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Binet's Log Gamma Formulas

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Binet's first formula for the log gamma function lnGamma(z), where Gamma(z) is a gamma function, is given by

lnGamma(z)=(z-1/2)lnz-z+1/2ln(2pi)+int_0^infty(1/2-1/t+1/(e^t-1))(e^(-tz))/tdt

for R[z]>0 (Erdélyi et al. 1981, p. 21; Whittaker and Watson 1990, p. 249).

Binet's second formula is

lnGamma(z)=(z-1/2)lnz-z+1/2ln(2pi)+2int_0^infty(tan^(-1)(t/z))/(e^(2pit)-1)dt

for R[z]>0 (Erdélyi et al. 1981, p. 22; Whittaker and Watson 1990, pp. 250-251).

See also

Gamma Function, Log Gamma Function, Malmstén's Formula, Stirling's Approximation

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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 logGamma(z) in Terms of an Infinite Integral" and "Binet's Second Expression for logGamma(z) 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.

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Binet's Log Gamma Formulas

Cite this as:

Weisstein, Eric W. "Binet's Log Gamma Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinetsLogGammaFormulas.html

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