gives the Kelvin function .


gives the Kelvin function .


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


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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  (26)

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 KelvinKer[0,x]:

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

Visualization  (3)

Plot the KelvinKer 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  (5)

The real domain of TemplateBox[{0, x}, KelvinKer2]:

The complex domain of TemplateBox[{0, x}, KelvinKer2]:

TemplateBox[{{-, {1, /, 2}}, x}, KelvinKer2] is defined for all real values greater than 0:

The complex domain is the whole plane except :

Approximate function range of TemplateBox[{0, x}, KelvinKer2]:

Approximate function range of TemplateBox[{1, x}, KelvinKer2]:

Functional identity:

TraditionalForm formatting:

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 Infinity:

Find the series expansion for an arbitrary symbolic direction :

The Taylor expansion at a generic point:

Generalizations & Extensions  (1)

KelvinKer 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  (3)

Use FullSimplify to simplify expressions involving Kelvin functions:

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

KelvinKer can be represented in terms of MeijerG:

Introduced in 2007