Conformal Maps Definition

Complex Analysis · Axiom Academy

LESSON What a Conformal Map Is One idea: it preserves angles. Wherever its derivative isn't zero, an analytic map carries crossing curves over at exactly the angle they came in. Draw two curves that cross at a point z_0 , meeting at angle . Send everything through a conformal map w=f(z) . The images bend into new shapes, but where they cross they still meet at exactly — same size, same turning sense. Near z_0 , an analytic f looks like its linear part: Write the derivative in polar form, . Every little displacement z-z_0 gets multiplied by the same number — which means each one is rotated by and scaled by r . Watch two displacements: both turn by the same , so the angle between them never changes. Both arguments shift by , so their difference — the angle between the vectors — is left alone. A genuine rotation + scaling. Angles preserved — the map is conformal at z_0 . The linear part collapses; this first-order picture fails. We'll see what happens next. The cleanest demonstration: feed a square grid into w=e^z . Since is never zero, the map is conformal everywhere . Horizontal lines become rays and vertical lines become circles — and because the grid lines met at right angles, the rays and circles do too. Every right-angle crossing of the original grid survives as a right-angle crossing of a ray and a circle — angle preservation, made completely visible.

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