Functions of a Complex Variable: Explained with Examples

A function of a complex variable assigns to each z in a domain D a complex number w = f(z). Writing z = x + iy and w = u + iv, every complex function splits into two real functions: f(z) = u(x, y) + iv(x, y). For instance f(z) = z² expands to (x² − y²) + i(2xy), so u = x² − y² and v = 2xy.

This split matters because complex functions model 2D phenomena economically: fluid flow, heat, and electric fields all live in the pair (u, v). It is also the doorway to deeper theory, since complex differentiability will impose a strong link between u and v.

Key Concepts

• Domain: the set D where f is defined, typically an open set like a disc.
• Components: u(x, y) = Re(f(x+iy)), v(x, y) = Im(f(x+iy)), ordinary real functions of two variables.
• Polynomials: a_nz^n + … + a_0 with complex coefficients; defined everywhere.
• Rational functions: p(z)/q(z), defined where q(z) ≠ 0.
• Mapping view: f deforms regions of the plane; tracking the deformation often beats staring at formulas.

How It Works

1. Expand: substitute z = x + iy, use i² = −1, and collect real and imaginary parts into u and v.
2. For f(z) = z²: (x + iy)² = (x² − y²) + i(2xy).
3. Evaluate either via the z-formula or via (u, v); both give the same w.
4. Read geometry: z² doubles angles and squares moduli, so the first quadrant maps onto the upper half-plane.

Worked Example

Evaluate f(z) = z² + 1 at z = 1 + i: (1 + i)² + 1 = (1 + 2i − 1) + 1 = 1 + 2i. Via components: u(1,1) = 1 − 1 + 1 = 1, v(1,1) = 2, confirming w = 1 + 2i. Sanity check on the mapping: |1 + i| = √2 so |z|² = 2, and |(1+i)²| = |2i| = 2, matching |z²| = |z|²; the +1 shifts the image right by one.

Common Misconceptions

• "Any pair (u, v) gives a differentiable complex f." Differentiability demands strong compatibility (Cauchy–Riemann); arbitrary pairs fail it.
• "You can graph it like y = f(x)." The graph is 4D, so we use domain coloring or watch regions deform instead.
• "f(z) = |z|² is harmless." It equals x² + y² + i·0, yet is complex-differentiable only at z = 0.

Real-World Applications

• Complex potentials w = φ + iψ encode velocity potential and streamlines of airflow around wings.
• Electrostatics derives fields from a complex potential, turning field-line sketches into function theory.
• Conformal maps simplify awkward geometries when solving heat and flow equations.

A complex function has two faces: the compact f(z) and the pair u + iv. Translating between them is mechanical; the payoff is geometric, seeing how f stretches and reshapes the plane.