Clausen Functions

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Thomas Clausen (1801–1885) was a Danish mathematician, astronomer, and geophysicist who introduced the functions , defined in terms of polylogarithms [1]; the case is called Clausen's integral. These functions are useful to define because some identities connect them with the Barnes function, polylogarithm and polygamma functions, and Dirichlet 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].

Contributed by: Enrique Zeleny (September 2014)
Open content licensed under CC BY-NC-SA



The functions are defined as



where is the polylogarithm function.

They can also be represented by the trigonometric expansions




[1] T. Clausen, "Über die Function sin ϕ + (1/22) sin 2ϕ + (1/32) sin 3ϕ + etc.," Journal für die reine und angewandte Mathematik, 8, 1832 pp. 298–300.

[2] Wikipedia. "Clausen Function." (Sep 9, 2014)

[3] F. Johansson. "Improved Incomplete Gamma and Exponential Integrals; Clausen Functions." (Sep 9, 2014)

[4] H. J. Lu and C. A. Perez. "Massless One-Loop Scalar Three-Point Integral and Associated Clausen, Glaisher and L-Functions." (May 1992)

[5] V. E. Wood, "Efficient Calculation of Clausen's Integral," Mathematics of Computation, 22(104), 1968 pp. 883–884.

[6] J. Wu, X. Zhang, and D. Liu, "An Efficient Calculation of the Clausen Functions ," BIT Numerical Mathematics, 50(1), 2010 pp. 193–206. doi:10.1007/s10543-009-0246-8.

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