a harmonic conjugate

In the plane there is a hidden bridge between potential theory and complex analysis, and the harmonic conjugate is the plank you walk across it. Given one harmonic function — say an electrostatic potential — there is a partner harmonic function whose level curves run exactly perpendicular to the first's, tracing the field lines. Pair the two and you get a single complex-analytic function. So every planar harmonic function is secretly the real part of an analytic one, with its conjugate as the imaginary part.

Precisely, if u is harmonic on a (simply connected) planar region, a harmonic conjugate v is a function such that u and v together satisfy the Cauchy-Riemann equations: u_x = v_y and u_y = -v_x. Then the combination f = u + i v is a complex-analytic (holomorphic) function of z = x + i y, and v is automatically harmonic too. You construct v by integration: from the Cauchy-Riemann equations you know both partial derivatives of v in terms of u's derivatives, so you integrate them up (consistency is guaranteed exactly because u is harmonic). The level curves of u and v form an orthogonal grid — in physics, u is the potential and v is the stream function, with v constant along streamlines.

This connection is the reason planar potential problems can be attacked with the full arsenal of complex analysis: conformal mapping transplants a hard domain to a simple one (a disk or half-plane) while preserving harmonicity, so you solve the Dirichlet problem on the easy domain and map back. It powers two-dimensional fluid flow (the complex potential), electrostatics, and heat conduction. The honest limitations: this is special to two dimensions — there is no analogous single complex companion in three dimensions — and on a domain with holes (not simply connected) the conjugate can fail to be single-valued, picking up a period each time you loop around a hole, just as the potential log r has the multivalued angle theta as its conjugate.