Gauss's Integral Form of Digamma Function

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Theorem

Let $z$ be a complex number with a positive real part.


Then:

$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t$

where $\psi$ is the digamma function.


Proof

From Extension of Harmonic Number to Non-Integer Argument, we have:

$\map H x = \gamma + \dfrac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }$

which is equivalent to:

$\ds \map \psi z = -\gamma + H_{z - 1}$


We aim to demonstrate:

$\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t = -\gamma + H_{z - 1}$


We have:

\(\ds \map \psi z\) \(=\) \(\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t\)
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {e^{-z t} } {1 - e^{-t} } \rd t\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {e^{-t} } {1 - e^{-t} } \rd t + \int_0^\infty \frac {e^{-t} } {1 - e^{-t} } \rd t - \int_0^\infty \frac {e^{-z t} } {1 - e^{-t} } \rd t\) adding $0$
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {e^{-t} } {1 - e^{-t} } \rd t + \int_0^\infty \frac {e^{-t} - e^{-t z } } {1 - e^{-t} } \rd t\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {e^{-t} } {1 - e^{-t} } \times \frac {e^t} {e^t} \rd t + \int_0^\infty \frac {e^{-t} - e^{-t z } } {1 - e^{-t} } \rd t\) multiplying by $1$
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac 1 {e^t - 1 } \rd t + \int_0^\infty \frac {e^{-t} - e^{-t z } } {1 - e^{-t} } \rd t\)
\(\ds \) \(=\) \(\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {e^t - 1 } } \rd t + \int_0^\infty \frac {e^{-t} - e^{-t z } } {1 - e^{-t} } \rd t\) Linear Combination of Integrals
\(\ds \) \(=\) \(\ds -\gamma + \int_0^\infty \paren {\frac {1 - e^{-t\paren {z - 1} } } {1 - e^{-t} } } e^{-t} \rd t\) Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x


Let:

\(\ds u\) \(=\) \(\ds e^{-t}\) Integration by Substitution
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d t}\) \(=\) \(\ds -e^{-t}\) Derivative of Exponential Function: Corollary 1


We also have:

\(\ds \lim_{t \mathop \to 0} e^{-t}\) \(=\) \(\ds 1\) Exponential of Zero
\(\ds \lim_{t \mathop \to \infty} e^{-t}\) \(=\) \(\ds 0\) Complex Exponential Tends to Zero


Then:

\(\ds \map \psi z\) \(=\) \(\ds -\gamma + \int_{\to 1}^{\to 0} \paren {\dfrac {1 - u^{z - 1} } {1 - u} } \paren {-\rd u}\)
\(\ds \) \(=\) \(\ds -\gamma + \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u^{z - 1} } {1 - u} } \rd u\) reversing limits of integration
\(\ds \) \(=\) \(\ds -\gamma + \int_{\to 0}^{\to 1} \sum_{k \mathop = 0}^{z - 2} u^k\) Sum of Geometric Sequence
\(\ds \) \(=\) \(\ds -\gamma + \sum_{k \mathop = 0}^{z - 2} \int_{\to 0}^{\to 1} u^k\) Tonelli's Theorem
\(\ds \) \(=\) \(\ds -\gamma + \sum_{k \mathop = 0}^{z - 2} \intlimits {\dfrac {u^{k + 1} } {k + 1} } 0 1\) Integral of Power
\(\ds \) \(=\) \(\ds -\gamma + \sum_{k \mathop = 0}^{z - 2} \frac 1 {k + 1}\)
\(\ds \) \(=\) \(\ds -\gamma + \sum_{k \mathop = 1}^{z - 1} \frac 1 k\) reindexing the sum
\(\ds \) \(=\) \(\ds -\gamma + H_{z - 1}\) Definition of Harmonic Numbers

$\blacksquare$


Source of Name

This entry was named for Carl Friedrich Gauss.


Also see


Sources

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