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Available4.7
10 reviews
ISBN 10: 3110597853
ISBN 13: 9783110597851
Author: Xiao Xiong Gan
Formal analysis is the study of formal power series, formal Laurent series, formal root series, and other formal series or formal functionals. This book is the first comprehensive presentation of the topic that systematically introduces formal analysis, including its algebraic, analytic, and topological structure, along with various applications.
1 Basics of formal power series
1.1 Basic algebraic operations
1.2 Composition of formal power series with nonunit
1.3 Right distributive law for composition
1.4 The matrix representation of formal power series
1.5 Almost unit formal power series
1.6 Algebraic structure of π
1.7 Analytic structure of π
2 Calculus of formal power series
2.1 Formal derivatives of formal power series
2.2 Formal differential equations and their applications
2.3 Ultrametric
2.4 Some topological structure for π
2.5 The umbral calculus and formal power series
3 Applications of formal power series, I
3.1 Riordan group and formal power series
3.2 Riordan involutions
3.3 Some subgroups of the Riordan group
3.4 Cayley–Hamilton theorem and Fermat’s little theorem
4 Applications of formal power series, II
4.1 Formal power series and some differential equations
4.2 Functional equations and formal power series
4.3 Lagrange inversion
4.4 Difference equations and formal power series
5 General composition
5.1 Introduction
5.2 Matrix representation of the general composition, I
5.3 Coefficients of fn(z)
5.4 The general composition theorem
5.5 The general chain rule
5.6 The general right distributive law
5.7 Matrix representation of the general composition, II
6 Formal analysis and classical analysis
6.1 Introduction to the boundary behavior
6.2 Boundary behavior and formal analysis
6.3 Banach spaces βp(β) and βp(β)
6.4 General formal logarithm
7 Formal Laurent series
7.1 Semiformal Laurent series
7.2 Reversed semiformal Laurent series
7.3 Basic algebra of formal Laurent series
7.4 Composition of formal Laurent series and formal power series
7.5 Canonical mapping and dot product of formal Laurent series
7.6 Canonical mapping and multiplication of formal Laurent series
7.7 Topological spaces of formal Laurent series
8 Iteration and iterative roots
8.1 Conjugacy of formal power series in π»
8.2 The iteration of formal power series
8.3 The iterative roots of formal series in π»
8.4 Schröder’s problem
9 Formal series and general exponents
9.1 Introduction
9.2 Formal root series and its algorithm
9.3 Power of formal power series with rational exponents
9.4 The real exponents of formal power series
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Tags: Xiao Xiong Gan, Analysis, Formal
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