harmonic conjugate
Suppose you are handed a harmonic function u — say the electrostatic potential in a region with no charge. Hidden alongside it is a partner, the harmonic conjugate v, which packages the same physics from a complementary point of view. Where u traces the equipotential lines, v traces the field lines that cross them at right angles; where u is the temperature, v is the heat-flow stream function. The harmonic conjugate is the second harmonic function that combines with the first to make an analytic whole.
Concretely, given a harmonic u(x, y), its harmonic conjugate v is the harmonic function such that u + i v is analytic, meaning the pair satisfies the Cauchy-Riemann equations: dv/dx = -du/dy and dv/dy = du/dx. You construct v by integrating these: read off dv/dx from -du/dy, integrate in x, then fix the leftover function of y using the other equation. On a simply connected region this always works and determines v up to an additive constant. The level curves of u and of v form an orthogonal grid — the curvilinear coordinate net of the flow.
This pairing is the engine of two-dimensional field theory. In fluid dynamics u is the velocity potential and v the stream function, and their combination w = u + i v, the complex potential, captures the entire ideal flow in one analytic function — streamlines are the contours v = const, and the flow speed is the magnitude of dw/dz. In electrostatics u gives equipotentials and v the lines of force. The honest caveat: on a region with a hole the conjugate may fail to be single-valued (think of the angle around a point), which is precisely how circulation and net charge enter the picture.
Let u = x^2 - y^2, which is harmonic. To find its conjugate v, use dv/dy = du/dx = 2x, integrate in y to get v = 2xy + g(x); then dv/dx = 2y + g'(x) must equal -du/dy = 2y, forcing g'(x) = 0. So v = 2xy (up to a constant), and indeed u + i v = (x + i y)^2 = z^2, analytic.
Integrating the Cauchy-Riemann equations reconstructs the missing partner v from a given harmonic u.
The relationship is not symmetric in sign: if v is the harmonic conjugate of u, then u is the conjugate of -v, not of v. The order matters, encoding which of the orthogonal families is the potential and which is the stream.