Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Compute average case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix KelvinKei function using MatrixFunction:

Values at zero:

Find the first positive maximum of KelvinKei[0,x]:

For some half-integer orders, KelvinKei evaluates to the same elementary functions:

Plot the KelvinKei function for integer () and half-integer () orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

The real domain of :

The complex domain of :

is defined for all real values greater than 0:

The complex domain is the whole plane except :

Approximate function range of :

Approximate function range of :

Recurrence relations:

is not an analytic function:

KelvinKei is neither non-decreasing nor non-increasing:

KelvinKei is not injective:

KelvinKei is neither non-negative nor non-positive:

KelvinKei has both singularity and discontinuity for z≤0:

KelvinKei is neither convex nor concave:

TraditionalForm formatting:

The first derivative with respect to z:

The first derivative with respect to z when n=1:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Formula for the derivative with respect to z:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

The definite integral:

More integrals:

Find the Taylor expansion using Series:

Plots of the first three approximations around :

The general term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

The Taylor expansion at a generic point: