# Integral Representation of Dirichlet Eta Function in terms of Gamma Function

## 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

$\blacksquare$