Integral Representation of Dirichlet Eta Function in terms of Gamma Function
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Theorem
- $\ds \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$
where:
- $s$ is a complex number with $\map \Re s > 0$
- $\eta$ denotes the Dirichlet eta function
- $\Gamma$ denotes the Gamma function.
Proof
\(\ds \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x\) | \(=\) | \(\ds \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {-e^{-x} } } \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty x^{s - 1} e^{-x} \paren {\sum_{n \mathop = 0}^\infty \paren {-e^{-x} }^n} \rd x\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \int_0^\infty x^{s - 1} e^{-\paren {n + 1} x} \rd x}\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Gamma s \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {n + 1}^s}\) | Laplace Transform of Complex Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Gamma s \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^s}\) | shifting the index | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Gamma s \map \eta s\) | Definition of Dirichlet Eta Function |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Exponential Functions: $15.82$