Wielandt theorem

In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers [math]\displaystyle{ z }[/math] for which [math]\displaystyle{ \mathrm{Re}\,z \gt 0 }[/math] by

[math]\displaystyle{ \Gamma(z)=\int_0^{+\infty} t^{z-1} \mathrm e^{-t}\,\mathrm dt, }[/math]

as the only function [math]\displaystyle{ f }[/math] defined on the half-plane [math]\displaystyle{ H := \{ z \in \Complex : \operatorname{Re}\,z \gt 0\} }[/math] such that:

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

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. .