Difference between revisions of "Polygamma"
From specialfunctionswiki
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File:Complexpolygamma,k=3plot.png|Domain coloring of $\psi^{(3)}(z)$. | File:Complexpolygamma,k=3plot.png|Domain coloring of $\psi^{(3)}(z)$. | ||
File:Complexpolygamma,k=4plot.png|Domain coloring of $\psi^{(4)}(z)$. | File:Complexpolygamma,k=4plot.png|Domain coloring of $\psi^{(4)}(z)$. | ||
| + | File:Complexpolygamma,k=5plot.png|Domain coloring of $\psi^{(5)}(z)$. | ||
</gallery> | </gallery> | ||
</div> | </div> | ||
Revision as of 19:14, 3 June 2016
The polygamma function of order $m$, $\psi^{(m)}(z)$, is defined by the formula $$\psi^{(m)}(z) = \dfrac{d^m}{dz^m} \log \Gamma(z),$$ where $\log$ denotes the logarithm and $\log \Gamma$ 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 $\psi^{(5)}(z)$.
