Editing Polygamma function (section)

{{Short description|Meromorphic function}}
{{For|Barnes's gamma  function|multiple gamma function}}
{{one source|date=August 2021}}{{Use American English|date = March 2019}}
[[File:Mplwp polygamma03.svg|thumb|300px|Graphs of the polygamma functions {{math|''ψ''}}, {{math|''ψ''<sup>(1)</sup>}}, {{math|''ψ''<sup>(2)</sup>}} and {{math|''ψ''<sup>(3)</sup>}} of real arguments]]
[[File:Plot of polygamma function in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1.svg| thumb | alt=Plot of the [[digamma function]], the first polygamma function, in the complex plane, with colors showing one cycle of phase shift around each pole and zero | Plot of the [[digamma function]], the first polygamma function, in the complex plane from −2−2i to 2+2i with colors created by Mathematica's function ComplexPlot3D showing one cycle of phase shift around each pole and the zero]]

In [[mathematics]], the '''polygamma function of order {{mvar|m}}''' is a [[meromorphic function]] on the [[complex numbers]] <math>\mathbb{C}</math> defined as the {{math|(''m'' + 1)}}th [[derivative of the logarithm]] of the [[gamma function]]:

:<math>\psi^{(m)}(z) := \frac{\mathrm{d}^m}{\mathrm{d}z^m} \psi(z) = \frac{\mathrm{d}^{m+1}}{\mathrm{d}z^{m+1}} \ln\Gamma(z).</math>

Thus

:<math>\psi^{(0)}(z) = \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)}</math>

holds where {{math|''ψ''(''z'')}} is the [[digamma function]] and {{math|Γ(''z'')}} is the [[gamma function]]. They are [[Holomorphic function|holomorphic]] on <math>\mathbb{C} \backslash\mathbb{Z}_{\le0}</math>. At all the nonpositive integers these polygamma functions have a [[isolated singularity|pole]] of order {{math|''m'' + 1}}. The function {{math|''ψ''<sup>(1)</sup>(''z'')}} is sometimes called the [[trigamma function]].

{|  style="text-align:center"
|+ '''The logarithm of the gamma function and the first few polygamma functions in the complex plane'''
|[[Image:Complex LogGamma.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 0.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 1.jpg|1000x140px|none]]
|-
|{{math|ln Γ(''z'')}}
|{{math|''ψ''<sup>(0)</sup>(''z'')}}
|{{math|''ψ''<sup>(1)</sup>(''z'')}}
|-
|[[Image:Complex Polygamma 2.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 3.jpg|1000x140px|none]]
|[[Image:Complex Polygamma 4.jpg|1000x140px|none]]
|-
|{{math|''ψ''<sup>(2)</sup>(''z'')}}
|{{math|''ψ''<sup>(3)</sup>(''z'')}}
|{{math|''ψ''<sup>(4)</sup>(''z'')}}
|}