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 KelvinBei function using MatrixFunction:

Values at zero:

Find the positive minimum of KelvinBei[0,x]:

For half-integer orders, KelvinBei evaluates to elementary functions:

Plot the KelvinBei 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 :

Function range of :

Approximate function range of :

is an even function:

is an odd function:

is an analytic function of :

KelvinBei is neither non-decreasing nor non-increasing:

KelvinBei is not injective:

KelvinBei is neither non-negative nor non-positive:

has singularities or discontinuities in the non-positive reals when is not an integer:

KelvinBei is neither convex nor concave:

TraditionalForm formatting:

The first derivative with respect to :

The first derivative with respect to when :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Formula for the derivative with respect to :

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 series expansion for an arbitrary symbolic direction :

The Taylor expansion at a generic point:

Functional identity:

Recurrence relations: