Difference between revisions of "Polygamma"
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=Properties= | =Properties= | ||
| − | + | [[Integral representation of polygamma]]<br /> | |
| − | + | [[Integral representation of polygamma 2]]<br /> | |
| − | + | [[Polygamma recurrence relation]]<br /> | |
| − | + | [[Polygamma reflection relation]]<br /> | |
| − | + | [[Polygamma series representation]]<br /> | |
| − | + | [[Relation between polygamma and Hurwitz zeta]]<br /> | |
=See Also= | =See Also= | ||
Revision as of 06:30, 11 June 2016
The polygamma function of order $m$, $\psi^{(m)}(z)$, is defined by the formula $$\psi^{(m)}(z) = \dfrac{\mathrm{d}^{m+1}}{\mathrm{d}z^{m+1}} \log \Gamma(z),$$ where $\log \Gamma$ denotes the loggamma function. The digamma function $\psi$ is the function $\psi^{(0)}(z)$ and the trigamma function is $\psi^{(1)}(z)$.
Domain coloring of $\psi^{(0)}(z)$.
Domain coloring of $\psi^{(1)}(z)$.
Domain coloring of $\psi^{(2)}(z)$.
Domain coloring of $\psi^{(3)}(z)$.
Domain coloring of $\psi^{(4)}(z)$.
Domain coloring of $\psi^{(5)}(z)$.
Properties
Integral representation of polygamma
Integral representation of polygamma 2
Polygamma recurrence relation
Polygamma reflection relation
Polygamma series representation
Relation between polygamma and Hurwitz zeta
