# KelvinKei

KelvinKei[z]

gives the Kelvin function .

KelvinKei[n,z]

gives the Kelvin function .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• For positive real values of parameters, . For other values, is defined by analytic continuation.
• KelvinKei[n,z] has a branch cut discontinuity in the complex z plane running from to .
• KelvinKei[z] is equivalent to KelvinKei[0,z].
• For certain special arguments, KelvinKei automatically evaluates to exact values.
• KelvinKei can be evaluated to arbitrary numerical precision.
• KelvinKei automatically threads over lists.

# Examples

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## Basic Examples(6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

## Scope(32)

### Numerical Evaluation(4)

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:

### Specific Values(3)

Values at zero:

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

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

### Visualization(3)

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 :

### Function Properties(11)

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 z0:

KelvinKei is neither convex nor concave:

### Differentiation(3)

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:

### Integration(3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

The definite integral:

More integrals:

### Series Expansions(5)

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:

Find the series expansion for an arbitrary symbolic direction :

The Taylor expansion at a generic point:

## Generalizations & Extensions(1)

KelvinKei can be applied to a power series:

## Applications(2)

Solve the Kelvin differential equation:

Plot the radial density profile for AC current in a hollow cylinder:

## Properties & Relations(4)

Use FullSimplify to simplify expressions involving Kelvin functions:

Use FunctionExpand to expand Kelvin functions of half-integer orders:

Integrate expressions involving Kelvin functions:

KelvinKei can be represented in terms of MeijerG: