Polygamma function

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In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers ${\displaystyle \mathbb {C} }$ defined as the (m + 1)th derivative of the logarithm of the gamma function:

${\displaystyle \psi ^{(m)}(z):={\frac {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\psi (z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z).}$

Thus

${\displaystyle \psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}}$

holds where ψ(z) is the digamma function and Γ(z) is the gamma function. They are holomorphic on ${\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}}$. 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.

The logarithm of the gamma function and the first few polygamma functions in the complex plane

ln Γ(z)	ψ(0)(z)	ψ(1)(z)
ψ(2)(z)	ψ(3)(z)	ψ(4)(z)