Integral Form of Polygamma Function/Corollary
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Corollary to Integral Form of Polygamma Function
Let $z$ be a complex number with a positive real part.
Then:
- $\ds \map {\psi_n} z= -\int_0^1 \frac {u^{z - 1} \paren {\ln u}^n } {1 - u} \rd u$
where $\map {\psi_n} z$ denotes the $n$th polygamma function.
Proof
\(\ds \map {\psi_n} z\) | \(=\) | \(\ds \paren {-1}^{n + 1} \int_0^\infty \frac {t^n e^{-z t} } {1 - e^{-t} } \rd t\) | Integral Form of Polygamma Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \int_0^\infty \frac {t^n e^{-\paren {z - 1} t} e^{-t} } {1 - e^{-t} } \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \int_1^0 \frac {u^{z - 1} \paren {-\ln u}^n } {1 - u} \paren {-\rd u}\) | substituting $e^{-t} \to u$, $t \to -\map \ln u$ and $-e^{-t}\rd t \to \rd u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^{n + 1} \int_1^0 \paren {-1}^{n + 1} \frac {u^{z - 1} \paren {\ln u}^n} {1 - u} \rd u\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int_0^1 \frac {u^{z - 1} \paren {\ln u}^n } {1 - u} \rd u\) | Reversal of Limits of Definite Integral |
$\blacksquare$