gives the digamma function .
gives the n derivative of the digamma function .
- Mathematical function, suitable for both symbolic and numerical manipulation.
- PolyGamma[z] is the logarithmic derivative of the gamma function, given by .
- PolyGamma[n,z] is given for positive integer by .
- For arbitrary complex n, the polygamma function is defined by fractional calculus analytic continuation.
- PolyGamma[z] and PolyGamma[n,z] are meromorphic functions of z with no branch cut discontinuities.
- For certain special arguments, PolyGamma automatically evaluates to exact values.
- PolyGamma can be evaluated to arbitrary numerical precision.
- PolyGamma automatically threads over lists.
Examplesopen allclose all
Basic Examples (7)
Series expansion at Infinity:
Numerical Evaluation (6)
Specific Values (6)
Plot PolyGamma for half-integer values of parameter :
Function Properties (2)
First derivative of PolyGamma:
Indefinite integral of PolyGamma:
Series Expansions (7)
PolyGamma can be applied to a power series:
Function Identities and Simplifications (5)
Plot of the absolute value of PolyGamma over the complex plane:
The electric field energy of a charge at a fraction of the distance between parallel conducting plates:
Expand near the left wall:
Final speed of a rocket with discrete propulsion events:
Final velocity in the limit of constant continuous propulsion:
Properties & Relations (7)
Use FullSimplify to simplify polygamma functions:
Express rational arguments through elementary functions:
Numerically find a root of a transcendental equation:
Sums and integrals:
Generate PolyGamma from integrals, sums, and limits:
Obtain as special cases of hypergeometric functions:
Possible Issues (3)
The one‐argument form evaluates to the two-argument form:
Large orders can give results too large to be computed explicitly:
Machine‐number inputs can give high‐precision results: