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Concept# Harmonic conjugate

Summary

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

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