Bohr–Mollerup theorem

The Bohr–Mollerup theorem characterizes the gamma function as the unique positive, log-convex function on the positive reals satisfying Γ(x + 1) = xΓ(x) and Γ(1) = 1.

Log-convex function: A positive function whose logarithm is convex. Log-convexity is the regularity condition that makes the theorem's uniqueness possible.
Convex function: A function whose value on a line segment lies at or below the segment joining its endpoint values. Log-convexity is ordinary convexity applied after taking a logarithm.
Christian Goldbach: An eighteenth-century mathematician known for work on number theory and a famous conjecture about even integers. His correspondence with Daniel Bernoulli posed the problem of interpolating factorial values.
Factorial interpolation: The extension of factorial values from integers to noninteger arguments. The theorem gives a natural uniqueness criterion for this extension.
Weierstrass definition of the gamma function: An infinite-product representation defining the reciprocal gamma function from its zeros and normalization. It constructs the gamma function directly, rather than characterizing it through log-convexity.
Gamma function: A function extending the factorial to positive real and complex arguments, with Γ(n) = (n − 1)! for positive integers n. The theorem identifies this function as the unique one satisfying its hypotheses.
Positive real number: A real number greater than zero. The theorem's domain excludes zero and negative arguments.
Daniel Bernoulli: An eighteenth-century mathematician and physicist from the Bernoulli family. He addressed Goldbach's question about extending factorials beyond integer arguments.
Stirling's approximation: An asymptotic formula estimating factorials and gamma-function values for large arguments. Log-convexity and recurrence help derive bounds related to this approximation.
Hölder's theorem for the gamma function: A uniqueness result characterizing the gamma function through its recurrence and convexity properties. It is a closely related formulation whose convexity assumptions can be stated differently.

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