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Psi

the Digamma and Polygamma functions

Calling Sequence

 Psi(x) $\mathrm{\Psi }\left(x\right)$ Psi(n,x) $\mathrm{\Psi }\left(n,x\right)$

Parameters

 x - expression n - expression

Description

 • Psi(x) is the digamma function,

$\mathrm{\Psi }\left(x\right)=\frac{ⅆ}{ⅆx}\mathrm{ln}\left(\mathrm{\Gamma }\left(x\right)\right)=\frac{\frac{ⅆ}{ⅆx}\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x\right)}$

 • Psi(n, x) is the nth polygamma function, which is the nth derivative of the digamma function.
 • You can enter the command Psi using either the 1-D or 2-D calling sequence.
 • If n is an integer greater than one, Psi(n) + gamma is a rational number. (gamma is Euler's constant.) For small values of n, Psi(n) computes as a sum of gamma and a rational number. To perform this computation for larger values of n, use expand.

$\mathrm{\Psi }\left(n,x\right)=\frac{{ⅆ}^{n}}{ⅆ{x}^{n}}\mathrm{\Psi }\left(x\right)$

$\mathrm{\Psi }\left(0,x\right)=\mathrm{\Psi }\left(x\right)$

 • Psi(n, x) is extended to complex n, including negative integer indices, by the formula

$\mathrm{\Psi }\left(w,z\right)=\frac{\mathrm{\zeta }\left(1,w+1,z\right)+\left(\mathrm{\gamma }+\mathrm{\Psi }\left(-w\right)\right)\mathrm{\zeta }\left(0,w+1,z\right)}{\mathrm{\Gamma }\left(-w\right)}$

 where $\mathrm{\zeta }$ is the Hurwitz zeta function.

Examples

 > $\mathrm{\Psi }\left(2\right)$
 ${1}{-}{\mathrm{\gamma }}$ (1)
 > $\mathrm{\Psi }\left(1,2\right)$
 ${-}{1}{+}\frac{{{\mathrm{\pi }}}^{{2}}}{{6}}$ (2)
 > $\mathrm{\Psi }\left(3.5+4.7I\right)$
 ${1.717883835}{+}{1.001470255}{}{I}$ (3)
 > $\mathrm{\Psi }\left(7,-2.2+3.3I\right)$
 ${-0.02713341434}{+}{0.003825068416}{}{I}$ (4)
 > $\mathrm{\Psi }\left(-2,1.543\right)$
 ${-0.7957394716}$ (5)
 > $\mathrm{\Psi }\left(1.342+I,3.5233\right)$
 ${-0.6988919005}{-}{0.7978763419}{}{I}$ (6)
 > $\mathrm{\Psi }\left(50\right)$
 $\frac{{13881256687139135026631}}{{3099044504245996706400}}{-}{\mathrm{\gamma }}$ (7)
 > $\mathrm{\Psi }\left(51\right)$
 ${\mathrm{\Psi }}{}\left({51}\right)$ (8)

Evaluating Psi(51) directly is faster than expanding and then evaluating.

 > $\mathrm{expand}\left(\mathrm{\Psi }\left(51\right)\right)$
 ${-}{\mathrm{\gamma }}{+}\frac{{13943237577224054960759}}{{3099044504245996706400}}$ (9)
 > $\mathrm{evalf}\left(\right)$
 ${3.921989673}$ (10)
 > $\mathrm{evalf}\left(\mathrm{\Psi }\left(51\right)\right)$
 ${3.921989673}$ (11)

References

 Espinosa, O., and Moll, V. "A Generalized Polygamma Function." Integral Transforms and Special Functions, (April 2004): 101-115.