Log Gamma Function

Min Max

Min Max Re Im

The plots above show the values of the function obtained by taking the natural logarithm of the gamma function, . Note that this introduces complicated branch cut structure inherited from the logarithm function.

Min Max Re Im

For this reason, the logarithm of the gamma function is sometimes treated as a special function in its own right, and defined differently from . This special "log gamma" function is implemented in the Wolfram Language as LogGamma[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 analytic multivalued 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 single branch cut discontinuity along the negative real axis. In particular, the log gamma function allows concise formulation of many identities related to the Riemann zeta function .

The log gamma function can be defined as

(1)

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

(2)

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

The second of Binet's log gamma formulas is

(3)

for (Whittaker and Watson 1990, p. 251). Another formula for is given by Malmstén's formula.

Integrals of include

			(4)
			(5)
			(6)

(OEIS A075700; Bailey et al. 2007, p. 179), which was known to Euler, and

(7)

(OEIS A102887; Espinosa and Moll 2002, 2004; Boros and Moll 2004, p. 203; Bailey et al. 2007, p. 179), where is the Euler-Mascheroni constant and is the derivative of the Riemann zeta function.

is considered by Espinosa and Moll (2006) who, however, were not able to establish a closed form (Bailey et al. 2006, p. 181).

Another integral is given by

(8)

where is the Glaisher-Kinkelin constant (Glaisher 1878).