Log Gamma Function

The log gamma function is the analytic branch of the logarithm of the gamma function on the complex plane cut along the nonpositive real axis, chosen to be real for positive real numbers. It is denoted here and implemented in the Wolfram Language as LogGamma[z]. Its values are plotted above.

The principal natural logarithm of , plotted above, has additional branch cuts wherever crosses the negative real axis. The two functions have identical real parts, but their imaginary parts can differ by integer multiples of . They are branches of the same analytic multivalued function. The log gamma branch allows concise formulation of many identities involving the Riemann zeta function .

The log gamma function can be defined as

(1)

for off the nonpositive real axis, with the principal branch of each logarithm on the right (Boros and Moll 2004, p. 204). Another sum is given by

(2)

for (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. 2007, p. 181).

Another integral is given by

(8)

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

An infinite series representation in terms of the upper incomplete gamma function is

(9)

where and is the upper incomplete gamma function on its principal branch (Reynolds and Stauffer 2021). This follows by differentiating their Hurwitz zeta function representation with respect to its first argument at zero.