Cauchy–Riemann equations

E239285

mathematical condition system of partial differential equations

The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).

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All labels observed (1)

Label	Occurrences
Cauchy–Riemann equations canonical	4

Statements (47)

Predicate	Object
instanceOf	mathematical condition ⓘ system of partial differential equations ⓘ
alternativeForm	polar coordinates ⓘ
appliesTo	complex-valued functions of a complex variable ⓘ
characterizes	complex differentiable functions ⓘ holomorphic functions ⓘ
conditionType	necessary condition for complex differentiability ⓘ sufficient condition for complex differentiability under mild regularity assumptions ⓘ
coordinateForm	cartesian coordinates ⓘ
domainVariable	z = x + i y ⓘ
ensures	infinite real differentiability of holomorphic functions ⓘ local power series expansion of holomorphic functions ⓘ real analyticity of holomorphic functions ⓘ
equivalentTo	existence of complex derivative at a point with continuity in a neighborhood ⓘ
failsFor	absolute value map z ↦ \|z\| ⓘ complex conjugation map z ↦ z̄ ⓘ
field	complex analysis ⓘ
generalization	CR-structures in several complex variables ⓘ Cauchy–Riemann–Fueter equations in quaternionic analysis ⓘ
historicalPeriod	19th century mathematics ⓘ
imaginaryPartNotation	v(x,y) ⓘ
implies	conformality at noncritical points ⓘ direction-independent complex derivative ⓘ harmonicity of imaginary part ⓘ harmonicity of real part ⓘ
mathematicalContext	functions from open subsets of ℂ to ℂ ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ Bernhard Riemann ⓘ
polarForm	∂u/∂r = (1/r) ∂v/∂θ ⓘ ∂v/∂r = −(1/r) ∂u/∂θ ⓘ
realPartNotation	u(x,y) ⓘ
regularityAssumption	continuity of first partial derivatives ⓘ
relatedConcept	Wirtinger derivatives ⓘ
relatedTo	Laplace equation ⓘ analytic functions ⓘ conformal mappings ⓘ
requires	real differentiability of component functions ⓘ
role	criterion for analyticity ⓘ foundational tool in complex function theory ⓘ
standardForm	∂u/∂x = ∂v/∂y ⓘ ∂u/∂y = −∂v/∂x ⓘ
usedIn	complex potential theory ⓘ proofs of analyticity of power series ⓘ two-dimensional elasticity theory ⓘ two-dimensional electrostatics ⓘ two-dimensional fluid dynamics ⓘ
wirtingerForm	∂f/∂z̄ = 0 for holomorphic functions ⓘ

How these facts were elicited

The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.

Instruction

You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10.

# Requirements
- If you don't know the subject at all, return an empty list.
- If the subject is not a named entity, return an empty list.
- Include at least one triple where predicate is "instanceOf".
- Do not get too wordy.
- Separate several objects into multiple triples with one object.

Input

Subject: Cauchy–Riemann equations
Description of subject: The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Augustin-Louis Cauchy → knownFor → Cauchy–Riemann equations ⓘ

Augustin-Louis → notableFor → Cauchy–Riemann equations ⓘ

subject surface form: Augustin-Louis Cauchy

Cauchy integral formula → relatedTo → Cauchy–Riemann equations ⓘ

Differential Analysis on Complex Manifolds → topic → Cauchy–Riemann equations ⓘ