Function Repository Resource:

LommelS

Evaluate the Lommel function

Contributed by: Jan Mangaldan
 ResourceFunction["LommelS"][μ,ν,z] gives the Lommel function Sμ,ν(z).

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
Sμ,ν(z) satisfies the differential equation .
ResourceFunction["LommelS"][μ,ν,z] has a branch cut discontinuity in the complex z plane running from - to 0.
For certain special arguments, ResourceFunction["LommelS"] automatically evaluates to exact values.
ResourceFunction["LommelS"] can be evaluated to arbitrary numerical precision.

Examples

Basic Examples (3)

Evaluate numerically:

 In[1]:=
 Out[1]=

Plot S-1/4,2/3(x) over a subset of the reals:

 In[2]:=
 Out[2]=

Series expansion at the origin:

 In[3]:=
 Out[3]=

Scope (4)

Evaluate for complex arguments:

 In[4]:=
 Out[4]=

Evaluate to high precision:

 In[5]:=
 Out[5]=

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

 In[6]:=
 Out[6]=

 In[7]:=
 Out[7]=

Simple exact values are generated automatically:

 In[8]:=
 Out[8]=

Properties and Relations (7)

Verify recurrence relations satisfied by LommelS:

 In[9]:=
 Out[9]=
 In[10]:=
 Out[10]=

Express the derivative of LommelS in terms of LommelS:

 In[11]:=
 Out[11]=
 In[12]:=
 Out[12]=
 In[13]:=
 Out[13]=

Indefinite integrals of the Bessel functions BesselJ and BesselY multiplied by a power function can be expressed in terms of Bessel functions and LommelS:

 In[14]:=
 Out[14]=
 In[15]:=
 Out[15]=

Express the Anger and Weber functions in terms of LommelS:

 In[16]:=
 Out[16]=
 In[17]:=
 Out[17]=

The associated Anger–Weber function AngerWeberA can be expressed in terms of LommelS:

 In[18]:=
 Out[18]=

The Neumann polynomial NeumannO can be expressed in terms of LommelS:

 In[19]:=
 Out[19]=

The Schläfli polynomial SchlaefliS can be expressed in terms of LommelS:

 In[20]:=
 Out[20]=

Version History

• 1.1.0 – 14 September 2021
• 1.0.0 – 07 January 2021