WebThe theorem characterizes the gamma function, defined for x > 0 by as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties: * f (1) = 1, and * f (x + 1) = x f (x) for x > 0 and * f is logarithmically convex. WebA Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: A Tutorial

A simple proof of Stirling’s formula for the gamma function

WebSep 30, 2024 · This volume develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger … Webthen clearly so is the function f =f (1) , which must be the gamma function by Bohr-Mollerup's Theorem 1.1. The following theorem provides a reformulation of the latter … top rated wireless security cameras

Bohr–Mollerup theorem for $x\gt 1$ - Mathematics Stack Exchange

WebOnce we know this, Theorem 1 follows in a simple and elegant way, as we now show: ... mally, this means curving upwards). So logΓ(x) is convex. The celebrated Bohr-Mollerup theorem states that the gamma function is the unique function f(x) with the property that logf(x) is convex, together with f(x+1) = xf(x) and f(1) = 1. For a proof, see [Jam1]. WebJul 27, 2024 · The Bohr–Mollerup theorem states that $f (x)=\int_ {0}^\infty t^ {x-1}e^ {-t}\, dt$ is the only function on $x\gt 0$ such that the following conditions are satisfied simultaneously: $f (1)=1,$ $\forall x\gt 0:\, f (x+1)=xf (x),$ $\forall x\gt 0: f (x)$ is logarithmically convex. WebAccording to Wikipedia, the Bohr-Mollerup Theorem (discussed previously on MO here) was first published in a textbook.It says the authors did that instead of writing a paper because they didn't think the theorem was new. What other examples are there of significant theorems that first saw the light of day in a textbook? top rated wireless routers for business