Harmonic conjugate
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 .