Analytic Functions & Cauchy–Riemann

Complex functions map the plane to itself. Learn limits, continuity and differentiability in ℂ, analytic (holomorphic) functions, the Cauchy–Riemann equations, harmonic functions and conjugates, elementary functions (eᶻ, sin z, log z), and conformal mapping in potential flow and electrostatics.

Complex differentiability is far stronger than real differentiability: an analytic function is infinitely differentiable, and its real and imaginary parts automatically solve Laplace's equation — a gift for 2D fluid flow, heat conduction and electrostatics.

An analogy

An analytic function is like a perfectly elastic rubber sheet transformation: tiny shapes are rotated and scaled but never distorted (conformal).

Complex differentiability

The limit [f(z + h) − f(z)]/h must be the same from every direction in the plane. z² and eᶻ pass; z̄ and |z|² fail.

Cauchy–Riemann test

If u, v have continuous partials satisfying CR, f is analytic. Polynomials, eᶻ, sin z are entire (analytic everywhere); 1/z is analytic except at 0.

Harmonic conjugates

Given harmonic u, find v from CR. In potential flow, u is the velocity potential and v the stream function — their level curves are orthogonal equipotentials and streamlines.

Conformal mapping

Analytic maps preserve angles, so hard geometries (flow around a cylinder, fields near an edge) can be mapped to simple ones — the Joukowski map gives aerofoil flow.

Formulas

• Cauchy–Riemann: u_x = v_y, u_y = −v_x
• Derivative: f′(z) = u_x + j v_x
• Harmonic: u_xx + u_yy = 0, v_xx + v_yy = 0
• Elementary functions: eᶻ = eˣ(cos y + j sin y); Log z = ln|z| + j Arg z

Worked example: CR check

Given: f = z² = (x² − y²) + j2xy

Find: Analytic?

• u_x = 2x = v_y; u_y = −2y = −v_x ✓
• f′ = 2x + j2y = 2z

Answer: Analytic everywhere, f′ = 2z

Worked example: Harmonic conjugate

Given: u = x³ − 3xy²

Find: v.

• u_x = 3x² − 3y² = v_y → v = 3x²y − y³ + g(x)
• u_y = −6xy = −v_x → g′ = 0

Answer: v = 3x²y − y³ (f = z³)

Common mistakes

• Wrong: Assuming real-differentiable ⇒ analytic. — Right: CR equations must hold.
• Wrong: Treating Log z as single-valued everywhere. — Right: Use branch cuts.

Applications

• Potential flow: Complex potential for 2D inviscid flow.
• Electrostatics: Field lines and equipotentials.
• Aerofoil theory: Joukowski conformal mapping.

Key takeaways

• Analytic ⇔ CR + continuous partials.
• Real and imaginary parts are harmonic.
• f′ = u_x + jv_x.
• Conformal maps solve 2D field problems.

Going further

• Schwarz–Christoffel: Mapping polygons to half-planes.
• Riemann surfaces: Multivalued functions.
• Maximum modulus principle: Extremes on boundaries.