Alphabeta MathAnalytic Majorants and the Cauchy–Kovalevskaya Theorem
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Variation and the Riemann–Stieltjes Integral
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Contour Integration
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Holomorphic Functions of Several Complex Variables
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Partial Differential Equations and Characteristics
- Partitions of Unity and Paracompactness
- pi: the Equivalent Characterizations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The Winding Number and the Global Cauchy Theorem
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Real analytic germs in several variables
Definition
Coefficientwise majorisation
Lemma
Operations preserving coefficient majorisation
Lemma
Geometric majorants for analytic germs
Theorem
Real analytic inverse and implicit functions
Lemma
Analytic ODE systems from majorants
Lemma
Analytic flattening and the normal principal coefficient
Lemma
Subtracting analytic Cauchy jets
Lemma
Formal recursion for solved analytic normal equations
Lemma
Positive majorants dominate the Cauchy recursion
Lemma
Convergence of the Goursat majorant
Theorem
Cauchy–Kovalevskaya for first-order analytic systems
Lemma
Reduction of higher-order normal form with jet compatibility
Theorem
Cauchy–Kovalevskaya on a noncharacteristic analytic hypersurface
Remark