Construction of conformal maps based on the locations of singularities for improving the double exponential formula

The double exponential formula, or DE formula, is a high-precision integration formula using a change of variables called a DE transformation; it has the disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky & Olver (2015, On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. SIAM J. Sci. Comput., 37, A676–A700) attempted to improve the formula by constructing conformal maps based on the locations of singularities. Based on their ideas, we construct a new transformation formula. Our method employs special types of the Schwarz–Christoffel transformation for which we can derive the explicit form. The new transformation formula can be regarded as a generalization of DE transformations. We confirm its effectiveness by numerical experiments.