Polygamma Function

A Polygamma Function is a gamma function that ...

• Example(s):
  • Trigamma Function.
  • …
• Counter-Example(s):
  • Digamma Function.
• See: Mixture Probability Function, Meromorphic, Logarithm Derivative, Holomorphic Function, Isolated Singularity.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/polygamma_function Retrieved:2015-2-14.
  • In mathematics, the 'polygamma function of order m is a meromorphic function on [math]\displaystyle{ \C }[/math] and defined as the (m+1)-th

     derivative of the logarithm of the gamma function: :[math]\displaystyle{ \psi^{(m)}(z) := \frac{d^m}{dz^m} \psi(z) = \frac{d^{m+1}}{dz^{m+1}} \ln\Gamma(z). }[/math]

    Thus :[math]\displaystyle{ \psi^{(0)}(z) = \psi(z) = \frac{\Gamma'(z)}{\Gamma(z)} }[/math]

    holds where ψ(z) is the digamma function and Γ(z) is the gamma function.

    They are holomorphic on [math]\displaystyle{ \C \setminus -\N_0 }[/math]. At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ⁽¹⁾(z) is sometimes called the trigamma function.

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