AngerJ
✖
AngerJ
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.
is defined by
.
- 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
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0enzr831t-fckaku

Plot over a subset of the reals:

https://wolfram.com/xid/0enzr831t-ngnftc

Plot over a subset of the complexes:

https://wolfram.com/xid/0enzr831t-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0enzr831t-ej957w

Scope (39)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0enzr831t-l274ju


https://wolfram.com/xid/0enzr831t-cksbl4


https://wolfram.com/xid/0enzr831t-b0wt9

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

https://wolfram.com/xid/0enzr831t-b392zk


https://wolfram.com/xid/0enzr831t-hfml09

Evaluate efficiently at high precision:

https://wolfram.com/xid/0enzr831t-di5gcr


https://wolfram.com/xid/0enzr831t-bq2c6r

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

https://wolfram.com/xid/0enzr831t-h0d6g


https://wolfram.com/xid/0enzr831t-dj6d9x

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0enzr831t-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0enzr831t-thgd2

Or compute the matrix AngerJ function using MatrixFunction:

https://wolfram.com/xid/0enzr831t-o5jpo

Specific Values (7)

https://wolfram.com/xid/0enzr831t-bdij6w


https://wolfram.com/xid/0enzr831t-bmqd0y


https://wolfram.com/xid/0enzr831t-e41pf2

AngerJ for symbolic ν and x:

https://wolfram.com/xid/0enzr831t-3wm4h

Find the first positive maximum of AngerJ:

https://wolfram.com/xid/0enzr831t-otdu3


https://wolfram.com/xid/0enzr831t-h177qi

AngerJ simplifies to BesselJ for integer orders:

https://wolfram.com/xid/0enzr831t-chhice

Simple exact values are generated automatically:

https://wolfram.com/xid/0enzr831t-eww5n

Evaluate AngerJ for half-integer orders:

https://wolfram.com/xid/0enzr831t-z8bw1

Visualization (3)
Plot the AngerJ function for integer () and half-integer (
) orders:

https://wolfram.com/xid/0enzr831t-ecj8m7


https://wolfram.com/xid/0enzr831t-b41shq


https://wolfram.com/xid/0enzr831t-f2w91s


https://wolfram.com/xid/0enzr831t-bcvb9v


https://wolfram.com/xid/0enzr831t-ue5vx

Function Properties (15)

https://wolfram.com/xid/0enzr831t-cl7ele


https://wolfram.com/xid/0enzr831t-de3irc

is defined for all real values:

https://wolfram.com/xid/0enzr831t-rrrwu

Complex domain is the whole plane:

https://wolfram.com/xid/0enzr831t-bthhjz

Approximate function range of :

https://wolfram.com/xid/0enzr831t-evf2yr

Approximate function range of :

https://wolfram.com/xid/0enzr831t-fphbrc


https://wolfram.com/xid/0enzr831t-ewxrep


https://wolfram.com/xid/0enzr831t-fu5czq

Use FullSimplify to simplify Anger functions:

https://wolfram.com/xid/0enzr831t-c8anmk

AngerJ threads elementwise over lists:

https://wolfram.com/xid/0enzr831t-kv466y


https://wolfram.com/xid/0enzr831t-gva6yl

AngerJ is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/0enzr831t-2ra8g


https://wolfram.com/xid/0enzr831t-c9npzh


https://wolfram.com/xid/0enzr831t-b5buvp


https://wolfram.com/xid/0enzr831t-patce


https://wolfram.com/xid/0enzr831t-bcrbvs

AngerJ is neither non-negative nor non-positive:

https://wolfram.com/xid/0enzr831t-dvzykj


https://wolfram.com/xid/0enzr831t-kpi9jh

AngerJ does not have either singularity or discontinuity:

https://wolfram.com/xid/0enzr831t-fyfbxx


https://wolfram.com/xid/0enzr831t-5vh4e

AngerJ is neither convex nor concave:

https://wolfram.com/xid/0enzr831t-l0srvu

TraditionalForm formatting:

https://wolfram.com/xid/0enzr831t-c6i1u7


https://wolfram.com/xid/0enzr831t-kk3key

Differentiation and Integration (5)
First derivatives with respect to z:

https://wolfram.com/xid/0enzr831t-krpoah


https://wolfram.com/xid/0enzr831t-ez6occ

Higher derivatives with respect to z:

https://wolfram.com/xid/0enzr831t-z33jv

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

https://wolfram.com/xid/0enzr831t-fxwmfc

Formula for the derivative with respect to z when ν=3:

https://wolfram.com/xid/0enzr831t-cb5zgj

Indefinite integral of AngerJ:

https://wolfram.com/xid/0enzr831t-kchfgu


https://wolfram.com/xid/0enzr831t-bponid


https://wolfram.com/xid/0enzr831t-cf948


https://wolfram.com/xid/0enzr831t-fcaobo

Series Expansions (3)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0enzr831t-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0enzr831t-binhar

General term in the series expansion using SeriesCoefficient:

https://wolfram.com/xid/0enzr831t-dznx2j

Taylor expansion at a generic point:

https://wolfram.com/xid/0enzr831t-jwxla7

Properties & Relations (2)Properties of the function, and connections to other functions
Use FunctionExpand to expand AngerJ into hypergeometric functions:

https://wolfram.com/xid/0enzr831t-3eynv

Relationships between the Anger and Weber functions:

https://wolfram.com/xid/0enzr831t-c8jmj4


https://wolfram.com/xid/0enzr831t-e9rhow

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
]}
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
]}