Wielandt theorem

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

$\mathrm{\Gamma }(z)=\underset{0}{\overset{+\mathrm{\infty }}{\int }}{t}^{z-1}{\mathrm{e}}^{-t}\mathrm{d}t,$

as the only function $f$ defined on the half-plane $H:=\{z\in \mathbb{ℂ}:\mathrm{Re}z>0\}$ such that:

• $f$ is holomorphic on $H$;
• $f(1)=1$;
• $f(z+1)=zf(z)$ for all $z\in H$ and
• $f$ is bounded on the strip $\{z\in \mathbb{ℂ}:1\le \mathrm{Re}z\le 2\}$.

This theorem is named after the mathematician Helmut Wielandt.

See also

• Bohr–Mollerup theorem
• Hadamard's gamma function

References

• Reinhold Remmert (1996). "Wielandt's theorem about the Γ-function". American Mathematical Monthly 103: 214–220. .