposes, definition can be given in the form of an infinite product [564, 565]:
(B.2)
Γ ( z ) = lim n c n c ! n c z 1 z ( z + 1 ) ( z + 2 ) ( z + n c 1 ) .
The factorial here can be substituted by its asymptotic expression according to Stirling's formula
(B.3)
n c ! ( n c e ) n c 2 π n c , ln ( n c ! ) ( n c + 1 2 ) ln n c n c + ln 2 π ,
that gives the asymptotic expression for the gamma function for large argument (z ≫ 1)
(B.4)
ln Γ ( z ) ( z 1 2 ) ln z z + 1 2 ln ( 2 π ) + 1 12 z .
Finally, let us report several useful partial values of the gamma function:
(B.5)
Γ ( n + 1 ) = n ! ,
(B.6)
Γ ( 1 2 ) = π ,
and mention the functional relation
(B.7)
Γ ( z ) Γ ( z ) = π z sin π z .

B.2 Digamma function and its derivatives

In the theory of fluctuations the fundamental role plays the logarithmic derivative of the Euler gamma function or, the so-called digamma function. Let us report some of their properties (see e.g. [564, 565]).

(p.373) The definition of the digamma function ψ(z) ≡ ψ (0)(z) is

(B.8)
ψ ( z ) d d z ln Γ ( z ) = lim n c { n = 0 n c 1 1 n + z + ln n c } .
It is analytic everywhere except the points zm = 0,−1,−2, ··· where, following the gamma function, it has the simple poles. High order derivatives of the digamma function can be found directly from (B.8). They are expressed by the convergent series
(B.9)
ψ ( N ) ( z ) = ( 1 ) N + 1 N ! n = 0 1 ( n + z ) N + 1 .
and can be related to the Hurwitz zeta functions [564, 565] with integer first argument:
(B.10)
ψ ( N ) ( z ) = d N d z N ψ ( z ) = ( 1 ) N N ! ζ ( N + 1 , z ) .
where
(B.11)
ζ ( q , z ) = n = 0 1 ( n + z ) q
is Hurwitz, or generalized Riemann, zeta function. For the case, important for further discussion z = 1/2 this relation can be simplified and expressed in terms of the Riemann zeta function ζ(z):
(B.12)
ψ ( N ) ( z ) = ( 1 ) N + 1 N ! ( 2 N + 1 1 ) ζ ( z ) .
For example
(B.13)
ψ ( 1 ) ( 1 2 ) = 3 ζ ( 1 2 ) = π 2 2 ,
(B.14)
ψ ( 2 ) ( 1 2 ) = 14 ζ ( 3 ) ,
(B.15)
ζ ( 3 ) = 1.21...

One can introduce the function ψ (−1)(z) generating digamma function and its derivatives, which with the accuracy of the constant coincides with the ln Γ(z):

ψ ( 1 ) ( z ) lim n c { n = 0 n c 1 ln ( n + z ) + ( n c 1 2 + z ) ln ( n c ) n c } = ln Γ ( z ) 2 π .

In conclusion let us present several asymptotic expressions and useful relations. For large argument z ≫ 1,

(B.16)
ψ ( z ) ln z 1 2 z 1 12 z 2 ,
(p.374)
(B.17)
ψ ( 1 ) ( z ) ζ ( 2 , z ) 1 z + 1 2 z 2 + 1 6 z 3 ,
(B.18)
ψ ( 1 2 + z ) ln z + 1 24 z 2 ,
(B.19)
ψ ( 1 ) ( z ) ( z 1 2 ) ln z z + 1 12 z ,
(B.20)
ψ ( z 0 ) = 1 z .

Functional relations

(B.21)
ψ ( 1 + z ) ψ ( z ) = 1 z ,
(B.22)
ψ ( 1 2 + i z ) ψ ( 1 2 i z ) = π i tanh π z ,
(B.23)
ψ ( 1 ) = C Euler = lim n c { n = 1 n c 1 1 n ln ( n c ) } = 0.577216 ,
(B.24)
ψ ( 1 2 ) = ln ( 4 γ E ) = 2 ln 2 C Euler .
We used the definition γ E = e C Euler = 1.78. .