Function Repository Resource:

# ClausenCl

Evaluate the Clausen function

Contributed by: Enrique Zeleny and Jan Mangaldan
 ResourceFunction["ClausenCl"][n,z] gives the Clausen function Cln(z).

## Details

Mathematical function, suitable for both symbolic and numerical manipulation.
The Clausen function is defined as if n is even and if n is odd.
Cln(z) is a periodic function in z with period 2π.
ResourceFunction["ClausenCl"] has branch cut discontinuities in the complex z plane running from 2n π-ⅈ ∞ to 0 and from 0 to 2n π+ⅈ ∞ for integers n.
For certain special arguments, ResourceFunction["ClausenCl"] automatically evaluates to exact values.
ResourceFunction["ClausenCl"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (2)

Evaluate numerically:

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Plot the first few Clausen functions:

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### Scope (5)

Evaluate for complex arguments and parameters:

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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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Simple exact values are generated automatically:

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Parity transformations and periodicity relations are automatically applied:

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### Applications (1)

Plots of the Clausen function in the complex plane:

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### Properties and Relations (4)

The Clausen function can be expressed in terms of PolyLog:

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The Clausen function appears in the reflection formula for BarnesG:

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Verify a relationship between the Clausen function and the inverse tangent integral:

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Verify the duplication theorem:

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### Neat Examples (1)

Values of the Clausen function at rational multiples of π can be expressed in terms of PolyGamma:

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## Requirements

Wolfram Language 11.3 (March 2018) or above

## Version History

• 1.1.0 – 31 August 2021
• 1.0.0 – 12 February 2019