Difference between revisions of "Relationship between Hurwitz zeta and gamma function"

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==Theorem==
<strong>[[Relationship between Hurwitz zeta and gamma function|Theorem]]:</strong> The following formula holds:
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The following formula holds:
 
$$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$
 
$$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$
 
where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Hurwitz zeta]] function.  
 
where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Hurwitz zeta]] function.  
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<strong>Proof:</strong> █
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==Proof==
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==References==
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[[Category:Theorem]]
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[[Category:Unproven]]

Latest revision as of 07:13, 16 June 2016

Theorem

The following formula holds: $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ where $\Gamma$ denotes the gamma function and $\zeta$ denotes the Hurwitz zeta function.

Proof

References