Harmonic functions and harmonic conjugates

Try

The real part of z² is x²−y². Its second x derivative is two and its second y derivative is minus two, so their sum is zero. Such a function is harmonic: it satisfies the planar Laplace equation.

Its companion imaginary part is 2xy. The pair is not arbitrary; their first derivatives satisfy the Cauchy–Riemann equations. Constructing a harmonic conjugate means recovering that companion from a real-valued harmonic function, when the domain permits it.

Predict

Predict whether every harmonic function has a conjugate near each point. On a small disk the answer is yes. Predict whether these local conjugates always combine into one function on a domain with holes. The function log|z| on the punctured plane provides a counterexample.

Its local conjugate is an angle. A full circuit around zero changes that angle by 2π, so it cannot be a single-valued global function there.

Experiment

Work through u=x²−y²+3x.

Requirement	Equation
Harmonicity	u_xx+u_yy=2−2=0
First conjugate derivative	v_y=u_x=2x+3
Second conjugate derivative	v_x=−u_y=2y
Conjugate	v=2xy+3y+C

Verify both derivative equations after integrating. Checking only one leaves an undetermined function, not merely an undetermined constant.

Understand

A real-valued C² function u on an open plane set is harmonic if Δu=u_xx+u_yy=0. If f=u+iv is holomorphic, analyticity gives the required smoothness. Differentiating u_x=v_y and u_y=−v_x and commuting mixed partials yields Δu=0; the same argument gives Δv=0.

Conversely, for harmonic u the desired conjugate must satisfy dv=−u_y dx+u_x dy. This C¹ differential is closed because ∂y(−u_y)=−u_yy=u_xx=∂x(u_x). On a simply connected domain, the standard path-independence theorem for closed planar differentials supplies a potential v. Then its partial derivatives satisfy Cauchy–Riemann, so u+iv is holomorphic. This exactness theorem can be proved using Green’s theorem on local patches and cancellation along a homotopy; the topological hypothesis is what globalizes the local construction.

On the punctured plane, u=log|z| is harmonic, but the conjugate differential is −y/(x²+y²)dx+x/(x²+y²)dy. Its unit-circle integral is 2π. An exact differential would integrate to zero around every closed curve, so no global conjugate exists.

Locally taking real parts of Cauchy’s mean formula shows that a harmonic function equals its circle average. This yields the harmonic maximum principle: a local maximum forces the function constant on each small circle by the mean equality, then on a neighborhood, and connectedness propagates the result.

Apply

Construct a conjugate for u=x³−3xy². Integrating v_y=3x²−3y² gives v=3x²y−y³+h(x). The other equation requires h′=0, so v=3x²y−y³+C. Recognize the result as the imaginary part of z³.

Diagnose “A harmonic function is holomorphic.” A real-valued harmonic function is one component of a holomorphic function locally. If a real-valued function itself were nonconstant and holomorphic on a domain, the open-mapping theorem would contradict its image lying on the real axis.

Continue

Poisson’s formula next reconstructs a harmonic function from boundary data on a disk. Optional depth: verify that composing a harmonic function with a conformal map remains harmonic where the composition is defined; the Laplacian transforms by the positive factor |f′|². Keep this separate from arbitrary real coordinate changes.