Polygamma Function

Polygamma Function

In mathematics, the polygamma function of order m is a meromorphic function on and defined as the (m+1)-th derivative of the logarithm of the gamma function:

Thus

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorph on . At all the nonnegative integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function.


The logarithm of the gamma function and the first few polygamma functions in the complex plane
\ln\Gamma(z) \psi^{(0)}(z) \psi^{(1)}(z)
\psi^{(2)}(z) \psi^{(3)}(z) \psi^{(4)}(z)

Read more about Polygamma Function:  Integral Representation, Recurrence Relation, Reflection Relation, Multiplication Theorem, Series Representation, Taylor Series, Asymptotic Expansion

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