Difference between revisions of "Relationship between Hurwitz zeta and gamma function"
From specialfunctionswiki
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ | $$\Gamma(s)\zeta(s,a) = \displaystyle\int_0^{\infty} \dfrac{x^{s-1}e^{-ax}}{1-e^{-x}} \mathrm{d}x,$$ | ||
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. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
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| + | [[Category:Theorem]] | ||
| + | [[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.
