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In mathematics, a real-valued function u {\displaystyle u} defined on a connected open set Ω ⊂ R 2 {\displaystyle \Omega \subset \mathbb {R} ^{2}} is said to have a conjugate v {\displaystyle v} if and only if they are respectively the real and imaginary parts of a holomorphic function f {\displaystyle f} 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 := u + i v {\displaystyle f:=u+iv} 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}.