Bochner–Martinelli formula
E613405
The Bochner–Martinelli formula is a fundamental integral representation in several complex variables that generalizes the Cauchy integral formula to higher dimensions.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Cauchy integral formula in several complex variables | 1 |
| Cauchy–Green formula | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
integral representation formula
ⓘ
mathematical formula ⓘ result in several complex variables ⓘ |
| appearsIn | advanced textbooks on several complex variables ⓘ |
| appliesTo |
domains in \mathbb{C}^n
ⓘ
holomorphic functions of several complex variables ⓘ |
| assumptionOnFunction | holomorphic in the interior of the domain ⓘ |
| category | theorem in complex analysis ⓘ |
| component |
d\sigma(\zeta) surface measure element
ⓘ
factor (n-1)! / (2\pi i)^n ⓘ |
| dimension | n \ge 1 ⓘ |
| domainCondition | bounded domain in \mathbb{C}^n with sufficiently smooth boundary ⓘ |
| expresses | function value f(z) as an integral over the boundary involving f(\zeta) ⓘ |
| field |
complex analysis
ⓘ
several complex variables ⓘ |
| generalizes | Cauchy integral formula NERFINISHED ⓘ |
| gives |
boundary integral representation
ⓘ
integral representation of holomorphic functions ⓘ |
| hasPrerequisite |
Cauchy integral formula
NERFINISHED
ⓘ
Stokes' theorem NERFINISHED ⓘ differential forms on \mathbb{C}^n ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies | Cauchy integral formula when n = 1 NERFINISHED ⓘ |
| integralType | surface integral ⓘ |
| integratesOver | boundary of the domain ⓘ |
| kernelProperty |
(n,n-1)-form
ⓘ
Cauchy–Fantappiè type kernel ⓘ |
| kernelType | Bochner–Martinelli kernel NERFINISHED ⓘ |
| mathematicalArea |
analysis
ⓘ
complex geometry ⓘ |
| namedAfter |
Enzo Martinelli
NERFINISHED
ⓘ
Salomon Bochner NERFINISHED ⓘ |
| property | invariant under biholomorphic changes of coordinates (up to natural factors) ⓘ |
| relatedConcept |
Cauchy–Fantappiè formula
NERFINISHED
ⓘ
Henkin kernel NERFINISHED ⓘ integral representation in complex manifolds ⓘ |
| relates | values of a holomorphic function inside a domain to its values on the boundary ⓘ |
| requires | orientation of the boundary of the domain ⓘ |
| requiresFunctionClass | C^1 functions on the closure of the domain ⓘ |
| role | fundamental tool in higher-dimensional complex analysis ⓘ |
| topicOf | research in several complex variables ⓘ |
| usedFor |
deriving estimates for holomorphic functions
ⓘ
extension problems in several complex variables ⓘ representation of solutions to boundary value problems ⓘ solving \bar{\partial}-equations ⓘ |
| variable |
\zeta \in \mathbb{C}^n
ⓘ
z \in \mathbb{C}^n ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cauchy–Green formula
this entity surface form:
Cauchy integral formula in several complex variables