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Log Cosine Function

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

C_n=int_0^(pi/2)[ln(cosx)]^ndx.
(1)

The first few cases are given by

C_1=-1/2piln2
(2)
C_2=1/(24)pi^3+1/2pi(ln2)^2
(3)
C_3=-1/8pi^3ln2-1/2pi(ln2)^3-3/4pizeta(3),
(4)

where zeta(z) is the Riemann zeta function.

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

C_n=1/2S_n
(5)

and the two are equal if the range of integration for S_n is restricted from 0 to pi to 0 to pi/2.

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

(32)/piint_0^(pi/2)(x^4)/(x^2+ln^2(2cosx))dx=12zeta(2)ln(2pi)-18zeta(2)gamma+4zeta(3)+2gamma^3+12zeta^'(0,1,1)+9zeta(2)-3/2gamma^2,
(6)

where zeta(z) is the Riemann zeta function, gamma is the Euler-Mascheroni constant, zeta(s,1,1) is a multivariate zeta function, and zeta^'(s,1,1) denotes dzeta(s,1,1)/ds|_(s=0). A closed form for zeta^'(s,1,1) in terms of more elementary functions is not known as of Apr. 2011, but it is numerically given by

zeta^'(s,1,1)=1.396989620926385869015999484472258...
(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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