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KelvinBei

gives the Kelvin function .

gives the Kelvin function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For positive real values of parameters, $TemplateBox[{n, z}, KelvinBei2]=Im(e^(npii)TemplateBox[{n, {z, , {e, ^, {(, {{-, pi}, , {i, /, 4}}, )}}}}, BesselJ])$ . For other values, is defined by analytic continuation.
KelvinBei[n,z] has a branch cut discontinuity in the complex z plane running from to .
KelvinBei[z] is equivalent to KelvinBei[0,z].
For certain special arguments, KelvinBei automatically evaluates to exact values.
KelvinBei can be evaluated to arbitrary numerical precision.
KelvinBei 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 (37)

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 KelvinBei function using MatrixFunction:

Specific Values (3)

Values at zero:

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

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

Visualization (3)

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 :

Function Properties (12)

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 z:

KelvinBei is neither non-decreasing nor non-increasing:

KelvinBei is not injective:

KelvinBei is neither non-negative nor non-positive:

KelvinBei has no singularities or discontinuities:

KelvinBei 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 series expansion for an arbitrary symbolic direction :

The Taylor expansion at a generic point:

Function Identities and Simplifications (2)

Functional identity:

Recurrence relations:

Generalizations & Extensions (1)

KelvinBei can be applied to a power series:

Applications (3)

Solve the Kelvin differential equation:

Plot the resistance of a wire with circular cross section versus AC frequency (skin effect):

For some specific values, HypergeometricPFQRegularized is represented with KelvinBei:

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:

KelvinBei can be represented in terms of MeijerG:

Possible Issues (1)

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

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