Wielandt theorem

{{Short description|Characterization of the gamma function}}

In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers $z$ for which $\backslash mathrm\{Re\}\backslash ,z\; >\; 0$ by

:$\backslash Gamma(z)=\backslash int\_0^\{+\backslash infty\}\; t^\{z-1\}\; \backslash mathrm\; e^\{-t\}\backslash ,\backslash mathrm\; dt,$

as the only function $f$ defined on the half-plane $H\; :=\; \backslash \{\; z\; \backslash in\; \backslash Complex\; :\; \backslash operatorname\{Re\}\backslash ,z\; >\; 0\backslash \}$ such that:

• $f$ is holomorphic on $H$;
• $f(1)=1$;
• $f(z+1)=z\backslash ,f(z)$ for all $z\; \backslash in\; H$ and
• $f$ is bounded on the strip $\backslash \{\; z\; \backslash in\; \backslash Complex\; :\; 1\; \backslash leq\; \backslash operatorname\{Re\}\backslash ,z\; \backslash leq\; 2\backslash \}$.

This theorem is named after the mathematician Helmut Wielandt.