Clausen Function

The Clausen functions , with a positive integer and real, are defined wherever the series converge using the Fourier series

			(1)
			(2)

as

(3)

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

Then the Clausen function can be given symbolically in terms of the polylogarithm as

(4)

For and not an integer multiple of , the function takes on the special form

(5)

and for , it becomes Clausen's integral, given for by

(6)

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

			(7)
			(8)
			(9)
			(10)
			(11)

for , except that the formula for holds only for . At and , the series defining is zero (Abramowitz and Stegun 1972, p. 1005).