Some trigonometric and elliptic integrals (Q934564)

Let \[ C(m,n;k)\equiv\int_{0}^{\frac{\pi}{2}}| \cos(m\theta+n\phi)| d\theta, \] \[ S(m,n;k)\equiv\int_{0}^{\frac{\pi}{2}}| \sin(m\theta+n\phi)| d\theta , \] where \(\phi\in[0,\frac{\pi}{2})\) and \(\tan\phi=k\tan\theta\). The main result is \[ C(m,n;k)=S(m,n;k)=1,\;m,n\in\mathbb{N},\;k\in\mathbb{R}^+. \] Simultaneous identities involving elliptic integrals of the first and second kinds, evaluated at the roots of a very simple trigonometric polynomial, are given. Only the cases \(m=1,2,3,4\) are treated for the last part, the remainder being offered as a problem. The results involved are too complicated to be included in this brief review.