Difference between revisions of "Polygamma"
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| − | [[Integral representation of polygamma]]<br /> | + | [[Integral representation of polygamma for Re(z) greater than 0]]<br /> |
[[Integral representation of polygamma 2]]<br /> | [[Integral representation of polygamma 2]]<br /> | ||
[[Polygamma recurrence relation]]<br /> | [[Polygamma recurrence relation]]<br /> | ||
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[[Polygamma multiplication formula]]<br /> | [[Polygamma multiplication formula]]<br /> | ||
[[Polygamma series representation]]<br /> | [[Polygamma series representation]]<br /> | ||
| + | [[Value of polygamma at 1]]<br /> | ||
| + | [[Value of polygamma at positive integer]]<br /> | ||
| + | [[Value of polygamma at 1/2]]<br /> | ||
| + | [[Value of derivative of trigamma at positive integer plus 1/2]]<br /> | ||
[[Relation between polygamma and Hurwitz zeta]]<br /> | [[Relation between polygamma and Hurwitz zeta]]<br /> | ||
Revision as of 21:09, 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 for Re(z) greater than 0
Integral representation of polygamma 2
Polygamma recurrence relation
Polygamma reflection formula
Polygamma multiplication formula
Polygamma series representation
Value of polygamma at 1
Value of polygamma at positive integer
Value of polygamma at 1/2
Value of derivative of trigamma at positive integer plus 1/2
Relation between polygamma and Hurwitz zeta
See Also
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 6.4.1
