Function Repository Resource:

# ArcTanIntegral

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.
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For certain special arguments, ResourceFunction["ArcTanIntegral"] automatically evaluates to exact values.
ResourceFunction["ArcTanIntegral"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

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Plot over a subset of the reals:

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Series expansion at the origin:

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### Scope (5)

Simple exact values are generated automatically:

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Evaluate for complex order and arguments:

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Evaluate to high precision:

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

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### Applications (1)

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

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### Properties and Relations (3)

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

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ArcTanIntegral can be expressed in terms of LerchPhi:

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

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## Version History

• 1.0.0 – 22 December 2020