Binet's Log Gamma Formulas

Binet's first formula for , where  is a gamma function, is given by

for  (Erdélyi et al. 1981, p. 21; Whittaker and Watson 1990, p. 249).

Binet's second formula is

for  (Erdélyi et al. 1981, p. 22; Whittaker and Watson 1990, pp. 250-251).

REFERENCES:

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, 1981.

Whittaker, E. T. and Watson, G. N. "Binet's First Expansion for  in Terms of an Infinite Integral" and "Binet's Second Expression for  in Terms of an Infinite Integral." §12.31 and 12.32 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 248-251, 1990.