Alphabeta Math

Complex Differentiability and the Cauchy–Riemann Equations

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✓ 21 results · all verified · 14 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Complex Analysis
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Remark

ℂ=ℝ[x]/(x²+1) as the Euclidean plane and as a normed real algebra: what the identification preserves

Definition

A complex domain is a nonempty connected open subset of ℂ

Definition

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Lemma

The complex derivative at a point is unique

Definition

The Wirtinger derivatives ∂_z f and ∂_z̄f, and antiholomorphic functions

Theorem

Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations

Corollary

Complex differentiability at a point implies continuity there

Theorem

Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set

Corollary

Cartesian and polar forms of the Cauchy–Riemann equations agree away from the origin

Theorem

Linearity, product, reciprocal, and quotient rules for complex derivatives

Theorem

The chain rule for complex derivatives

Theorem

The Wirtinger chain rule for compositions of real-differentiable complex-valued maps

Theorem

A conjugate difference quotient characterizes antiholomorphic maps

Corollary

A holomorphic function with continuous complex derivative has C¹ real and imaginary components

Theorem

A continuous local inverse has derivative reciprocal to a nonzero complex derivative

Theorem

Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero

Theorem

The complex exponential is entire and its complex derivative is itself

Theorem

A holomorphic function with zero derivative on a domain is constant

Corollary

A real-valued holomorphic function on a domain is constant

Corollary

A holomorphic function of constant modulus on a domain is constant

Corollary

If both f and f̄ are holomorphic on a domain, then f is constant

Definition

Orientation-preserving conformality for a real-differentiable complex map at a point

Lemma

Plane similarities are complex or conjugate-complex multiplications; the orientation-preserving ones are exactly the nonzero complex multiplications

Theorem

A real-differentiable complex map is orientation-preserving conformal at a point exactly when it is complex differentiable there with nonzero derivative

Corollary

The Jacobian determinant of a holomorphic map is |f'|² and is positive exactly where f'≠0

Theorem

The C² real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair

Theorem

If a holomorphic function has C² components, then its derivative is holomorphic

Corollary

The Hessian determinant of each C² holomorphic component is nonpositive; a nondegenerate critical point is a saddle

Sources