Dirichlet's Integral Form of Digamma Function
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Theorem
Let $z$ be a complex number with a positive real part.
Then:
- $\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {t \paren {1 + t}^z } } \rd t$
where $\psi$ is the digamma function.
Proof
We have:
| \(\ds \map \psi z\) | \(=\) | \(\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t\) | Gauss's Integral Form of Digamma Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {\paren {e^{-t} }^z } {1 - e^{-t} } \rd t\) | Linear Combination of Integrals |
Let:
| \(\ds \frac 1 {\paren {1 + x} }\) | \(=\) | \(\ds e^{-t}\) | Integration by Substitution | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {1 + x}\) | \(=\) | \(\ds e^t\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \ln {1 + x}\) | \(=\) | \(\ds t\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 {\paren {1 + x} } \rd x\) | \(=\) | \(\ds \d t\) |
We also have:
| \(\ds \lim_{x \mathop \to 0} \map \ln {1 + x}\) | \(=\) | \(\ds 0\) | Natural Logarithm of 1 is 0 | |||||||||||
| \(\ds \lim_{x \mathop \to \infty} \map \ln {1 + x}\) | \(=\) | \(\ds \infty\) | Logarithm Tends to Infinity |
Then:
| \(\ds \map \psi z\) | \(=\) | \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {\paren {e^{-t} }^z} {1 - e^{-t} } \rd t\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {\paren {\frac 1 {\paren {1 + x} } }^z} {1 - \frac 1 {\paren {1 + x} } } \paren {\frac 1 {\paren {1 + x} } \rd x}\) | substituting in second integral | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac {\paren {\frac 1 {\paren {1 + x} } }^z} {\paren {1 + x} - 1} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac 1 {x \paren {1 + x}^z} \rd x\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \frac {e^{-t} } t \rd t - \int_0^\infty \frac 1 {t \paren {1 + t}^z} \rd t\) | $x \to t$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac 1 {t \paren {1 + t}^z} } \rd t\) | Linear Combination of Integrals |
$\blacksquare$
Source of Name
This entry was named for Johann Peter Gustav Lejeune Dirichlet.
Also see
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