Introduction to multipoints summation method Modern applied mathematics that connects here and the infinite beyond

This English translation version book is the world's first introductory and technical book on the multi-point summation method and the method of function analysis read by the author. It describes the method developed by the author when I obtained my doctorate from the Graduate School of Science, Osaka University, and its further development and application.
Beginning with a review of necessary middle school and high school mathematics, we will first explain the Padé approximant from the Taylor development, which is the content of university mathematics. As a result, the target audience ranges from aspiring middle school students to high school students, non-specialists, and professionals.
Asymptotic expansions such as Taylor expansions and Padé approximations are function analysis methods that incorporate one point of information. The two-point Padé approximation and the two-point Borel-Pade analysis, which extend this and incorporate two-point information, are kindly and carefully introduced. Asymptotic analysis will show that there is no need to truncate on a particular term.
It also touches on how to extend the multipoint summation method by incorporating information about singular points.
Specific examples of using the multi-point summation method are shown for a large number of examples such as trigonometric functions, hyperbolic functions, Γ functions, and Lambert W functions. Therefore, through this book, you will gain experience in actually dealing with multi-point summation methods.
Also, as applications, it also touches on the approximate solution of differential equations and the approximate analysis of the zeros of the Riemann zeta function.