Computation of the orders of singularity of sectionally analytic functions (Q1186989)

Functions of the following form are considered: \(\Phi(z)=(z- a)^{\gamma_ 1}(z-b)^{\gamma_ 2}F(z)\), where the orders of singularity \(\gamma_ 1\) and \(\gamma_ 2\) at the tips \(a\) and \(b\) of the arc of discontinuity \(L\) satisfy the relation \(\gamma_ 1+\gamma_ 2=-1\). The orders of singularity in question are computed by numerical evaluation of a complex contour integral of the logarithmic derivative \(\Phi'/\Phi\); the contour contains \(L\). The estimation of the integrals of \(z^ k\Phi'/\Phi\) improves the numerical precision. The convergence is rapid.