Complex Functions Summary

Complex Analysis · Axiom Academy

Unit 3 recap — extending functions to the complex plane, where exponentials turn periodic, logs go multi-valued, and everything connects through e^ z . A complex function is a mapping w=f(z)=u+iv that sends the z -plane to the w -plane — there is no single graph, so we picture how regions move. The complex exponential is the master function: it is periodic with period and is never zero . Trig and hyperbolic functions are just exponentials in disguise; and become unbounded , and bridges the two families. The complex logarithm is multi-valued ; choosing a branch (the principal ) and a branch cut makes it single-valued. Complex powers are defined through the log, , giving surprises like the real value . Core Concept Functions as Mappings A complex function takes a point z=x+iy in the z -plane and returns a point w in the w -plane . With two input and two output dimensions there is no ordinary graph, so we study how grids, lines, and regions are transformed. Picture it as: a transformation of the plane, not a curve. Watch out for: f carries both a real part u and an imaginary part v , each a function of x and y . Core Concept Polynomials & Rational Functions By the Fundamental Theorem of Algebra , a degree- n polynomial has exactly n roots in (with multiplicity), so it always factors completely. A rational function is a quotient of polynomials whose zeros of q are poles . When to use: FTA guarantees complex roots even when real ones are missing.

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