The zⁿ Maps

Complex Analysis · Axiom Academy

One rule runs every power map: it multiplies the angle by n and raises the modulus to the n-th power — opening a sector wide. 1. Modulus to the Power, Angle Times n Take a point and a fixed sweep of its angle from 0 upward. Watch the input ray on the left and its image ray on the right: the image turns n times faster, so a wedge of opening becomes a wedge of opening . (Here n = 2 , so the image always sits at double the angle.) Modulus is raised to the n-th power 2. A Sector Opens Into a Wider One Because angles multiply by n , a whole sector of opening is carried to a sector of opening . The map below squares the wedge : each ray's angle doubles and its length is squared, so the input sector spreads out to a image sector (the first quadrant). A ray at angle lands at angle — the rays fan out. A point at distance r lands at distance r^n from the origin. Opening becomes opening ; the sector gets n times wider. An opening of already fills the entire plane. Take z = 1 + i , so and . Then z^2 = 2i : the modulus is and the angle is — exactly on the positive imaginary axis. Send z once around the unit circle, . Since the image angle is , the image races around n times — for n = 2 , the output circle is traced twice while the input goes around once. So n distinct inputs share each output value: the map is n -to-one away from 0 .

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