Research on the double exponential formula for indefinite integrals

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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.