Wielandt theorem

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

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

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

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

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..