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

(1)

The first few cases are given by

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 ).

See also Clausen's Integral ,

Log
Gamma Function ,

Log Sine Function
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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." Referenced on Wolfram|Alpha Log Cosine Function
Cite this as:
Weisstein, Eric W. "Log Cosine Function."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/LogCosineFunction.html

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