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- Harmonic_conjugate abstract "In mathematics, a function defined on some open domain is said to have as a conjugate a function if and only if they are respectively real and imaginary part of a holomorphic function of the complex variable That is, is conjugate to if is holomorphic on As a first consequence of the definition, they are both harmonic real-valued functions on . Moreover, the conjugate of if it exists, is unique up to an additive constant. Also, is conjugate to if and only if is conjugate to . Equivalently, is conjugate to in if and only if and satisfy the Cauchy–Riemann equations in As an immediate consequence of the latter equivalent definition, if is any harmonic function on the function is conjugate to , for then the Cauchy–Riemann equations are just and the symmetry of the mixed second order derivatives, Therefore an harmonic function admits a conjugated harmonic function if and only if the holomorphic function has a primitive in in which case a conjugate of is, of course, So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain. There is an operator taking a harmonic function u on a simply connected region in R2 to its harmonic conjugate v (putting e.g. v(x0)=0 on a given x0 in order to fix the indeterminacy of the conjugate up to constants). This is well known in applications as (essentially) the Hilbert transform; it is also a basic example in mathematical analysis, in connection with singular integral operators. Conjugate harmonic functions (and the transform between them) are also one of the simplest examples of a Bäcklund transform (two PDEs and a transform relating their solutions), in this case linear; more complex transforms are of interest in solitons and integrable systems.Geometrically u and v are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function; the contours on which u and v are constant cross at right angles. In this regard, u+iv would be the complex potential, where u is the potential function and v is the stream function.".
- Harmonic_conjugate wikiPageExternalLink HarmonicRatio.shtml.
- Harmonic_conjugate wikiPageID "910519".
- Harmonic_conjugate wikiPageRevisionID "578461438".
- Harmonic_conjugate hasPhotoCollection Harmonic_conjugate.
- Harmonic_conjugate id "p/c025020".
- Harmonic_conjugate title "Conjugate function".
- Harmonic_conjugate subject Category:Harmonic_functions.
- Harmonic_conjugate subject Category:Partial_differential_equations.
- Harmonic_conjugate type Abstraction100002137.
- Harmonic_conjugate type Communication100033020.
- Harmonic_conjugate type DifferentialEquation106670521.
- Harmonic_conjugate type Equation106669864.
- Harmonic_conjugate type Function113783816.
- Harmonic_conjugate type HarmonicFunctions.
- Harmonic_conjugate type MathematicalRelation113783581.
- Harmonic_conjugate type MathematicalStatement106732169.
- Harmonic_conjugate type Message106598915.
- Harmonic_conjugate type PartialDifferentialEquation106670866.
- Harmonic_conjugate type PartialDifferentialEquations.
- Harmonic_conjugate type Relation100031921.
- Harmonic_conjugate type Statement106722453.
- Harmonic_conjugate comment "In mathematics, a function defined on some open domain is said to have as a conjugate a function if and only if they are respectively real and imaginary part of a holomorphic function of the complex variable That is, is conjugate to if is holomorphic on As a first consequence of the definition, they are both harmonic real-valued functions on . Moreover, the conjugate of if it exists, is unique up to an additive constant. Also, is conjugate to if and only if is conjugate to .".
- Harmonic_conjugate label "Harmonic conjugate".
- Harmonic_conjugate sameAs m.03p2jv.
- Harmonic_conjugate sameAs Q5659251.
- Harmonic_conjugate sameAs Q5659251.
- Harmonic_conjugate sameAs Harmonic_conjugate.
- Harmonic_conjugate wasDerivedFrom Harmonic_conjugate?oldid=578461438.
- Harmonic_conjugate isPrimaryTopicOf Harmonic_conjugate.