Function Repository Resource:

# EpsteinHubbellOmega

Evaluate the Epstein–Hubbell integral

Contributed by: Jan Mangaldan
 ResourceFunction["EpsteinHubbellOmega"][n,m] gives the Epstein–Hubbell integral Ωn(m).

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For and positive integer n, the Epstein–Hubbell integral satisfies .
ResourceFunction["EpsteinHubbellOmega"][n,m] has branch cut discontinuities in the complex m plane running from - to -1 and +1 to +.
For certain special arguments, ResourceFunction["EpsteinHubbellOmega"] automatically evaluates to exact values.
ResourceFunction["EpsteinHubbellOmega"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

 In[1]:=
 Out[1]=

Plot Ω2(m):

 In[2]:=
 Out[2]=

Series at the origin:

 In[3]:=
 Out[3]=

### Scope (4)

Evaluate for complex arguments and orders:

 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]=

### Applications (1)

Log plot of a family of Epstein–Hubbell integrals:

 In[8]:=
 Out[8]=

### Properties and Relations (5)

For integer n, EpsteinHubbellOmega can be expressed in terms of EllipticE and EllipticK:

 In[9]:=
 Out[9]=

For integer n, EpsteinHubbellOmega can be expressed in terms of half-integer order LegendreP:

 In[10]:=
 Out[10]=

For integer n, EpsteinHubbellOmega can be expressed in terms of LegendreQ:

 In[11]:=
 Out[11]=

Express an Epstein–Hubbell integral of noninteger order in terms of simpler functions:

 In[12]:=
 Out[12]=

Compare EpsteinHubbellOmega with the integral definition:

 In[13]:=
 Out[13]=

## Version History

• 1.0.0 – 04 February 2021