Integrals of some trigonometric functions (Q752245)

General explicit expressions for the integrals \[ \int^{\infty}_{0}(\sin^ mat \sin^ nbt \sin^ kct)t^{- (m+n+k)}dt \] with integers \(m>0\), \(n\geq 0\), \(k\geq 0\) are given. This is the whole fact, but it is a highly remarkable fact: Only very few cases for \(k=0\) or \(k=n=0\) have been known, most of them long since Bierens de Haan's ``Nouvelles tables d'intégrals definies'', published more than a century ago. And the second remarkable fact is the method to prove the expressions: It is based on the theory of convex sets, especially of slicing cubes in \(R^ n\) and of other convex bodies, founded by \textit{D. Hensley} [Proc. Am. Math. Soc. 73,95-100 (1979; Zbl 0394.52006), ibid. 79, 619-625 (1980; Zbl 0439.52008)] and \textit{K. Ball} [ibid. 97, 465-473 (1986; Zbl 0601.52005), and Lect. Notes Math. 1317, 224-231 (1988, Zbl 0651.52010)].