Wielandt theorem

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

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

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

• $f$ 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 $\{z\in \mathbb {C} :1\leq \operatorname {Re} \,z\leq 2\}$.

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. JSTOR 2975370..