Alphabeta Math

Complex Differentiability and the Cauchy–Riemann Equations: Examples and Counterexamples

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✓ 13 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 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Complex Analysis
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

z↦ z² is entire with derivative 2z, directly from the complex difference quotient

Example

The complex exponential satisfies the Cauchy–Riemann equations in Cartesian and polar form

Example

z↦1/z is holomorphic on ℂ∖{0} with derivative -1/z², directly from the difference quotient

Example

A Möbius map (az+b)/(cz+d) with ad-bc≠0 is conformal wherever cz+d≠0

Example

The square map sends the Cartesian grid lines off the coordinate axes to two orthogonal families of parabolas

False statement

FALSE: real differentiability as a map ℝ²→ℝ² implies complex differentiability; conjugation is the counterexample

Counterexample

z↦|z|² is complex differentiable exactly at 0, with derivative 0, but is holomorphic on no neighbourhood of 0

Counterexample

z mapstoRez is real differentiable but nowhere complex differentiable

Counterexample

z↦|z| is nowhere complex differentiable

Example

x³+3xy²+i(y³+3x²y) is complex differentiable exactly on the coordinate axes but holomorphic nowhere

False statement

FALSE: the Cauchy–Riemann equations at one point imply complex differentiability there

False statement

FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy

False statement

FALSE: a holomorphic function with zero derivative on an arbitrary open set is constant

Sources