Research on the double exponential formula for indefinite integrals

双指数变换（简称DE）是1974年由高桥和森喜朗两位首席研究员首次提出的，目的是为了有效地求定积分。到目前为止，它已被应用于各个科学和技术领域，并被纳入几个著名的数学软件，如Mathematica和Maple。另一方面，近年来人们发现DE变换也可以应用于不定积分的有效数值求值。事实上，在2003年，首席研究员和他的小组成员提出了一个将DE变换和Sinc方法（DE-Sinc方法）结合起来的不定积分公式，并在2003年和2005年发表了两篇关于新公式的论文。作为不定积分公式的一个应用，他们提出了一个新的迭代积分有效求值的DE公式，并发表了两篇论文。他们还将DE变换应用于第二类Volterra积分方程的数值解，并于2005年发表了一篇论文。由于给定的常微分方程可以转化为Volterra积分方程，他们提出了一种基于DE-Sinc法数值求解常微分方程初值问题的新方法。此外，他们还将这一思想应用于常微分方程边值问题的求解。2004年9月，在京都大学数学科学研究所举办了一个名为“三十年的双指数变换”的国际研讨会，这是一个很好的机会，使DE变换更受日本以外的人的欢迎。

The double exponential (abbreviated as DE) transformation was first proposed in 1974 by Takahasi and Mori, the head investigator, in order to evaluate definite integrals efficiently. Until now it has come to be used in various fields of science and technology and also incorporated in several famous mathematical softwares such as Mathematica and Maple. On the other hand recently it has been found that the DE transformation can also be applied to efficient numerical evaluation of indefinite integrals. In fact in 2003 the head investigator and people in his group proposed a formula for indefinite integration combining the DE transformation and the Sinc method (DE-Sinc method) and they published two papers on the new formula in 2003 and 2005. As an application of the formula for indefinite integrals they proposed a new DE formula for efficient evaluation of iterated integrals and published also two papers. Also they applied the DE transformation to numerical solution of Volterra integral equations of the second kind and published a paper in 2005. Since a given ordinary differential equation can be transformed into a Volterra integral equation they proposed a new method based on the DE-Sinc method to solve numerically the initial value problem of ordinary differential equation. Furthermore they applied their idea to the solution of boundary value problems of ordinary differential equations. In September 2004 an international workshop titled "Thirty years of the double exponential transforms' was held in Research Institute for Mathematical Sciences, Kyoto University and it was a good opportunity to make the DE transformation more popular also to people from outside Japan.