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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 AngerWeberA

Source Notebook

Evaluate the associated Anger–Weber function

Contributed by: Jan Mangaldan

ResourceFunction["AngerWeberA"][ν,z]

gives the associated Anger–Weber function .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
is defined by .
ResourceFunction["AngerWeberA"][ν,z] has a branch cut discontinuity in the complex z plane running from -∞ to 0.
For certain special arguments, ResourceFunction["AngerWeberA"] automatically evaluates to exact values.
ResourceFunction["AngerWeberA"] can be evaluated to arbitrary numerical precision.
ResourceFunction["AngerWeberA"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate numerically:

In[1]:=
ResourceFunction["AngerWeberA"][1/2, 5.0]
Out[1]=

Plot :

In[2]:=
Plot[ResourceFunction["AngerWeberA"][1/2, x], {x, 0, 9}, PlotRange -> All]
Out[2]=

Scope (4) 

Evaluate for complex arguments:

In[3]:=
ResourceFunction["AngerWeberA"][1/3, 2.5 + I/2]
Out[3]=

Evaluate to arbitrary precision:

In[4]:=
N[ResourceFunction["AngerWeberA"][1/3, 1], 50]
Out[4]=

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

In[5]:=
ResourceFunction["AngerWeberA"][1/3, 1.000000000000000000000000000]
Out[5]=

Simple exact values are generated automatically:

In[6]:=
ResourceFunction["AngerWeberA"][3, 0]
Out[6]=

AngerWeberA threads elementwise over lists:

In[7]:=
ResourceFunction["AngerWeberA"][1/2, {0.1, 0.2, 0.3}]
Out[7]=

Properties and Relations (7) 

Use FunctionExpand to expand AngerWeberA into hypergeometric functions:

In[8]:=
FunctionExpand[ResourceFunction["AngerWeberA"][n, x]]
Out[8]=

Compare AngerWeberA with the integral definition:

In[9]:=
With[{n = 5, z = 11/3, prec = 20}, {N[ResourceFunction["AngerWeberA"][n, z], prec], NIntegrate[Exp[-n t - z Sinh[t]], {t, 0, Infinity}, WorkingPrecision -> prec]/Pi}]
Out[9]=

Express AngerWeberA in terms of the Lommel function LommelS:

In[10]:=
FullSimplify[ ResourceFunction["AngerWeberA"][n, z] == 1/\[Pi] (ResourceFunction["LommelS"][0, n, z] - n ResourceFunction["LommelS"][-1, n, z])]
Out[10]=

Verify a differential equation for AngerWeberA:

In[11]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/a70713bf-bc5a-41cc-9edc-dddc2e6ee7f7"]
Out[11]=

Verify a recurrence identity for AngerWeberA:

In[12]:=
Table[ResourceFunction["AngerWeberA"][n + 1, z] + ResourceFunction["AngerWeberA"][n - 1, z] == 2/(\[Pi] z) - (2 n)/z ResourceFunction["AngerWeberA"][n, z] /. z -> RandomComplex[WorkingPrecision -> 20], {n, 9}]
Out[12]=

An identity relating AngerWeberA and AngerJ:

In[13]:=
AngerJ[n, z] == BesselJ[n, z] + Sin[\[Pi] n] ResourceFunction["AngerWeberA"][n, z] // FunctionExpand // FullSimplify
Out[13]=

An identity relating AngerWeberA and WeberE:

In[14]:=
WeberE[n, z] == -BesselY[n, z] - Cos[\[Pi] n] ResourceFunction["AngerWeberA"][n, z] - ResourceFunction["AngerWeberA"][-n, z] // FunctionExpand // FullSimplify
Out[14]=

Version History

  • 1.1.0 – 14 September 2021

Source Metadata

  • Citation:
    • "Anger–Weber Functions." §11.10 in F. W. J. Olver et al. (eds.), NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/11.10. Release 1.1.1 of 2021-03-15.

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