Thomas Clausen (1801–1885) was a Danish mathematician, astronomer, and geophysicist who introduced the functions , defined in terms of polylogarithms ; 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 . They can also be used to evaluate some divergent Fourier series  and in the computation of singular integrals in quantum field theory . For complex arguments, they are related to zeta functions. Efficient methods to calculate Clausen functions can be found in [5, 6].
The functions are defined as
where is the polylogarithm function.
They can also be represented by the trigonometric expansions
 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.
 Wikipedia. "Clausen Function." (Sep 9, 2014) en.wikipedia.org/wiki/Clausen_function.
 F. Johansson. "Improved Incomplete Gamma and Exponential Integrals; Clausen Functions." (Sep 9, 2014) fredrik-j.blogspot.mx/2009/07/improved-incomplete-gamma-and.html.
 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.
 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.
 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.