# Gauss's Integral Form of Digamma Function

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## Contents

## Theorem

Let $z$ be a complex number with a positive real part, then:

- $\displaystyle \psi \left({z}\right) = \int_0^\infty \left({\frac{ e^{-t} } t - \frac {e^{-zt } } {1 - e^{-t} } }\right) \rd t$

where $\psi$ is the digamma function.

## Proof

## Source of Name

This entry was named for Carl Friedrich Gauss.

## Sources

- 1920: E.T. Whittaker and G.N. Watson:
*A Course of Modern Analysis*(3rd ed.): $12.3$: Gauss's expression for the logarithmic derivate of the Gamma function as an infinite integral