AngerJ

AngerJ[ν,z]

gives the Anger function .

AngerJ[ν,μ,z]

gives the associated Anger function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • satisfies the differential equation .
  • is defined by .
  • AngerJ[ν,z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, AngerJ automatically evaluates to exact values.
  • AngerJ can be evaluated to arbitrary numerical precision.
  • AngerJ automatically threads over lists.

Examples

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Basic Examples  (4)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Scope  (28)

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  (7)

Limiting value at infinity:

Values at zero:

AngerJ for symbolic ν and x:

Find the first positive maximum of AngerJ:

AngerJ defines as BesselJ for integer orders:

Simple exact values are generated automatically:

Evaluate AngerJ for half-integer orders:

Visualization  (3)

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

Plot the real part of TemplateBox[{0, {x, +, {i,  , y}}}, AngerJ2]:

Plot the imaginary part of TemplateBox[{0, {x, +, {i,  , y}}}, AngerJ2]:

Plot the real part of TemplateBox[{{-, {1, /, 4}}, {x, +, {i,  , y}}}, AngerJ2]:

Plot the imaginary part of TemplateBox[{{-, {1, /, 4}}, {x, +, {i,  , y}}}, AngerJ2]:

Function Properties  (8)

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

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

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

Complex domain is the whole plane:

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

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

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

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

Use FullSimplify to simplify Anger functions:

AngerJ threads elementwise over lists:

TraditionalForm formatting:

Differentiation  (3)

First derivatives with respect to z:

Higher derivatives with respect to z:

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

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

Series Expansions  (3)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

General term in the series expansion using SeriesCoefficient:

Taylor expansion at a generic point:

Properties & Relations  (1)

Use FunctionExpand to expand AngerJ into hypergeometric functions:

Introduced in 2008
 (7.0)