Evaluation of \(\int_0^{2\pi}\exp(iax+b\cos mx+c\cos nx)\,dx\) and \(\int_0^{2\pi}\exp(iax+b\sin mx+c\cos nx)\,dx\) for \(a,m,n\in\mathbb Z\) and \(b,c\in\mathbb C\) in terms of modified Bessel function of the first kind. (English) Zbl 1142.33001
The authors work out and define integrals involving modified Bessel function of first kind \(I_\nu(x)\). The integral is
\[
\int^{2\pi}_0 \exp[iax+ b\cos mx+ c\cos nx]\,dx
\]
\(\forall a,m,n\in\mathbb{Z}\) and \(b,c\in \mathbb{C}\). Particular cases provide results derived earlier.
Reviewer: Ajendra Nath Srivastava (Puna)
MSC:
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 33C65 | Appell, Horn and Lauricella functions |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
