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mathematical physics > special function > polygamma function of order m

mathematical analysis > functional analysis > special function > polygamma function of order m

mathematical analysis > complex analysis > meromorphic function > polygamma function of order m

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polygamma function of order m

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:

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

Thus

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

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

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Polygamma_function)

fonction polygamma d'ordre m

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http://data.loterre.fr/ark:/67375/PSR-QK86QW05-V

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