Rational Functions

Complex Analysis · Axiom Academy

Quotients of polynomials — and what happens at the points where the denominator vanishes. 1. Zeros, Poles, and the Blow-Up Write f(z) = p(z)/q(z) with p and q polynomials. Two kinds of special point control its behavior. A zero is where f vanishes — a root of the top, p(z_0)=0 . A pole is where f runs off to infinity — a root of the bottom, q(z_0)=0 . The function is not defined at its poles: the domain is all of except the roots of q . pole of f — a root of q , where Probing along the imaginary axis . The simplest non-polynomial rational function is f(z) = 1/z : one pole at z = 0 , and no zeros at all (the top is the constant 1 ). Watch what it does to a point. Its image w = 1/z has reciprocal size and opposite angle — so as z moves out , w moves in . The unit circle maps to itself, and inside and outside swap. |w| = 1/|z| . A point at distance 2 lands at distance ; a point at lands at 3 . — a reflection across the real axis. So . If |z| = 1 then |w| = 1 . Points on the circle stay on the circle. , and vice versa. The disk turns inside out. Take z = 1 + i , which has at angle . Then : size at angle . Same line through the origin, reciprocal distance, mirrored below the real axis. Send z toward the pole at the origin. As the size |1/z| has no ceiling — it grows past every bound. That is exactly what "pole" means: get arbitrarily close to z = 0 and |f(z)| becomes arbitrarily large. The single missing point z=0 is sent all the way out to .

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