## 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

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)
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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)
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The factorial here can be substituted by its asymptotic expression according to Stirling's formula
(B.3)
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that gives the asymptotic expression for the gamma function for large argument (z ≫ 1)
(B.4)
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Finally, let us report several useful partial values of the gamma function:
(B.5)
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(B.6)
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and mention the functional relation
(B.7)
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# 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)
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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)
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and can be related to the Hurwitz zeta functions [564, 565] with integer first argument:
(B.10)
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where
(B.11)
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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)
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For example
(B.13)
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(B.14)
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(B.15)
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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):

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

(B.16)
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(p.374)
(B.17)
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(B.18)
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(B.19)
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(B.20)
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Functional relations

(B.21)
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(B.22)
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(B.23)
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(B.24)
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We used the definition $γ E = e C Euler = 1.78.$.