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


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Details

The functions are defined as

,

,

where is the polylogarithm function.

They can also be represented by the trigonometric expansions

,

.

References

[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. gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0008&IDDOC=268720.

[2] Wikipedia. "Clausen Function." (Sep 9, 2014) en.wikipedia.org/wiki/Clausen_function.

[3] F. Johansson. "Improved Incomplete Gamma and Exponential Integrals; Clausen Functions." (Sep 9, 2014) fredrik-j.blogspot.mx/2009/07/improved-incomplete-gamma-and.html.

[4] H. J. Lu and C. A. Perez. "Massless One-Loop Scalar Three-Point Integral and Associated Clausen, Glaisher and L-Functions." (May 1992) www.learningace.com/doc/121222/a5bcfe501dfce2e2c58770e1fcea369a/slac-pub-5809.

[5] V. E. Wood, "Efficient Calculation of Clausen's Integral," Mathematics of Computation, 22(104), 1968 pp. 883–884. www.ams.org/journals/mcom/1968-22-104/S0025-5718-1968-0239733-9/S0025-5718-1968-0239733-9.pdf.

[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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