Verifying Conformality

Complex Analysis · Axiom Academy

EXAMPLE Verifying Conformality Use the derivative test to find exactly where a map preserves angles The map f(z) = z^3 - 3z is entire (analytic on all of ), so it preserves angles almost everywhere. Find the exact set of points where it is conformal — and the points where it is not . The z-plane. At a generic point such as z = 2 (green), two crossing curves keep their right angle under the map — it is conformal there. At the two red critical points z = −1 and z = +1 the derivative vanishes and angles are not preserved. Nicely done. You applied the one test that settles conformality for any analytic map: conformal exactly where . The test: an analytic f is conformal at every point of its domain where ; it fails precisely at the critical points , the zeros of f' . What happens at a critical point: near z = 1 , f(z) - f(1) = 3(z-1)^2 + (z-1)^3 , so the leading term is quadratic — angles are doubled , not preserved. This example: f(z) = z^3 - 3z is conformal on . Same recipe, other maps: e^z has , so it is conformal on all of ; any Möbius map has , so it is conformal everywhere it is defined. Checking conformality is never about the formula's complexity — it is one derivative and one equation: differentiate, then ask where f' vanishes.

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