Wielandt theorem

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

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

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

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

This theorem is named after the mathematician Helmut Wielandt.

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