Power Maps

Complex Analysis · Axiom Academy

The map w = z^n has one job: it multiplies every angle by n — so it pries sectors wide open. 1. The Rule: Multiply the Angle, Power the Length Write a point in polar form , . Raising it to the n th power is mechanical: the modulus is raised to the n th power and the angle is multiplied by n . length scales as a power; angle scales by a factor Take z = 1 + i , which has and . Then z^2 has modulus and angle — that is exactly 2i . The length squared, the angle doubled. 2. Sectors Fan Open by a Factor of n If every angle is multiplied by n , then a whole sector — the region between two rays from the origin — opens up by that same factor. A wedge of opening angle becomes a wedge of opening angle . A sector under z^3 opens to — the entire upper half-plane. To flatten a sector of angle into the upper half-plane, apply w = z^n . z^2 doubles angles: the first quadrant ( ) becomes the upper half-plane ( ). A sector under z^n wraps all the way around to the full plane. 3. Conformal Everywhere — Except the Origin A map is conformal where it preserves the angle between crossing curves. Power maps are analytic, so they pass this test wherever the derivative is nonzero. But angle-multiplying must show up somewhere — and it shows up at the one point the rays all pass through. For this vanishes only at z = 0 . So z^n is conformal for every , and the origin is its lone critical point .

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