Cauchy integral formula
E241728
The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Cauchy integral formula canonical | 21 |
| Cauchy estimates | 3 |
| Cauchy integral | 1 |
| Cauchy integral formula for derivatives | 1 |
| Cauchy integral formula for higher derivatives | 1 |
| Cauchy integral formula when n = 1 | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2171645 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Cauchy integral formula Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy integral formula]
-
A.
Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
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B.
Cauchy residue theorem
The Cauchy residue theorem is a fundamental result in complex analysis that relates contour integrals of analytic functions around singularities to the sum of their residues, greatly simplifying the evaluation of many complex and real integrals.
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C.
Cauchy principal value
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
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D.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
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E.
Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Cauchy integral formula Target entity description: The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
-
A.
Cauchy integral theorem
The Cauchy integral theorem is a fundamental result in complex analysis stating that the integral of a holomorphic function over any closed contour in a simply connected domain is zero.
-
B.
Cauchy residue theorem
The Cauchy residue theorem is a fundamental result in complex analysis that relates contour integrals of analytic functions around singularities to the sum of their residues, greatly simplifying the evaluation of many complex and real integrals.
-
C.
Cauchy principal value
The Cauchy principal value is a method in mathematical analysis for assigning finite values to certain improper or divergent integrals and series by symmetrically balancing their singularities.
-
D.
Riemann mapping theorem
The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
-
E.
Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in complex function theory
ⓘ
theorem in complex analysis ⓘ |
| appliesTo | functions holomorphic on an open set containing a closed disk and its boundary ⓘ |
| assumes |
function is holomorphic on and inside the contour
ⓘ
point lies in the interior of the contour ⓘ |
| category | complex integration theorems ⓘ |
| componentOf |
standard curriculum in graduate complex analysis
ⓘ
standard curriculum in undergraduate complex analysis ⓘ |
| domain |
analytic functions
ⓘ
holomorphic functions ⓘ |
| expresses | f(z0) as an integral of f over a contour enclosing z0 ⓘ |
| field | complex analysis ⓘ |
| hasConsequence |
mean value property for holomorphic functions
ⓘ
removable singularity theorem (via related arguments) ⓘ |
| hasGeneralization |
Cauchy integral formula for derivatives
ⓘ
linked to:
Cauchy integral formula
Cauchy integral formula for higher derivatives ⓘ
linked to:
Cauchy integral formula
Cauchy integral formula in several complex variables ⓘ
linked to:
Bochner–Martinelli formula
Cauchy integral theorem ⓘ Cauchy–Green formula ⓘ
linked to:
Bochner–Martinelli formula
Cauchy–Pompeiu formula ⓘ |
| historicalPeriod | 19th-century mathematics ⓘ |
| holdsIn | simply connected domains (with appropriate hypotheses) ⓘ |
| implies |
Cauchy estimates
ⓘ
Liouville's theorem ⓘ holomorphic functions are analytic ⓘ holomorphic functions are infinitely differentiable ⓘ identity theorem for holomorphic functions ⓘ maximum modulus principle ⓘ |
| involves |
closed curves in the complex plane
ⓘ
contour integrals ⓘ simple closed positively oriented contours ⓘ |
| language | mathematical notation ⓘ |
| mathematicalExpression | f(z0) = (1/(2πi)) ∮_Γ f(z)/(z−z0) dz ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| relatedTo |
Cauchy–Riemann equations
ⓘ
Morera's theorem ⓘ Taylor series in the complex plane ⓘ residue theorem ⓘ
linked to:
Cauchy residue theorem
|
| relates | values of a holomorphic function inside a contour to values on the contour ⓘ |
| requires |
orientation of the contour
ⓘ
positively oriented boundary of a disk or region ⓘ |
| statement | If f is holomorphic on an open set containing a simple closed contour Γ and its interior, then for any z0 in the interior, f(z0) = (1/(2πi)) ∮_Γ f(z)/(z−z0) dz ⓘ |
| usedFor |
bounding derivatives of holomorphic functions
ⓘ
computing Taylor coefficients of holomorphic functions ⓘ deriving power series expansions ⓘ evaluating contour integrals ⓘ proving uniqueness of analytic continuation ⓘ solving problems in mathematical physics involving analytic 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 integral formula Description of subject: The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
Referenced by (28)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Augustin-Louis
linked to: Cauchy integral formula
linked to: Cauchy integral formula
linked to: Cauchy integral formula
subject linked to:
Montel theorem
linked to: Cauchy integral formula
linked to: Cauchy integral formula
Le calcul des résidus et ses applications à la théorie des fonctions
→
usesConcept
→
Cauchy integral formula
ⓘ
subject linked to:
Complex Analysis
linked to: Cauchy integral formula
subject linked to:
Functions of One Complex Variable by John B. Conway
subject linked to:
Liouville's theorem
subject linked to:
Liouville's theorem
linked to: Cauchy integral formula
subject linked to:
Liouville's theorem