Alphabeta MathContour Integration
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 11 results · all verified · 5 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
Linear Algebra
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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- 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 Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Remark
Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries
Definition
Rectifiable complex contours, reversal, concatenation, closedness, and orientation
Definition
The complex line integral over a rectifiable path as a componentwise Riemann–Stieltjes integral
Definition
The absolute line integral over a rectifiable path using its arc-length function
Theorem
Continuous integrands have complex and absolute line integrals along every rectifiable path
Theorem
For piecewise-C1 contours the Riemann–Stieltjes integral agrees with the parametric complex integral and the published real line integrals
Proposition
Complex line integrals are linear in the integrand
Proposition
Complex line integrals change sign under reversal and add under concatenation
Theorem
Complex and absolute line integrals are invariant under increasing continuous reparametrization
Theorem
The fundamental inequality: the modulus of the integral is at most the absolute line integral for rectifiable contours
Corollary
ML estimate: a contour integral is bounded by a supremum bound times path length
Corollary
The absolute line integral of the constant function 1 is the length of the path
Definition
A primitive of a complex function on an open set
Theorem
The line integral of a continuous function admitting a primitive is that primitive's endpoint increment along every rectifiable path
Corollary
The integral of a continuous complex derivative over every closed rectifiable contour is zero
Corollary
The contour integral of a constant c is c times the endpoint displacement
Theorem
For a continuous function on a complex domain, endpoint independence, zero closed-contour integrals, and existence of a primitive are equivalent
Theorem
A uniformly convergent sequence of continuous integrands on a fixed contour permits passage of the limit through the complex line integral
Theorem
On a positively oriented circle about a, the integral of (z-a)ᵐ is zero for every integer m except -1, and is 2π i for m=-1
Corollary
The normalized integral around a positively oriented circle centred at a is 1
False statement