Digamma function: Difference between revisions

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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}zdz}\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>