Harmonic functions

1. A real-valued function \( f = f(x,y) \) of real variables \( x \) and \( y \) is said to be harmonic in an open set \( D \subset \mathbb{C} \) if it has continuous partial derivatives of second order and satisfies Laplace's Equation: \[ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \]
2. A complex-valued function is harmonic in an open set if both its real and imaginary parts are harmonic.
3. Let \( D \) be an open subset of \( \mathbb{C} \). Then the real and imaginary parts of an analytic function in \( D \) are harmonic in \( D \).
4. A domain \( D \subset \mathbb{C} \) is simply connected if its complement with respect to \( \mathbb{C}_\infty \) is a connected subset of \( \mathbb{C}_\infty \).
5. Let \( D \) be a simply connected domain and let \( f \) be harmonic in \( D \). Then \( f \) has a harmonic conjugate in \( D \).
6. If \(u\) and \(v\) are harmonic conjugate to each other in some domain then \(u\) and \(v\) must be constant there.
7. Laplace equation in polar form: \[ u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta\theta} = 0 \]
8. An analytic function \( F \) in a simply connected domain \( D \) has an analytic function \( H \) such that \( H'(z) = F(z) \), where \( H \) is called the primitive (or antiderivative) of \( F \).