Bohr-Mollerup Theorem

If a function phi:(0,infty)->(0,infty) satisfies

1. ln[phi(x)] is convex,

2. phi(x+1)=xphi(x) for all x>0, and

3. phi(1)=1,

then phi(x) is the gamma function Gamma(x). Therefore, by analytic continuation, Gamma(z) is the only meromorphic function on C satisfying the functional equation


with Gamma(1)=1 and which is logarithmically convex on the positive real axis.

See also

Gamma Function

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Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 56-57, 2003.Krantz, S. G. "The Bohr-Mollerup Theorem." §13.1.10 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 157, 1999.

Referenced on Wolfram|Alpha

Bohr-Mollerup Theorem

Cite this as:

Weisstein, Eric W. "Bohr-Mollerup Theorem." From MathWorld--A Wolfram Web Resource.

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