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Clausen Function

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ClausenFunction

Define

S_n(x)=sum_(k=1)^(infty)(sin(kx))/(k^n)
(1)
C_n(x)=sum_(k=1)^(infty)(cos(kx))/(k^n),
(2)

then the Clausen functions are defined by

Cl_n(x)={S_n(x)=sum_(k=1)^(infty)(sin(kx))/(k^n) n even; C_n(x)=sum_(k=1)^(infty)(cos(kx))/(k^n) n odd,
(3)

sometimes also written as psi_n(x) (Arfken 1985, p. 783).

Then the Clausen function Cl_n(x) can be given symbolically in terms of the polylogarithm as

Cl_n(x)={1/2i[Li_n(e^(-ix))-Li_n(e^(ix))] n even; 1/2[Li_n(e^(-ix))+Li_n(e^(ix))] n odd.
(4)

For n=1, the function takes on the special form

Cl_1(x)=C_1(x)=-ln|2sin(1/2x)|
(5)

and for n=2, it becomes Clausen's integral

Cl_2(x)=S_2(x)=-int_0^xln[2sin(1/2t)]dt.
(6)

The symbolic sums of opposite parity are summable symbolically, and the first few are given by

C_2(x)=1/6pi^2-1/2pix+1/4x^2
(7)
C_4(x)=1/(90)pi^4-1/(12)pi^2x^2+1/(12)pix^3-1/(48)x^4
(8)
S_1(x)=1/2(pi-x)
(9)
S_3(x)=1/6pi^2x-1/4pix^2+1/(12)x^3
(10)
S_5(x)=1/(90)pi^4x-1/(36)pi^2x^3+1/(48)pix^4-1/(240)x^5
(11)

for 0<=x<=2pi (Abramowitz and Stegun 1972).

See also

Clausen's Integral, Lobachevsky's Function, Polygamma Function, Polylogarithm

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Clausen's Integral and Related Summations" §27.8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1005-1006, 1972.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, pp. 89-90, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, p. 27, 2004.Borwein, J. M.; Broadhurst, D. J.; and Kamnitzer, J. "Central Binomial Sums, Multiple Clausen Values and Zeta Functions." Exp. Math. 10, 25-41, 2001.Clausen, R. "Über die Zerlegung reeller gebrochener Funktionen." J. reine angew. Math. 8, 298-300, 1832.Grosjean, C. C. "Formulae Concerning the Computation of the Clausen Integral Cl_2(alpha)." J. Comput. Appl. Math. 11, 331-342, 1984.Jolley, L. B. W. Summation of Series. London: Chapman, 1925.Lewin, L. Dilogarithms and Associated Functions. London: Macdonald, pp. 170-180, 1958.Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.Wheelon, A. D. A Short Table of Summable Series. Report No. SM-14642. Santa Monica, CA: Douglas Aircraft Co., 1953.

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Clausen Function

Cite this as:

Weisstein, Eric W. "Clausen Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ClausenFunction.html

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