Functions of Complex Variables: Complete Beginner's Guide

A complex variable is a number of the form z = x plus i y, where i is the square root of minus 1. Functions of a complex variable, like f(z) = z squared, take such numbers in and return complex numbers out, stretching, rotating, and bending the complex plane.

They matter far beyond pure mathematics. AC circuits are analyzed with complex impedances, fluid flow around wings is modeled with complex potentials, and signal processing leans on complex exponentials.

Key Concepts

• Modulus: |z| = square root of (x squared plus y squared), the distance from the origin.
• Argument: arg(z), the angle from the positive real axis, usually between minus pi and pi.
• Euler's formula: e^(i theta) = cos theta plus i sin theta, uniting exponentials with trigonometry.
• Analytic (holomorphic) function: differentiable at every point of some region. Analytic implies infinitely differentiable, which is false for real functions.
• Cauchy-Riemann equations: the test for analyticity, relating the partial derivatives of the real and imaginary parts.
• Singularity: a point where the function fails to be analytic, such as a pole where it blows up.
• Residue: a number capturing the behavior of f near an isolated singularity; residues evaluate difficult real integrals.

Formulas

• Polar form: z = r (cos theta plus i sin theta) = r e^(i theta), with r = |z| and theta = arg(z).
• Cauchy-Riemann equations: writing f = u plus i v, analyticity requires partial u by partial x = partial v by partial y, and partial u by partial y = minus partial v by partial x.
• Cauchy integral formula: f(a) = (1 divided by 2 pi i) times the closed contour integral of f(z) divided by (z minus a) dz. Values inside a curve are fixed by values on the curve.
• Residue theorem: a closed contour integral equals 2 pi i times the sum of residues inside the contour.

Worked Example

Check whether f(z) = z squared is analytic using the Cauchy-Riemann equations.

1. Write z = x plus i y. Then z squared = x squared minus y squared plus i (2 x y), so u = x squared minus y squared and v = 2 x y.
2. partials: partial u by partial x = 2x = partial v by partial y; partial u by partial y = minus 2y = minus partial v by partial x.
3. Both Cauchy-Riemann equations hold everywhere, so z squared is analytic on the whole plane, with derivative f'(z) = 2z, just like real calculus.

Common Misconceptions

• Imaginary numbers are not really useful. The name is historical baggage. Complex numbers model real AC circuits, real fluid flows, and real signal spectra.
• |z1 plus z2| equals |z1| plus |z2|. Only when the numbers point the same way. In general the triangle inequality says |z1 plus z2| is less than or equal to |z1| plus |z2|.
• Complex differentiability is like real differentiability. It is far stricter: the limit must be the same from every direction in the plane, which is why analytic functions are so rigid and powerful.

Real-World Applications

• AC circuits: voltage and current become phasors, and impedance Z = R plus i X simplifies circuit algebra.
• Aerodynamics: conformal maps like the Joukowski transform turn flow around a cylinder into flow around a wing.
• Signal processing: the Fourier transform is built on complex exponentials e^(i omega t).
• Control theory: root locus and Nyquist plots live in the complex s-plane.

Complex variables pack two real dimensions into one elegant object. Euler's formula turns rotation into multiplication, the Cauchy-Riemann equations test differentiability, and the residue theorem converts nightmare real integrals into simple arithmetic with poles.