Infinite series and continued fractions

Author: Gao Jianfu
Publisher:
Publish Date: 2005-12-01
Features: This book systematically introduces the technical tool role of infinite series in mathematics and the construction of Minkowski functions using continued fraction analytical theory, as well as its extension to the complex domain. It particularly discusses and demonstrates several summability methods of infinite divergent series in a practical manner. Of course, this reflects the mathematical thinking of the book. Chapter 1 mainly introduces the technical tool role of infinite convergent series in classical and modern mathematics, while Chapter 2 focuses on numerical calculations using infinite divergent series as asymptotic series for certain functions and solving numerical solutions of differential equations. It also elucidates various summability methods of infinite divergent series to varying degrees. Chapter 3 discusses the relationship between continued fractions and infinite series, as well as the analytical theory of continued fractions. Chapter 4 applies its analytical theory of continued fractions, particularly Denjoy's lemma, to construct Minkowski functions, which exhibit distinct characteristics. Incidentally, it analytically extends this function to a certain region of the complex plane, providing a general representation form.