On , the Gamma function is the unique positive function satisfying , , and the condition that is a convex function. This last condition concerns the natural logarithm of , rather than itself, and removes the interpolation freedom left by the Gamma function recurrence.[ref]
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The Bohr–Mollerup theorem is a result in mathematical analysis that characterizes the gamma function among other functions. Specifically, it provides a characterization of the gamma function using properties of a specific class of functions. The theorem states that if a function \( f : (0, \infty) \to \mathbb{R} \) satisfies the following conditions: 1. \( f(x) \) is continuous on \( (0, \infty) \).