multiplication theorem (Q98831)

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theorem

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Language	Label	Description	Also known as
default for all languages	No label defined	–
English	multiplication theorem	theorem

Statements

instance of

theorem

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part of

list of theorems

language of work or name

English

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defining formula

${\displaystyle {\frac {d\xi }{dt}}+\nabla \cdot f(\xi )=0}$

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${\displaystyle \Gamma (z)\;\Gamma \left(z+{\frac {1}{2}}\right)=2^{1-2z}\;{\sqrt {\pi }}\;\Gamma (2z).\,\!}$

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${\displaystyle \Gamma (z)\;\Gamma \left(z+{\frac {1}{k}}\right)\;\Gamma \left(z+{\frac {2}{k}}\right)\cdots \Gamma \left(z+{\frac {k-1}{k}}\right)=(2\pi )^{\frac {k-1}{2}}\;k^{1/2-kz}\;\Gamma (kz)\,\!}$

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${\displaystyle k^{m}\psi ^{(m-1)}(kz)=\sum _{n=0}^{k-1}\psi ^{(m-1)}\left(z+{\frac {n}{k}}\right)}$

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${\displaystyle k\left[\psi (kz)-\log(k)\right]=\sum _{n=0}^{k-1}\psi \left(z+{\frac {n}{k}}\right).}$

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${\displaystyle k^{s}\zeta (s)=\sum _{n=1}^{k}\zeta \left(s,{\frac {n}{k}}\right),}$

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${\displaystyle k^{s}\,\zeta (s,kz)=\sum _{n=0}^{k-1}\zeta \left(s,z+{\frac {n}{k}}\right)}$

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${\displaystyle \zeta (s,kz)=\sum _{n=0}^{\infty }{s+n-1 \choose n}(1-k)^{n}z^{n}\zeta (s+n,z).}$

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${\displaystyle F(s;q)=\sum _{m=1}^{\infty }{\frac {e^{2\pi imq}}{m^{s}}}=\operatorname {Li} _{s}\left(e^{2\pi iq}\right)}$

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${\displaystyle 2^{-s}F(s;q)=F\left(s,{\frac {q}{2}}\right)+F\left(s,{\frac {q+1}{2}}\right).}$

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${\displaystyle k^{-s}F(s;kq)=\sum _{n=0}^{k-1}F\left(s,q+{\frac {n}{k}}\right).}$

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maintained by WikiProject

WikiProject Mathematics

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Identifiers

Freebase ID

/m/027g78j

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Microsoft Academic ID

75090586

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Sitelinks

Wikipedia(5 entries)

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• arwiki مبرهنة الجداء (دالة غاما)
• cawiki Teorema de multiplicació
• enwiki Multiplication theorem
• eswiki Teorema de multiplicación
• jawiki 乗法定理

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