Improvement of the double exponential formula with conformal maps based on the locations of singularities

Abstract

The double exponential formula, or the DE formula, is a high-precision integration formula using a change of variables called a DE transformation. However, it has a disadvantage that it is sensitive to singularities of an integrand near the real axis, and Slevinsky and Olver (SIAM J. Sci. Comput., 2015) attempted to improve it by constructing conformal maps based on the locations of singularities. In this letter, we construct a new transformation formula based on their ideas. This can be regarded as a generalization of the DE transformations. We confirm the effectiveness of this by a numerical experiment.