Wirtinger derivatives
E825427
Wirtinger derivatives are complex differential operators that treat a complex variable and its conjugate as independent, providing a convenient formalism for expressing and analyzing holomorphicity and the Cauchy–Riemann equations.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
complex differential operator
ⓘ
mathematical concept ⓘ |
| advantage |
compact notation for complex differentiation
ⓘ
separate handling of holomorphic and antiholomorphic parts ⓘ |
| appliesTo |
complex-valued functions of complex variables
ⓘ
vector-valued complex functions ⓘ |
| assumeDecomposition | z = x + i y ⓘ |
| assumeVariables | x and y real ⓘ |
| category | differential operators on \mathbb{C} ⓘ |
| characterizesHolomorphicityBy | holomorphic iff \partial f / \partial \bar{z} = 0 ⓘ |
| definedOn | functions of a complex variable ⓘ |
| definition |
\partial / \partial \bar{z} = \tfrac12(\partial / \partial x + i\,\partial / \partial y)
ⓘ
\partial / \partial z = \tfrac12(\partial / \partial x - i\,\partial / \partial y) ⓘ |
| equivalentTo | Cauchy–Riemann equations in two real variables ⓘ |
| field | complex analysis ⓘ |
| framework | treats z and \bar{z} as formally independent ⓘ |
| generalizationDefinition |
\partial / \partial \bar{z}_j = \tfrac12(\partial / \partial x_j + i\,\partial / \partial y_j)
ⓘ
\partial / \partial z_j = \tfrac12(\partial / \partial x_j - i\,\partial / \partial y_j) ⓘ |
| generalizesTo | several complex variables z_1,\dots,z_n ⓘ |
| mathematicalDomain | analysis ⓘ |
| namedAfter | Wilhelm Wirtinger NERFINISHED ⓘ |
| notation |
\partial / \partial \bar{z}
ⓘ
\partial / \partial z ⓘ |
| property |
\partial \bar{z} / \partial \bar{z} = 1
ⓘ
\partial \bar{z} / \partial z = 0 ⓘ \partial z / \partial \bar{z} = 0 ⓘ \partial z / \partial z = 1 ⓘ |
| relatedConcept |
Cauchy–Riemann operator
NERFINISHED
ⓘ
Dolbeault operator \bar{\partial} NERFINISHED ⓘ antiholomorphic function ⓘ complex gradient ⓘ holomorphic function ⓘ |
| satisfies |
Leibniz rule for products
ⓘ
chain rule for compositions ⓘ linearity in the function argument ⓘ |
| treatsAsIndependentVariables |
complex conjugate \bar{z}
ⓘ
complex variable z ⓘ |
| usedFor |
characterizing holomorphic functions
ⓘ
expressing Cauchy–Riemann equations ⓘ simplifying calculations in complex analysis ⓘ |
| usedIn |
Wirtinger calculus in machine learning
ⓘ
complex differential geometry ⓘ complex potential theory ⓘ complex signal processing ⓘ distribution theory on the complex plane ⓘ optimization with complex variables ⓘ theory of several complex variables ⓘ |
| usedToDefine | complex Laplacian via \partial and \bar{\partial} ⓘ |
| usedToExpress | harmonicity conditions in complex form ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.