Editing Digamma function (section)

{{Short description|Mathematical function}}
{{For|Barnes' gamma function of two variables |double gamma function}}

[[File:Digamma.png|thumb|300px|The digamma function <math>\psi(z)</math>,<br>visualized using [[domain coloring]]]]
[[File:Mplwp polygamma03.svg|thumb|300px|Plots of the digamma and the next three polygamma functions along the real line (they are real-valued on the real line)]]

In [[mathematics]], the '''digamma function''' is defined as the [[logarithmic derivative]] of the [[gamma function]]:<ref name="AbramowitzStegun"/><ref name="DLMF5"/><ref name="Weissstein"/>

:<math>\psi(z) = \frac{\mathrm{d}}{\mathrm{d}z}\ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}.</math>

It is the first of the [[polygamma function]]s. This function is [[Monotonic function|strictly increasing]] and [[Concave function|strictly concave]] on <math>(0,\infty)</math>,<ref>{{Cite journal |last1=Alzer |first1=Horst |last2=Jameson |first2=Graham |date=2017 |title=A harmonic mean inequality for the digamma function and related results |url=https://core.ac.uk/download/pdf/228202664.pdf |journal=Rendiconti del Seminario Matematico della Università di Padova |volume=137 |pages=203–209|doi=10.4171/RSMUP/137-10 }}</ref> and it [[Asymptotic analysis|asymptotically behaves]] as<ref>{{cite web |url=https://dlmf.nist.gov/5.11 |title=NIST. Digital Library of Mathematical Functions (DLMF), 5.11.}}</ref>

:<math>\psi(z) \sim \ln{z} - \frac{1}{2z},</math>

for complex numbers with large modulus (<math>|z|\rightarrow\infty</math>) in the [[Circular sector|sector]] <math>|\arg z|<\pi-\varepsilon</math> for any <math>\varepsilon > 0</math>.

The digamma function is often denoted as <math>\psi_0(x), \psi^{(0)}(x) </math> or {{math|Ϝ}}<ref>{{cite book |last=Pairman |first=Eleanor |author-link=Eleanor Pairman |date=1919 |title=Tables of the Digamma and Trigamma Functions |url=https://archive.org/details/cu31924001468416/page/n9/mode/2up |publisher=Cambridge University Press |page=5}}</ref> (the uppercase form of the archaic Greek [[consonant]] [[digamma]] meaning [[Gamma|double-gamma]]).
Gamma.