In mathematics, a real-valued function u ( x , y ) {\displaystyle u(x,y)} defined on a connected open set Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} is said to have a conjugate (function) v ( x , y ) {\displaystyle v(x,y)} if and only if they are respectively the real and imaginary parts of a holomorphic function f ( z ) {\displaystyle f(z)} of the complex variable z := x + i y ∈ Ω . {\displaystyle z:=x+iy\in \Omega .} That is, v {\displaystyle v} is conjugate to u {\displaystyle u} if f ( z ) := u ( x , y ) + i v ( x , y ) {\displaystyle f(z):=u(x,y)+iv(x,y)} is holomorphic on Ω . {\displaystyle \Omega .} As a first consequence of the definition, they are both harmonic real-valued functions on Ω {\displaystyle \Omega } . Moreover, the conjugate of u , {\displaystyle u,} if it exists, is unique up to an additive constant. Also, u {\displaystyle u} is conjugate to v {\displaystyle v} if and only if v {\displaystyle v} is conjugate to − u {\displaystyle -u} .
Description Equivalently, v {\displaystyle v} is conjugate to u {\displaystyle u} in Ω {\displaystyle \Omega } if and only if u {\displaystyle u} and v {\displaystyle v} satisfy the Cauchy–Riemann equations in Ω . {\displaystyle \Omega .} As an immediate consequence of the latter equivalent definition, if u {\displaystyle u} is any harmonic function on Ω ⊂ R 2 , {\displaystyle \Omega \subset \mathbb {R} ^{2},} the function − u y {\displaystyle -u_{y}} is conjugate to u x {\displaystyle u_{x}} for then the Cauchy–Riemann equations are just Δ u = 0 {\displaystyle \Delta u=0} and the symmetry of the mixed second order derivatives, u x y = u y x . {\displaystyle u_{xy}=u_{yx}.} Therefore, a harmonic function u {\displaystyle u} admits a conjugated harmonic function if and only if the holomorphic function g ( z ) := u x ( x , y ) − i u y ( x , y ) {\displaystyle g(z):=u_{x}(x,y)-iu_{y}(x,y)} has a primitive f ( z ) {\displaystyle f(z)} in Ω , {\displaystyle \Omega ,} in which case a conjugate of u {\displaystyle u} is, of course, Im f ( x + i y ) . {\displaystyle \operatorname {Im} f(x+iy).} 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 R 2 {\displaystyle \mathbb {R} ^{2}} 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 zeros 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.
Examples For example, consider the function u ( x , y ) = e x sin y . {\displaystyle u(x,y)=e^{x}\sin y.}
Since
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