Log Cosine Function

By analogy with the log sine function, define the log cosine function by

(1)

The first few cases are given by

			(2)
			(3)
			(4)

where  is the Riemann zeta function.

The log cosine function is related to the log sine function by

(5)

and the two are equal if the range of integration for  is restricted from 0 to  to 0 to .

Oloa (2011) computed an exact value of the log cosine integral

(6)

where  is the Riemann zeta function,  is the Euler-Mascheroni constant,  is a multivariate zeta function, and  denotes . A closed form for  in terms of more elementary functions is not known as of Apr. 2011, but it is numerically given by

(7)

(Oloa 2011; OEIS A189272).

REFERENCES:

Oloa, O. "A Log-Cosine Integral Involving a Derivative of a MZV." Preprint. Apr. 18, 2011.

Sloane, N. J. A. Sequence A189272 in "The On-Line Encyclopedia of Integer Sequences."