Evaluating f(z) = z²

Complex Analysis · Axiom Academy

Square a complex number, read off its component functions, and see the geometry: the angle doubles and the modulus squares. Evaluate the function f(z) = z^2 at z = 3 + 2i . In other words, compute (3 + 2i)^2 , then identify the component functions u(x, y) and v(x, y) for f(z) = z^2 . Nice work! You evaluated f(z) = z^2 two ways and connected the algebra to the geometry of the complex plane. The result: (3 + 2i)^2 = 5 + 12i , found by expanding with i^2 = -1 . Component functions: for f(z) = z^2 , the real and imaginary parts are u(x, y) = x^2 - y^2 and v(x, y) = 2xy . At (3, 2) they give u = 5 and v = 12 . The geometry: squaring SQUARES the modulus ( |z^2| = |z|^2 = 13 ) and DOUBLES the argument ( ) — a preview of De Moivre's theorem. Whether you FOIL the algebra or read off u and v , you get the same point — and in polar form, squaring is just "double the angle, square the length."

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