#analysis
58. Global Analysis, Analysis on Manifolds
This volume studies analysis on manifolds, combining differential geometry, functional analysis, and partial differential equations. It focuses on global properties of differential operators and geometric structures. Part I. Foundations Chapter 1. Manifolds and Smooth Structures 1.1 Smooth manifolds 1.2 Charts and atlases 1.3 Smooth maps 1.4 Examples 1.5 Basic properties Chapter 2. Vector Bundles 2.1 Definitions 2.2 Sections 2.3 Bundle maps 2.4 Examples 2.5 Applications Chapter 3. Differential Operators 3.1...
49. Calculus of Variations and Optimal Control; Optimization
This volume studies optimization of functionals and systems. It connects analysis, geometry, control theory, and computation. Part I. Foundations of Optimization Chapter 1. Optimization Problems 1.1 Definitions and examples 1.2 Objective functions and constraints 1.3 Local and global optima 1.4 Existence of solutions 1.5 Examples Chapter 2. Convexity 2.1 Convex sets 2.2 Convex functions 2.3 Properties 2.4 Jensen’s inequality 2.5 Applications Chapter 3. Optimality Conditions 3.1 First-order conditions 3.2 Second-order...
44. Integral Transforms, Operational Calculus
This volume studies integral transforms as tools for solving equations, analyzing signals, and transforming problems into more tractable forms. It emphasizes both theoretical properties and computational methods. Part I. Foundations Chapter 1. Integral Transforms 1.1 Definitions and examples 1.2 Kernel functions 1.3 Linearity and basic properties 1.4 Inversion problems 1.5 Applications overview Chapter 2. Operational Calculus 2.1 Algebraic manipulation of operators 2.2 Differential operators 2.3 Integral operators 2.4 Symbolic methods...
45. Integral Equations
This volume studies equations where the unknown function appears under an integral. It connects functional analysis, differential equations, and applied mathematics. Part I. Foundations Chapter 1. Introduction to Integral Equations 1.1 Definitions and examples 1.2 Types: Fredholm and Volterra 1.3 Linear vs nonlinear equations 1.4 Kernel functions 1.5 Applications overview Chapter 2. Function Spaces 2.1 Spaces of continuous functions 2.2 Lp spaces 2.3 Norms and metrics 2.4 Completeness 2.5 Examples...
46. Functional Analysis
This volume studies infinite-dimensional vector spaces and linear operators. It provides the analytical framework underlying modern analysis, PDEs, probability, and quantum theory. Part I. Normed and Banach Spaces Chapter 1. Normed Vector Spaces 1.1 Definitions and examples 1.2 Norms and metrics 1.3 Convergence 1.4 Continuous linear maps 1.5 Examples Chapter 2. Banach Spaces 2.1 Completeness 2.2 Examples: ℓᵖ, Lᵖ 2.3 Subspaces and quotients 2.4 Linear operators 2.5 Applications Chapter 3....
43. Abstract Harmonic Analysis
This volume extends harmonic analysis to general locally compact groups. It replaces classical Fourier series and transforms with representation-theoretic and measure-theoretic frameworks. Part I. Locally Compact Groups Chapter 1. Topological Groups 1.1 Definitions and examples 1.2 Continuity and group operations 1.3 Subgroups and quotient groups 1.4 Compact and locally compact groups 1.5 Examples Chapter 2. Haar Measure 2.1 Existence and uniqueness 2.2 Left and right invariance 2.3 Integration on groups...
47. Operator Theory
This volume studies linear operators on Banach and Hilbert spaces, with emphasis on spectral properties, structure, and applications in analysis and physics. Part I. Foundations Chapter 1. Linear Operators 1.1 Definitions and examples 1.2 Domains and ranges 1.3 Bounded vs unbounded operators 1.4 Operator norms 1.5 Basic properties Chapter 2. Classes of Operators 2.1 Bounded operators 2.2 Compact operators 2.3 Self-adjoint operators 2.4 Normal operators 2.5 Examples Chapter 3. Operator...
41. Approximations and Expansions
This volume studies approximation of functions and data by simpler objects such as polynomials, splines, and rational functions. It develops both theoretical bounds and computational methods. Part I. Foundations of Approximation Chapter 1. Approximation Problems 1.1 Function approximation setup 1.2 Norms and error measures 1.3 Best approximation 1.4 Existence and uniqueness 1.5 Examples Chapter 2. Normed Function Spaces 2.1 Metric and normed spaces 2.2 Uniform norm 2.3 Lp norms 2.4...
35. Partial Differential Equations
This volume studies equations involving partial derivatives of functions in several variables. It develops theory, methods, and applications across analysis, physics, and geometry. Part I. Foundations Chapter 1. Introduction to PDEs 1.1 Definitions and examples 1.2 Classification: elliptic, parabolic, hyperbolic 1.3 Linear vs nonlinear equations 1.4 Initial and boundary conditions 1.5 Modeling contexts Chapter 2. First-Order PDEs 2.1 Linear equations 2.2 Method of characteristics 2.3 Nonlinear first-order equations 2.4 Hamilton–Jacobi...
40. Sequences, Series, Summability
This volume studies convergence, divergence, and summation methods for sequences and series. It extends classical analysis with refined convergence concepts and summability techniques. Part I. Sequences Chapter 1. Basic Properties of Sequences 1.1 Definitions and examples 1.2 Convergence and divergence 1.3 Boundedness and monotonicity 1.4 Subsequences 1.5 Limit superior and inferior Chapter 2. Cauchy Sequences 2.1 Definition 2.2 Completeness of ℝ 2.3 Equivalence with convergence 2.4 Examples 2.5 Applications Chapter...
42. Harmonic Analysis
This volume studies representation of functions via oscillatory components such as Fourier series and transforms. It connects analysis, PDEs, number theory, and signal processing. Part I. Fourier Analysis on ℝ Chapter 1. Fourier Transform 1.1 Definition 1.2 Basic properties 1.3 Inversion formula 1.4 Plancherel theorem 1.5 Examples Chapter 2. Convolution 2.1 Definition 2.2 Properties 2.3 Convolution theorem 2.4 Approximate identities 2.5 Applications Chapter 3. Distributions (Overview) 3.1 Generalized functions 3.2...
33. Special Functions
This volume studies classical and modern special functions arising as solutions to differential equations, integral transforms, and representation theory. It emphasizes structure, identities, and computational aspects. Part I. Foundations Chapter 1. What Are Special Functions 1.1 Definition and scope 1.2 Historical origins 1.3 Functions as solutions to equations 1.4 Orthogonality and completeness 1.5 Examples Chapter 2. Series and Integral Representations 2.1 Power series definitions 2.2 Integral representations 2.3 Generating functions...
39. Difference and Functional Equations
This volume studies equations defined by discrete steps and functional relations. It complements differential equations by focusing on iteration, recursion, and functional identities. Part I. Difference Equations Chapter 1. Discrete Dynamical Systems 1.1 Sequences as dynamical systems 1.2 Iteration of maps 1.3 Fixed points 1.4 Stability 1.5 Examples Chapter 2. First-Order Difference Equations 2.1 Linear equations 2.2 Nonlinear equations 2.3 Explicit solutions 2.4 Recurrence relations 2.5 Applications Chapter 3. Higher-Order...
34. Ordinary Differential Equations
This volume studies differential equations involving functions of a single variable. It develops existence theory, qualitative behavior, and analytical and numerical methods. Part I. First-Order Equations Chapter 1. Basic Concepts 1.1 Definitions and examples 1.2 Solutions and integral curves 1.3 Initial value problems 1.4 Direction fields 1.5 Modeling examples Chapter 2. Separable and Exact Equations 2.1 Separable equations 2.2 Exact equations 2.3 Integrating factors 2.4 Applications 2.5 Examples Chapter 3....
37. Dynamical Systems and Ergodic Theory
This volume studies systems that evolve over time, focusing on long-term behavior, stability, and statistical properties. It connects analysis, topology, geometry, and probability. Part I. Foundations of Dynamical Systems Chapter 1. Dynamical Systems 1.1 Discrete vs continuous systems 1.2 State space and evolution rules 1.3 Examples from physics and biology 1.4 Orbits and trajectories 1.5 Invariant sets Chapter 2. Flows and Maps 2.1 Iterated maps 2.2 Continuous flows 2.3 Fixed...
28. Measure and Integration
This volume develops measure theory and integration in a general setting. It extends classical calculus to more flexible notions of size, convergence, and integrability. Part I. Measure Spaces Chapter 1. Sigma-Algebras 1.1 Definitions and examples 1.2 Generated sigma-algebras 1.3 Measurable sets 1.4 Borel sets 1.5 Constructions Chapter 2. Measures 2.1 Definition of a measure 2.2 Finite and sigma-finite measures 2.3 Counting and Lebesgue measures 2.4 Measure properties 2.5 Examples Chapter...
26. Real Functions
This volume studies functions of real variables with an emphasis on limits, continuity, differentiation, integration, and fine properties of functions. It builds the rigorous core of real analysis. Part I. Real Numbers and Limits Chapter 1. The Real Number System 1.1 Ordered fields 1.2 Completeness axiom 1.3 Suprema and infima 1.4 Sequences in ℝ 1.5 Convergence Chapter 2. Limits of Sequences 2.1 Definition of limit 2.2 Limit laws 2.3 Monotone...
32. Several Complex Variables and Analytic Spaces
This volume studies functions of several complex variables, complex manifolds, and analytic spaces. It extends complex analysis into higher dimensions, where geometry and sheaf methods become central. Part I. Functions of Several Complex Variables Chapter 1. Complex Euclidean Space 1.1 Coordinates in ℂⁿ 1.2 Domains and open sets 1.3 Holomorphic functions 1.4 Complex differentiability 1.5 Examples Chapter 2. Power Series 2.1 Several-variable power series 2.2 Domains of convergence 2.3 Holomorphic...
30. Complex Analysis
This volume studies functions of a complex variable. It develops analyticity, contour integration, and the rich structure that arises from complex differentiability. Part I. Complex Numbers and Functions Chapter 1. Complex Numbers 1.1 Algebra of complex numbers 1.2 Geometry in the complex plane 1.3 Polar form and Euler’s formula 1.4 Roots of unity 1.5 Basic properties Chapter 2. Complex Functions 2.1 Functions of a complex variable 2.2 Limits and continuity...
31. Potential Theory
This volume studies harmonic, subharmonic, and superharmonic functions, along with potentials and their applications to analysis, geometry, and physics. It connects closely with partial differential equations and complex analysis. Part I. Foundations Chapter 1. Harmonic Functions 1.1 Definition via Laplace equation 1.2 Mean value property 1.3 Maximum and minimum principles 1.4 Uniqueness results 1.5 Examples Chapter 2. Subharmonic and Superharmonic Functions 2.1 Definitions 2.2 Basic properties 2.3 Comparison principles 2.4...