WOLFRAM

AngerJ[ν,z]

gives the Anger function TemplateBox[{nu, z}, AngerJ2].

AngerJ[ν,μ,z]

gives the associated Anger function TemplateBox[{nu, mu, z}, AngerJ].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{nu, z}, AngerJ2] satisfies the differential equation .
  • TemplateBox[{nu, z}, AngerJ2] is defined by TemplateBox[{nu, z}, AngerJ2]=1/piint_0^picos(theta nu-z sin(theta))dtheta.
  • AngerJ[ν,z] is an entire function of z with no branch cut discontinuities.
  • TemplateBox[{nu, mu, z}, AngerJ] is defined by TemplateBox[{nu, mu, z}, AngerJ]=1/piint_0^pi(2sin(theta))^mucos(theta nu-z sin(theta))dtheta.
  • For certain special arguments, AngerJ automatically evaluates to exact values.
  • AngerJ can be evaluated to arbitrary numerical precision.
  • AngerJ automatically threads over lists.
  • AngerJ can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (4)Summary of the most common use cases

Evaluate numerically:

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Plot TemplateBox[{{1, /, 2}, x}, AngerJ2] over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Scope  (39)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

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Evaluate to high precision:

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The precision of the output tracks the precision of the input:

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Complex number inputs:

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Evaluate efficiently at high precision:

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Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

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Or compute average-case statistical intervals using Around:

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Compute the elementwise values of an array:

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Or compute the matrix AngerJ function using MatrixFunction:

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Specific Values  (7)

Limiting value at infinity:

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Values at zero:

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AngerJ for symbolic ν and x:

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Find the first positive maximum of AngerJ:

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AngerJ simplifies to BesselJ for integer orders:

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Simple exact values are generated automatically:

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Evaluate AngerJ for half-integer orders:

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Visualization  (3)

Plot the AngerJ function for integer () and half-integer () orders:

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Plot the real part of TemplateBox[{0, z}, AngerJ2]:

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Plot the imaginary part of TemplateBox[{0, z}, AngerJ2]:

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Plot the real part of TemplateBox[{{-, {1, /, 4}}, z}, AngerJ2]:

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Plot the imaginary part of TemplateBox[{{-, {1, /, 4}}, z}, AngerJ2]:

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Function Properties  (15)

Real domain of TemplateBox[{0, x}, AngerJ2]:

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Complex domain of TemplateBox[{0, z}, AngerJ2]:

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TemplateBox[{{-, {1, /, 2}}, x}, AngerJ2] is defined for all real values:

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Complex domain is the whole plane:

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Approximate function range of TemplateBox[{0, x}, AngerJ2]:

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Approximate function range of TemplateBox[{1, x}, AngerJ2]:

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TemplateBox[{0, x}, AngerJ2] is an even function:

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TemplateBox[{1, x}, AngerJ2] is an odd function:

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Use FullSimplify to simplify Anger functions:

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AngerJ threads elementwise over lists:

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TemplateBox[{2, x}, AngerJ2] is an analytic function of :

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AngerJ is neither non-decreasing nor non-increasing:

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TemplateBox[{2, x}, AngerJ2] is not injective:

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TemplateBox[{2, x}, AngerJ2] is not surjective:

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AngerJ is neither non-negative nor non-positive:

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AngerJ does not have either singularity or discontinuity:

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AngerJ is neither convex nor concave:

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TraditionalForm formatting:

Differentiation and Integration  (5)

First derivatives with respect to z:

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Higher derivatives with respect to z:

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Plot the higher derivatives with respect to z when ν=1/4:

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Formula for the ^(th) derivative with respect to z when ν=3:

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Indefinite integral of AngerJ:

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More integrals:

Series Expansions  (3)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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General term in the series expansion using SeriesCoefficient:

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Taylor expansion at a generic point:

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Properties & Relations  (2)Properties of the function, and connections to other functions

Use FunctionExpand to expand AngerJ into hypergeometric functions:

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Relationships between the Anger and Weber functions:

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Wolfram Research (2008), AngerJ, Wolfram Language function, https://reference.wolfram.com/language/ref/AngerJ.html.
Wolfram Research (2008), AngerJ, Wolfram Language function, https://reference.wolfram.com/language/ref/AngerJ.html.

Text

Wolfram Research (2008), AngerJ, Wolfram Language function, https://reference.wolfram.com/language/ref/AngerJ.html.

Wolfram Research (2008), AngerJ, Wolfram Language function, https://reference.wolfram.com/language/ref/AngerJ.html.

CMS

Wolfram Language. 2008. "AngerJ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AngerJ.html.

Wolfram Language. 2008. "AngerJ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/AngerJ.html.

APA

Wolfram Language. (2008). AngerJ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AngerJ.html

Wolfram Language. (2008). AngerJ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/AngerJ.html

BibTeX

@misc{reference.wolfram_2025_angerj, author="Wolfram Research", title="{AngerJ}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/AngerJ.html}", note=[Accessed: 07-June-2025 ]}

@misc{reference.wolfram_2025_angerj, author="Wolfram Research", title="{AngerJ}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/AngerJ.html}", note=[Accessed: 07-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_angerj, organization={Wolfram Research}, title={AngerJ}, year={2008}, url={https://reference.wolfram.com/language/ref/AngerJ.html}, note=[Accessed: 07-June-2025 ]}

@online{reference.wolfram_2025_angerj, organization={Wolfram Research}, title={AngerJ}, year={2008}, url={https://reference.wolfram.com/language/ref/AngerJ.html}, note=[Accessed: 07-June-2025 ]}