Difference between revisions of "Relationship between Hurwitz zeta and gamma function"
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<strong>[[Relationship between Hurwitz zeta and gamma function|Theorem]]:</strong> The following formula holds: | <strong>[[Relationship between Hurwitz zeta and gamma function|Theorem]]:</strong> The following formula holds: | ||
| − | $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} | + | $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ |
where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Hurwitz zeta]] function. | where $\Gamma$ denotes the [[gamma function]] and $\zeta$ denotes the [[Hurwitz zeta]] function. | ||
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Revision as of 23:42, 23 May 2016
Theorem: The following formula holds: $$\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: █
