Harmonic function

A real-valued function $u(x,y)$ (or $u(x,y,z)$) is harmonic if it has continuous second partial derivatives and satisfies Laplace’s equation:

${\nabla }^{2}u=u_{xx}+u_{yy}=0\text{(2D)}$

${\nabla }^{2}u=u_{xx}+u_{yy}+u_{zz}=0\text{(3D)}$

The Laplacian ${\nabla }^2$ (also written $\Delta$) is the divergence of the Gradient: ${\nabla }^{2}u=\nabla \cdot \nabla u$.

Why they show up everywhere in physics

Laplace’s equation describes equilibrium / steady state in a lot of physical systems:

• Steady-state electrostatic potentials in regions without charge. The electric field is $\mathbf{E}=-\nabla u$; Gauss’s law gives $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$. With $\rho =0$, ${\nabla }^{2}u=0$.
• Irrotational incompressible fluid flow. Velocity $\mathbf{v}=\nabla u$ (irrotational), $\nabla \cdot \mathbf{v}=0$ (incompressible). So ${\nabla }^{2}u=0$.
• Steady heat conduction. Heat equation ${\partial }_{t}u=\alpha {\nabla }^{2}u$ in steady state gives ${\nabla }^{2}u=0$.
• Membrane displacements under tension (in equilibrium).
• Gravitational potential in vacuum regions.

Find the right harmonic function with the right boundary conditions and you’ve solved the problem.

Harmonic = real/imaginary part of analytic

The result that ties complex analysis to physics:

Theorem. If $f=u+iv$ is analytic on a domain $\Omega$, then $u$ and $v$ are both harmonic on $\Omega$.

Proof. From the Cauchy-Riemann equations $u_x=v_y$ and $u_y=-v_x$, differentiate the first w.r.t. $x$ and the second w.r.t. $y$:

$u_{xx}=v_{yx},u_{yy}=-v_{xy}.$

By Clairaut’s theorem $v_{yx}=v_{xy}$, so $u_{xx}+u_{yy}=0$. Same argument for $v$. ∎

(One technical catch: this proof uses Clairaut’s equality of mixed partials, which requires $v$ to be $C^2$. With just our definition of analyticity (once complex differentiable on an open set) we don’t yet know that. The fact that analytic functions are automatically infinitely differentiable, hence $C^{\infty }$, comes from the Cauchy integral formula. No circularity: the CIF only uses analyticity, not harmonicity.)

So analytic functions automatically solve Laplace’s equation in 2D. That’s the bridge between complex analysis and 2D steady-state physics.

Harmonic conjugate

If $u$ is harmonic on a simply connected domain, there exists a harmonic function $v$ (the harmonic conjugate of $u$) such that $f=u+iv$ is analytic.

Recipe to find $v$ from $u$:

1. From $v_y=u_x$, integrate: $v=\int u_{x}dy+g(x)$.
2. From $v_x=-u_y$, get an equation for $g^{\prime }(x)$.
3. Integrate.

Same shape as finding a potential function for a conservative field. The Cauchy–Riemann equations are exactly the cross-partial conditions of vector calculus for the field $⟨u,-v⟩$ to be both conservative and source-free.

Worked example. $u(x,y)=x^2-y^2+x$. Check harmonic: $u_{xx}=2$, $u_{yy}=-2$, sum $=0$. ✓

Find $v$: $v_y=u_x=2x+1$, integrate w.r.t. $y$: $v=2xy+y+g(x)$. Then $v_x=2y+g^{\prime }(x)$ must equal $-u_y=2y$, so $g^{\prime }(x)=0$, $g=0$. Stream function: $v=2xy+y$.

Combined: $f(z)=(x^2-y^2+x)+i(2xy+y)=(x^2-y^2+2ixy)+(x+iy)=z^2+z$.

So $u=x^2-y^2+x$ is the real part of the analytic function $f(z)=z^2+z$.

Properties of harmonic functions

These mirror the properties of analytic functions:

• Maximum principle. A non-constant harmonic function attains its maximum and minimum on the boundary of its domain, never in the interior. (Same statement as the Maximum modulus principle for analytic functions, since $|f|$ is the modulus of an analytic function.)
• Mean value property. The value at the center of any disk equals the average over the bounding circle.
• Uniqueness from boundary values. A harmonic function on a bounded domain is determined by its boundary values (Dirichlet problem). That’s why boundary conditions are enough to solve the steady-state problems above.
• Infinitely differentiable. Any harmonic function is automatically $C^{\infty }$.

These all follow from the connection to analytic functions via harmonic conjugates and the Cauchy integral formula.

In context

This viewpoint is the basis for 2D conformal mapping in physics: transform a hard region to an easy one via an analytic function, solve Laplace’s equation on the easy region (often a disk or half-plane), pull back via the inverse map. Laplace’s equation is preserved by conformal maps, which makes this work.