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Log Gamma Function

The plots above show the values of the function obtained by taking thenatural logarithm of thegamma function, lnGamma(z) . Note that this introduces complicatedbranch cut structure inherited from the logarithm function.

For this reason, the logarithm of the gamma function is sometimes treated as a special function in its own right, and defined differently from lnGamma(z) . This special "log gamma" function is implemented in theWolfram Language asLogGamma[z], plotted above. As can be seen, the two definitions have identical real parts, but differ markedly in their imaginary components. Most importantly, although the log gamma function and are equivalent as analyticmultivalued functions, they have different branch cut structures and a different principal branch, and the log gamma function is analytic throughout the complex-plane except for a singlebranch cut discontinuity along the negativereal axis. In particular, the log gamma function allows concise formulation of many identities related to theRiemann zeta function zeta(z) .

The log gamma function can be defined as

lnGamma(z)=-gammaz-lnz+sum_(k=1)^infty[z/k-ln(1+z/k)].

(1)

(Boros and Moll 2004, p. 204). Another sum is given by

lnGamma(z)=(z-1/2)lnz-z+1/2ln(2pi)+1/2sum_(n=2)^infty((n-1))/(n(n+1))zeta(n,z+1)

(2)

(Whittaker and Watson 1990, p. 261), where zeta(s,a) is aHurwitz zeta function.

The second ofBinet's log gamma formulasis

lnGamma(a)=(a-1/2)lna-a+1/2ln(2pi)+2int_0^infty(tan^(-1)(z/a))/(e^(2piz)-1)dz

(3)

for R[a]>0 (Whittaker and Watson 1990, p. 251). Another formula for lnGamma(z) is given byMalmstén's formula.

Integrals of lnGamma(x) include

			(4)
			(5)
			(6)

(OEISA075700; Baileyet al.2007, p. 179),which was known to Euler, and

int_0^1[lnGamma(x)]^2dx=1/(12)gamma^2+1/(48)pi^2+1/6gammaln(2pi)+1/3[ln(2pi)]^2 -[gamma+ln(2pi)](zeta^'(2))/(pi^2)+(zeta^('')(2))/(2pi^2),

(7)

(OEISA102887; Espinosa and Moll 2002, 2004; Boros and Moll 2004, p. 203; Baileyet al.2007, p. 179), where gamma is theEuler-Mascheroni constant and zeta^'(z) is the derivative of theRiemann zeta function.

int_0^1[lnGamma(x)]^3dx is considered by Espinosa and Moll (2006) who, however, were not able to establish a closed form (Baileyet al.2006, p. 181).

Another integral is given by

int_0^(1/2)ln[Gamma(x+1)]dx=-1/2-7/(24)ln2+1/4lnpi+3/2lnA,

(8)

where is theGlaisher-Kinkelin constant (Glaisher 1878).

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http://functions.wolfram.com/GammaBetaErf/LogGamma/

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References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H.Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Boros, G. and Moll, V. "The Expansion of the Loggamma Function." §10.6 inIrresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, pp. 201-203, 2004.Espinosa, O. and Moll, V. "On Some Definite Integrals Involving the Hurwitz Zeta Function. Part I."Ramanujan J.6, 159-188, 2002.Espinosa, O. and Moll, V. "A Generalized Polygamma Function."Integral Transforms Spec. Funct.15, 101-115, 2004.Espinosa, O. and Moll, V. "The Evaluation of Tornheim Double Sums. I."J. Number Th.116, 200-229, 2006.Glaisher, J. W. L. "On the Product 1^1.2^2.3^3...n^n

."Messenger Math.7, 43-47, 1878.Sloane, N. J. A. SequencesA075700 andA102887 in "The On-Line Encyclopedia of Integer Sequences."Whittaker, E. T. and Watson, G. N.A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Log Gamma Function

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Weisstein, Eric W. "Log Gamma Function."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/LogGammaFunction.html

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Log Gamma Function

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