Log Cosine Function
By analogy with the log sine function, define the log cosine function by
![画像: C_n=int_0^(pi/2)[ln(cosx)]^ndx.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LogCosineFunction/NumberedEquation1.svg){kind=link}
The first few cases are given by
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 FunctionExplore with Wolfram|Alpha
More things to try:
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 FunctionCite this as:
Weisstein, Eric W. "Log Cosine Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LogCosineFunction.html