Alphabeta MathJensen Theory and Nevanlinna's First Main Theorem: Examples and Counterexamples
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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
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- 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
- Holomorphic Functions of Several Complex Variables
- Isolated Singularities and Laurent Series
- Jensen Theory and Nevanlinna's First Main Theorem
- The Gamma Function
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Vector Spaces, Linear Subspaces, Span and Direct Sums
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
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- 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 Real Gamma and Beta Functions
- 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
Example
A repeated zero contributes its Green kernel twice
Example
A centre a-point requires regularised counting
Example
Characteristics of a monomial, an exponential and a tangent
Example
Characteristic under a target Möbius change
Example
Reciprocal Gamma has order one and characteristic of size r log r
Example
Rational degree appears as logarithmic characteristic
Sources
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §1
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §1, equation (2) and footnotes 1–2
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §§2,4
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §2, Exercise 2*
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §6, Theorems 6.1–6.2 and Corollary (6.26)
- Alexandre Eremenko, Lectures on Nevanlinna Theory, §§1–3
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §6, equation (6.8)
- Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 2 §5, reciprocal-Gamma exercise, printed pp. 80–81
- K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 7 §6, Stirling formula, printed pp. 60–62
- 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)