Clausen Functions

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2D

density

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n

1

f

Re

Im

Abs

Arg

points

Thomas Clausen (1801–1885) was a Danish mathematician, astronomer, and geophysicist who introduced the functions

Cl

n

(x)

, defined in terms of polylogarithms [1]; the case

Cl

2

(x)

is called Clausen's integral. These functions are useful to define because some identities connect them with the Barnes

G

function, polylogarithm and polygamma functions, and Dirichlet

L

functions [2]. They can also be used to evaluate some divergent Fourier series [3] and in the computation of singular integrals in quantum field theory [4]. For complex arguments, they are related to zeta functions. Efficient methods to calculate Clausen functions can be found in [5, 6].