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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Wielandt theorem characterizes the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, defined for all complex numbers z for which \mathrm\,z > 0 by :\Gamma(z)=\int_0^ t^ \mathrm e^\,\mathrm dt, as the only function f defined on the half-plane H := \ such that: * f is
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
on H; * f(1)=1; * f(z+1)=z\,f(z) for all z \in H and * f is bounded on the strip \. This theorem named after the mathematician
Helmut Wielandt __NOTOC__ Helmut Wielandt (19 December 1910 – 14 February 2001) was a German mathematician who worked on permutation groups. He was born in Niedereggenen, Lörrach, Germany. He gave a plenary lecture ''Entwicklungslinien in der Strukturtheorie d ...
.


See also

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Bohr–Mollerup theorem In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for by :\Gamma(x)=\int_0^\infty t^ e^\,dt as the '' ...
*
Hadamard's gamma function In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function. This function, with its argument shifted down by 1, interpolates the factorial and e ...


References

* {{cite journal, author=
Reinhold Remmert Reinhold Remmert (22 June 1930 – 9 March 2016) was a German mathematician. Born in Osnabrück, Lower Saxony, he studied mathematics, mathematical logic and physics in Münster. He established and developed the theory of complex-analytic spac ...
, title=Wielandt's theorem about the {{math, Γ-function, journal=
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...
, volume=103, year=1996, pages=214–220, jstor=2975370. Gamma and related functions Theorems in complex analysis