Harmonicity
Definition. A real-valued \(C^2\) function \(u\) on a plane domain is harmonic if \(\Delta u=u_{xx}+u_{yy}=0\). Real and imaginary parts of holomorphic functions are harmonic because the Cauchy-Riemann equations imply the Laplace equation.
Harmonic conjugates
Proposition. On a simply connected domain, every harmonic function \(u\) has a harmonic conjugate \(v\) such that \(u+iv\) is holomorphic. The one-form \(-u_y\,\mathrm{d}x+u_x\,\mathrm{d}y\) is closed, and simply connectedness makes it exact.
Mean value property
Theorem. A harmonic function equals its average over every sufficiently small circle centered at the point. Conversely, a continuous function with the mean value property is harmonic.
Maximum principle
Corollary. A nonconstant harmonic function has no interior maximum or minimum. Apply the mean value property: if the average equals an extreme value, the function must be locally constant.
Poisson integral formula
Theorem. For continuous boundary data \(\varphi\) on the unit circle, the harmonic solution in the disc is \[ u(re^{i\theta})=\frac{1}{2\pi}\int_0^{2\pi}P_r(\theta-t)\varphi(e^{it})\,\mathrm{d}t, \] where \(P_r\) is the Poisson kernel.
Dirichlet problem
Remark. Harmonic functions connect complex analysis with potential theory. Boundary values determine interior values in well-behaved domains, mirroring Cauchy's boundary representation for holomorphic functions.
Holomorphic and harmonic functions
Proposition. If \(f=u+iv\) is holomorphic, then \[ \Delta u=\Delta v=0. \] Conversely, every harmonic \(u\) on a simply connected domain has a harmonic conjugate \(v\), unique up to a constant, such that \(u+iv\) is holomorphic.
Proof. Differentiate Cauchy--Riemann: \(u_{xx}+u_{yy}=v_{yx}-v_{xy}=0\), and similarly for \(v\). Conversely, the one-form \[ \omega=-u_y\,\mathrm{d}x+u_x\,\mathrm{d}y \] is closed because \(\mathrm{d}\omega=(u_{xx}+u_{yy})\,\mathrm{d}x\wedge\mathrm{d}y=0\). On a simply connected domain it is exact, \(\omega=\mathrm{d}v\), giving Cauchy--Riemann.
Mean value and maximum principles
Theorem. A harmonic function satisfies \[ u(z_0)=\frac1{2\pi}\int_0^{2\pi}u(z_0+re^{it})\,\mathrm{d}t \] on every contained disc. Conversely, a continuous function with the mean-value property on all sufficiently small discs is harmonic.
The forward statement is the real part of Cauchy's formula. For the converse, compare with the harmonic solution having the same boundary values on a disc; their difference has zero boundary values and the mean-value maximum principle forces it to vanish. Smoothness and \(\Delta u=0\) follow.
Poisson kernel and the disc Dirichlet problem
Theorem. For continuous \(\varphi\) on \(\partial\mathbb D\), \[ u(re^{i\theta}) =\frac1{2\pi}\int_0^{2\pi} P_r(\theta-t)\varphi(e^{it})\,\mathrm{d}t, \] \[ P_r(\alpha)=\frac{1-r^2}{1-2r\cos\alpha+r^2}, \] is harmonic in \(\mathbb D\), extends continuously with boundary value \(\varphi\), and is the unique such solution.
Proof. \(P_r\) is the real part of \((e^{it}+z)/(e^{it}-z)\), so the integral is harmonic. It is positive, integrates to one, and concentrates near \(t=\theta\) as \(r\uparrow1\); uniform continuity of \(\varphi\) proves boundary convergence. Uniqueness is the maximum principle applied to the difference.
Harnack inequalities
Theorem (Harnack). If \(u\ge0\) is harmonic on \(\mathbb D\), then \[ \frac{1-r}{1+r}u(0) \le u(re^{i\theta}) \le\frac{1+r}{1-r}u(0). \]
Proof. Apply the Poisson representation on a larger concentric disc and bound its kernel between \((1-r)/(1+r)\) and \((1+r)/(1-r)\), then let the larger radius approach one.
Harnack's principle implies locally bounded sequences of positive harmonic functions have compactness properties parallel to Montel's theorem.