Evaluate numerically:

Evaluate for integer arguments of any size:

Evaluate for complex arguments and orders:

Evaluate to any precision:

The precision of the output tracks the precision of the input:

Evaluate PolyGamma efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix PolyGamma function using MatrixFunction:

Some singular points of PolyGamma:

Values at infinity:

Find a zero of :

Use FunctionExpand to expand higher-order polygamma functions:

Special case:

Evaluate at exact arguments:

Plot the digamma function:

Plot the real part of :

Plot the imaginary part of :

Plot PolyGamma for half-integer values of parameter :

Real domain of PolyGamma:

Approximate function ranges of PolyGamma for half-integer parameters:

PolyGamma is not an analytic function:

It has both singularities and discontinuities:

is a meromorphic function:

For fixed non-negative integer , is a meromorphic function of :

It is not meromorphic for other values of :

is neither nondecreasing nor nonincreasing:

is not injective:

is surjective:

is neither non-negative nor non-positive:

is neither convex nor concave:

First derivative of PolyGamma:

Higher derivatives of the digamma function:

Formula for the derivative:

Indefinite integral of PolyGamma:

Indefinite integral involving a power function:

Definite integral :

Taylor expansion for the digamma function around :

Plot the first three approximations for the Euler gamma function around :

General term in the series expansion of the digamma function:

Series expansion at infinities:

Series expansion at poles:

Series expansion at a generic point:

Series expansion near a singularity:

PolyGamma can be applied to a power series:

Use FullSimplify to simplify polygamma functions:

PolyGamma identity :

PolyGamma of a double argument:

Other argument simplifications:

Recurrence relation:

Digamma function definition:

Integral representation:

PolyGamma can be represented as a DifferenceRoot:

TraditionalForm formatting: