Section 1
Harmonic Functions and the Dirichlet Problem
Let $f$ and $g$ be harmonic on a set $D$ of points in the plane. Show that $f+g$ is harmonic, as well as $\alpha f$ for any real number $\alpha$.
Show that the following functions are harmonic on the entire plane:(a) $x^3-3 x y^2$(b) $3 x^2 y-y^3$(c) $x^4-6 x^2 y^2+y^4$(d) $4 x^3 y-4 x y^3$(e) $\sin (x) \cosh (y)$(f) $\cos (x) \sinh (y)$(g) $e^{-x} \cos (y)$
Show that $\ln \left(x^2+y^2\right)$ is harmonic on the plane with the origin removed.
Show that $r^n \cos (n \theta)$ and $r^n \sin (n \theta)$, in polar coordinates, are harmonic on the plane, for any positive integer n.
Show that, for any positive integer $n, r^{-n} \cos (n \theta)$ and $r^{-n} \sin (n \theta)$ are harmonic on the plane with the origin removed.