Editing Digamma function (section)

==Integral representations==
If the real part of {{mvar|z}} is positive then the digamma function has the following [[integral]] representation due to Gauss:<ref name="Whittaker and Watson, 12.3">Whittaker and Watson, 12.3.</ref>
:<math>\psi(z) = \int_0^\infty \left(\frac{e^{-t}}{t} - \frac{e^{-zt}}{1-e^{-t}}\right)\,dt.</math>

Combining this expression with an integral identity for the [[Euler–Mascheroni constant]] <math>\gamma</math> gives:
:<math>\psi(z + 1) = -\gamma + \int_0^1 \left(\frac{1-t^z}{1-t}\right)\,dt.</math>
The integral is Euler's [[harmonic number]] <math>H_z</math>, so the previous formula may also be written
:<math>\psi(z + 1) = \psi(1) + H_z.</math>
A consequence is the following generalization of the recurrence relation:
:<math>\psi(w + 1) - \psi(z + 1) = H_w - H_z.</math>

An integral representation due to Dirichlet is:<ref name="Whittaker and Watson, 12.3"/>
:<math>\psi(z) = \int_0^\infty \left(e^{-t} - \frac{1}{(1 + t)^z}\right)\,\frac{dt}{t}.</math>

Gauss's integral representation can be manipulated to give the start of the asymptotic expansion of <math>\psi</math>.<ref>Whittaker and Watson, 12.31.</ref>
:<math>\psi(z) = \log z - \frac{1}{2z} - \int_0^\infty \left(\frac{1}{2} - \frac{1}{t} + \frac{1}{e^t - 1}\right)e^{-tz}\,dt.</math>
This formula is also a consequence of Binet's first integral for the gamma function.  The integral may be recognized as a [[Laplace transform]].

Binet's second integral for the gamma function gives a different formula for <math>\psi</math> which also gives the first few terms of the asymptotic expansion:<ref>Whittaker and Watson, 12.32, example.</ref>
:<math>\psi(z) = \log z - \frac{1}{2z} - 2\int_0^\infty \frac{t\,dt}{(t^2 + z^2)(e^{2\pi t} - 1)}.</math>

From the definition of <math>\psi</math> and the integral representation of the gamma function, one obtains
:<math>\psi(z) = \frac{1}{\Gamma(z)} \int_0^\infty  t^{z-1} \ln (t) e^{-t}\,dt,</math>
with <math>\Re z > 0</math>.<ref>{{cite web | url=https://dlmf.nist.gov/5.9 | title=NIST. Digital Library of Mathematical Functions (DLMF), 5.9.}}</ref>