gives the Kelvin function TemplateBox[{z}, KelvinKei].


gives the Kelvin function TemplateBox[{n, z}, KelvinKei2].


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For positive real values of parameters, TemplateBox[{n, z}, KelvinKei2]=Im(e^(-npii/2)TemplateBox[{n, {z, , {e, ^, {(, {pi, , {i, /, 4}}, )}}}}, BesselK]). 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.


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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 TemplateBox[{0, x}, KelvinKei2]:

The complex domain of TemplateBox[{0, x}, KelvinKei2]:

TemplateBox[{{-, {1, /, 2}}, x}, KelvinKei2] is defined for all real values greater than 0:

The complex domain is the whole plane except :

Approximate function range of TemplateBox[{0, x}, KelvinKei2]:

Approximate function range of TemplateBox[{1, x}, KelvinKei2]:

Recurrence relations:

TemplateBox[{n, z}, KelvinKei2] 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:

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 ^(th) 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)

KelvinKei can be applied to a power series:

Applications  (3)

Solve the Kelvin differential equation:

Plot the radial density profile for alternating current within a hollow cylinder:

For some values, expressions with MeijerG are represented with KelvinKei:

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:

Possible Issues  (1)

The oneargument form evaluates to the two-argument form:

Wolfram Research (2007), KelvinKei, Wolfram Language function,


Wolfram Research (2007), KelvinKei, Wolfram Language function,


Wolfram Language. 2007. "KelvinKei." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2007). KelvinKei. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_kelvinkei, author="Wolfram Research", title="{KelvinKei}", year="2007", howpublished="\url{}", note=[Accessed: 20-July-2024 ]}


@online{reference.wolfram_2024_kelvinkei, organization={Wolfram Research}, title={KelvinKei}, year={2007}, url={}, note=[Accessed: 20-July-2024 ]}