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This documents harmonic functions in the complex domain.
Let $f(z): \mathbb{C} \rightarrow \mathbb{R}$ be a function that takes in a complex number and returns a real one. We can see $f(z)$ as a function of $(x, y) \in \mathbb{R}^2$ since $z = x + iy$.
Then we say it is harmonic if it satisfies the Laplace equation:
\[\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0\]A function $f(z): \mathbb{C} \rightarrow \mathbb{C}$ can be expressed as two functions $u(z), v(z): \mathbb{C} \rightarrow \mathbb{R}$, one for the real part and one for the imaginary part:
\[f(z) = u(z) + i v(z)\]If $f(z)$ is holomorphic, then its real part $u(z)$ and imaginary part $v(z)$ are harmonic. They’re also conjugate harmonic because they’re connected by the Cauchy-Riemann equations:
\[\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}\]