Evaluation of four irrational definite sine integrals using residue theory (Q913049)

The integrals are \[ \int^{2\pi}_{0}(a+\sin \sigma)^{\pm} d\sigma \quad (a>1),\quad and\quad \] \[ \int^{2\pi}_{0}(a+\sin^ 2 \sigma)^{\pm} d\sigma \quad (a>0). \] The integrals, which are equivalent to the complete elliptic integrals of the second and first kinds, are evaluated as power series in \(a^{-2}\) and \((a+)^{-2}\) respectively. The integrands are written as functions of a complex variable which are expanded by the binomial series. The coefficients are calculated by residue theory. Comparisons are made with the aid of numerical quadrature (Simpson's rule) when \(a=2\).