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KelvinKei

gives the Kelvin function .

gives the Kelvin function .

Details

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.

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 (34)

Numerical Evaluation (6)

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:

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 z≤0:

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 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 one‐argument form evaluates to the two-argument form:

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