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mathematical physics > special function > polygamma function of order m > trigamma function

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

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

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trigamma function

In mathematics, the trigamma function, denoted $ψ 1 (z)$ or $ψ (1) (z)$ , is the second of the polygamma functions, and is defined by
${\\displaystyle \\psi _{1}(z)={\ rac {d^{2}}{dz^{2}}}\\ln \\Gamma (z)}$ .
It follows from this definition that
${\\displaystyle \\psi _{1}(z)={\ rac {d}{dz}}\\psi (z)}$
where $ψ (z)$ is the digamma function. It may also be defined as the sum of the series
${\\displaystyle \\psi _{1}(z)=\\sum _{n=0}^{\\infty }{\ rac {1}{(z+n)^{2}}},}$
making it a special case of the Hurwitz zeta function
${\\displaystyle \\psi _{1}(z)=\\zeta (2,z).}$
Note that the last two formulas are valid when $1 - z$ is not a natural number.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Trigamma_function)

polygamma function of order m

fonction trigamma

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