Wielandt theorem

In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers $$z$$ for which $$\mathrm{Re}\,z > 0$$ by
 * $$\Gamma(z)=\int_0^{+\infty} t^{z-1} \mathrm e^{-t}\,\mathrm dt,$$

as the only function $$f$$ defined on the half-plane $$H := \{ z \in \Complex : \operatorname{Re}\,z > 0\}$$ such that: This theorem is named after the mathematician Helmut Wielandt.
 * $$f$$ is holomorphic on $$H$$;
 * $$f(1)=1$$;
 * $$f(z+1)=z\,f(z)$$ for all $$z \in H$$ and
 * $$f$$ is bounded on the strip $$\{ z \in \Complex : 1 \leq \operatorname{Re}\,z \leq 2\}$$.