Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix HurwitzZeta function using MatrixFunction:

Simple exact values are generated automatically:

HurwitzZeta[s,a] for symbolic a:

HurwitzZeta[s,a] for symbolic s:

Values at zero:

Find a value of s for which HurwitzZeta[s,1]=1.05:

Plot HurwitzZeta as a function of its parameter for positive :

Plot the real and imaginary parts as a function of its parameter for negative noninteger :

Plot HurwitzZeta as a function of for various positive integer values of :

Do it for nonpositive integer :

Plot the real and imaginary parts as a function of its parameter for noninteger :

Plot the real part of HurwitzZeta function:

Plot the imaginary part of HurwitzZeta function:

Real domain of :

For positive , this is simply :

For negative integer , the domain is just the negative integers:

Complex domain:

For positive , this is again :

Approximate function range of :

HurwitzZeta threads elementwise over lists:

HurwitzZeta is not an analytic function:

Nor is it meromorphic:

is neither non-decreasing nor non-increasing:

is not injective:

is surjective:

is not surjective:

is neither non-negative nor non-positive:

has both singularity and discontinuity for negative integers:

is neither convex nor concave:

TraditionalForm formatting:

First derivative with respect to a:

Higher derivatives with respect to a:

Plot the higher derivatives with respect to a when s=3:

Formula for the derivative with respect to a:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

HurwitzZeta is defined through the identity:

Recurrence identity: