Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 ArcTanIntegral

Source Notebook

Evaluate the inverse tangent integral

Contributed by: Jan Mangaldan

ResourceFunction["ArcTanIntegral"][n,z]

gives the inverse tangent integral function Tin(z).

Details and Options

Mathematical function, suitable for both symbolic and numerical manipulation.
.
For certain special arguments, ResourceFunction["ArcTanIntegral"] automatically evaluates to exact values.
ResourceFunction["ArcTanIntegral"] can be evaluated to arbitrary numerical precision.
ResourceFunction["ArcTanIntegral"] automatically threads over lists.

Examples

Basic Examples (3) 

Evaluate numerically:

In[1]:=
ResourceFunction["ArcTanIntegral"][3, 0.6]
Out[1]=

Plot over a subset of the reals:

In[2]:=
Plot[ResourceFunction["ArcTanIntegral"][3, x], {x, -2, 2}]
Out[2]=

Series expansion at the origin:

In[3]:=
Series[ResourceFunction["ArcTanIntegral"][5, x], {x, 0, 7}]
Out[3]=

Scope (5) 

Simple exact values are generated automatically:

In[4]:=
ResourceFunction["ArcTanIntegral"][5, 1]
Out[4]=

Evaluate for complex order and arguments:

In[5]:=
ResourceFunction["ArcTanIntegral"][1. - I, 3. + 4. I]
Out[5]=

Evaluate to high precision:

In[6]:=
N[ResourceFunction["ArcTanIntegral"][4, -22/10], 30]
Out[6]=

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

In[7]:=
ResourceFunction["ArcTanIntegral"][4, -2.20000000000000000000000]
Out[7]=

ArcTanIntegral threads elementwise over lists:

In[8]:=
ResourceFunction["ArcTanIntegral"][4, {0.2, 0.5, 0.7}]
Out[8]=

Applications (1) 

Plot of the absolute value of the second-order inverse tangent integral in the complex plane:

In[9]:=
ContourPlot[ Abs[ResourceFunction["ArcTanIntegral"][2, x + I y]], {x, -2, 2}, {y, -2, 2}]
Out[9]=

Properties and Relations (3) 

For integer orders less than 2, ArcTanIntegral can be expressed in terms of elementary functions:

In[10]:=
{ResourceFunction["ArcTanIntegral"][1, z], ResourceFunction["ArcTanIntegral"][-4, z]}
Out[10]=

ArcTanIntegral can be expressed in terms of LerchPhi:

In[11]:=
With[{n = 7}, ResourceFunction["ArcTanIntegral"][n, z] == z/2^n LerchPhi[-z^2, n, 1/2] // FullSimplify]
Out[11]=

Derivatives of ArcTanIntegral can be expressed in terms of an ArcTanIntegral of lower order:

In[12]:=
With[{n = 7}, D[ResourceFunction["ArcTanIntegral"][n, z], z] == ResourceFunction["ArcTanIntegral"][n - 1, z]/z // FullSimplify]
Out[12]=

Version History

  • 1.0.0 – 22 December 2020

Source Metadata

  • Citation:
    • Lewin L., Polylogarithms and Associated Functions. New York: North Holland, 1981.

Related Resources

License Information