Finding a Harmonic Conjugate

Complex Analysis · Axiom Academy

EXAMPLE Finding a Harmonic Conjugate Building an analytic function from a harmonic function with the Cauchy–Riemann equations The function u(x,y) = x^2 - y^2 is harmonic. Find a harmonic conjugate v(x,y) — a partner so that f(z) = u + iv is analytic. We will earn each line of the solution by answering one question at a time. For f = u + iv to be analytic, the real and imaginary parts must lock together exactly: Nice work — you reconstructed the analytic function f(z) = z^2 from just its real part. The conjugate is forced: Cauchy–Riemann pins down v from u , so once u is given, v is determined up to an additive constant C . The recipe: get u_x, u_y , set v_y = u_x and v_x = -u_y , integrate one of them, then differentiate and match the other to find g . Always verify: here f = (x^2 - y^2) + i(2xy) = (x + iy)^2 = z^2 — the conjugate rebuilt a clean function of z alone. It generalizes: the same steps turn into f = e^z and u = x^3 - 3xy^2 into f = z^3 . Every solution of Laplace's equation on a simply connected region is the real part of some analytic function — and this is exactly how you find it.

This is the written version of the interactive lesson above. See the full Complex Analysis course.