By analogy with the log sine function, define the log cosine function by
(1)
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The first few cases are given by
(2)
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(3)
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(4)
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where is the Riemann zeta function.
The log cosine function is related to the log sine function by
(5)
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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)
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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)
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(Oloa 2011; OEIS A189272).