Alphabeta MathJensen Theory and Nevanlinna's First Main Theorem
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Complex Analysis
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Harmonic Functions and the Poisson Integral
- Isolated Singularities and Laurent Series
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Linear Algebra
Measure Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- 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 Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Theorem
Poisson–Jensen formula for a meromorphic function on a disc
Definition
Counting, chordal proximity and characteristic
Theorem
Well-definedness and radius conventions for Nevanlinna quantities
Lemma
Meromorphic Jensen identity with a zero or pole at the centre
Theorem
Ahlfors–Shimizu area form of the characteristic
Theorem
Nevanlinna’s First Main Theorem with exact centre constant
Theorem
Elementary characteristic laws and fixed rational composition
Definition
Order and lower order from the Nevanlinna characteristic
Proposition
Entire-function order agrees with maximum-modulus order
Theorem
Rational functions are exactly those with logarithmic characteristic
Sources
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §1
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §§1–2
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §§1–3
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §§1–2, 4, 6–7; Ch. 2 §§1, 5
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §4
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §2
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §3, equations (9)–(12)
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §2, Theorem 2.6; §4, Theorem 4.2
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §1, equation (2), and §3, equations (9)–(14)
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §4, Theorem 4.1 and proof
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §2, algebraic properties (a)–(d)
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §6, equations (6.1)–(6.8), Theorems 6.1–6.2
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 2 §1, definition of order and lower order
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §7, Theorem 7.1; Ch. 2 §1, Theorem 1.3
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §2, Exercise 1
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §6, Theorem 6.1 and Corollary (6.26); §7, Theorem 7.1