# 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(35)

### 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 :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

### Function Properties(15)

Real domain of :

Complex domain of : is defined for all real values:

Complex domain is the whole plane:

Approximate function range of :

Approximate function range of : is an even function: is an odd function:

Use FullSimplify to simplify Anger functions: is an analytic function of :

AngerJ is neither non-decreasing nor non-increasing: is not injective: is not surjective:

AngerJ is neither non-negative nor non-positive:

AngerJ does not have either singularity or discontinuity:

AngerJ is neither convex nor concave:

### 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  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: