Integral Representation of Dirichlet Eta Function in terms of Gamma Function

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Theorem

$\displaystyle \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$

where:

$s$ is a complex number with $\map \Re s > 0$
$\eta$ denotes the Dirichlet eta function
$\Gamma$ denotes the Gamma function.


Proof

\(\displaystyle \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x\) \(=\) \(\displaystyle \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {-e^{-x} } } \rd x\)
\(\displaystyle \) \(=\) \(\displaystyle \int_0^\infty x^{s - 1} e^{-x} \paren {\sum_{n \mathop = 0}^\infty \paren {-e^{-x} }^n} \rd x\) Sum of Geometric Sequence
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \paren {\paren {-1}^n \int_0^\infty x^{s - 1} e^{-\paren {n + 1} x} \rd x}\) Fubini's Theorem
\(\displaystyle \) \(=\) \(\displaystyle \map \Gamma s \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {n + 1}^s}\) Laplace Transform of Complex Power
\(\displaystyle \) \(=\) \(\displaystyle \map \Gamma s \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^s}\) shifting the index
\(\displaystyle \) \(=\) \(\displaystyle \map \Gamma s \map \eta s\) Definition of Dirichlet Eta Function

$\blacksquare$


Sources