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

## TL;DR

This paper introduces a new conformal mapping technique based on Schwarz-Christoffel transformations to enhance the double exponential formula's accuracy near singularities, validated through numerical experiments.

## Contribution

It develops a generalized transformation formula using explicit Schwarz-Christoffel maps to improve the DE formula's handling of singularities.

## Key findings

- Enhanced accuracy in numerical integration near singularities.
- The new transformation generalizes existing DE transformations.
- Numerical experiments confirm improved performance.

## Abstract

The double exponential formula, or the DE formula, is a high-precision integration formula using a change of variables called a DE transformation; whereas there is a disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky and Olver (SIAM J. Sci. Comput., 2015) attempted to improve it 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 transformations for which we can derive their explicit form. Then, the new transformation formula can be regarded as a generalization of the DE transformations. We confirm its effectiveness by numerical experiments.