KelvinBei
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
open all close allBasic 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)
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 TemplateBox[{0, x}, KelvinBei2]:
The complex domain of TemplateBox[{0, x}, KelvinBei2]:
TemplateBox[{{-, {1, /, 2}}, x}, KelvinBei2] is defined for all real values greater than 0:
The complex domain is the whole plane except :
Function range of TemplateBox[{0, x}, KelvinBei2]:
Approximate function range of TemplateBox[{1, x}, KelvinBei2]:
TemplateBox[{0, x}, KelvinBei2] is an even function:
TemplateBox[{1, x}, KelvinBei2] is an odd function:
TemplateBox[{0, z}, KelvinBei2] 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 ^(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 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:
Possible Issues (1)
The one‐argument form evaluates to the two-argument form:
Tech Notes
Related Guides
Related Links
History
Text
Wolfram Research (2007), KelvinBei, Wolfram Language function, https://reference.wolfram.com/language/ref/KelvinBei.html.
CMS
Wolfram Language. 2007. "KelvinBei." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/KelvinBei.html.
APA
Wolfram Language. (2007). KelvinBei. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KelvinBei.html
BibTeX
@misc{reference.wolfram_2025_kelvinbei, author="Wolfram Research", title="{KelvinBei}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/KelvinBei.html}", note=[Accessed: 06-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_kelvinbei, organization={Wolfram Research}, title={KelvinBei}, year={2007}, url={https://reference.wolfram.com/language/ref/KelvinBei.html}, note=[Accessed: 06-January-2026]}