Harmonic conjugate

In mathematics, a real-valued function ${\displaystyle u(x,y)}$ defined on a connected open set ${\displaystyle \Omega \subset \mathbb {R} ^{2}}$ is said to have a conjugate (function) ${\displaystyle v(x,y)}$ if and only if they are respectively the real and imaginary parts of a holomorphic function ${\displaystyle f(z)}$ of the complex variable ${\displaystyle z:=x+iy\in \Omega .}$ That is, ${\displaystyle v}$ is conjugate to ${\displaystyle u}$ if ${\displaystyle f(z):=u(x,y)+iv(x,y)}$ is holomorphic on ${\displaystyle \Omega .}$ As a first consequence of the definition, they are both harmonic real-valued functions on ${\displaystyle \Omega }$. Moreover, the conjugate of ${\displaystyle u,}$ if it exists, is unique up to an additive constant. Also, ${\displaystyle u}$ is conjugate to ${\displaystyle v}$ if and only if ${\displaystyle v}$ is conjugate to ${\displaystyle -u}$.