Integral representations of \(\zeta(m)\) (Q6569635)
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scientific article; zbMATH DE number 7878677
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Integral representations of \(\zeta(m)\) |
scientific article; zbMATH DE number 7878677 |
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Integral representations of \(\zeta(m)\) (English)
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9 July 2024
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The main purpose of this short note is to prove the following elegant formula for any positive integer \(n\) \N\[\N\int_{-\infty}^{+\infty} x^{2n}\frac{\ln (1+e^x)}{1+e^{x}} dx=\frac{(2n+2)!}{2n+1}\left(1-\frac{1}{2^{2n+1}}\right)\zeta(2n+2),\N\]\Nwhere \(\zeta(s)\) is the Riemann zeta function.
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Riemann zeta function
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generating function
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beta integral
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digamma function
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