Cauchy Riemann Equations: A Complete Student Guide

Write f(z) = u(x, y) + iv(x, y). If f is differentiable at z_0, letting h → 0 along the real axis gives f′(z_0) = u_x + iv_x, while along the imaginary axis it gives f′(z_0) = v_y − iu_y. Equating yields the Cauchy–Riemann equations: u_x = v_y and u_y = −v_x.

They convert complex differentiability into a concrete calculus test on u and v, and they bridge to physics: differentiating them shows u and v satisfy Laplace's equation, which is why complex analysis governs electrostatics, heat flow, and ideal fluids.

Key Concepts

• Partials: u_x, u_y, v_x, v_y, the ordinary calculus ingredients.
• Cauchy–Riemann: u_x = v_y and u_y = −v_x, necessary for complex differentiability.
• Sufficiency: continuous partials satisfying the equations in a neighborhood imply analyticity there.
• Derivative formula: f′(z) = u_x + iv_x = v_y − iu_y.
• Polar form: u_r = (1/r)v_θ and v_r = −(1/r)u_θ.

Formulas

• ∂u/∂x = ∂v/∂y; ∂u/∂y = −∂v/∂x.
• Differentiating gives u_xx + u_yy = 0: u and v are harmonic.
• f′(z) = u_x + iv_x computes the derivative from components.
• For z^n (n positive integer) the equations hold everywhere and f′ = nz^{n−1}.

Worked Example

Test f(z) = e^z = e^x(cos y + i sin y): u = e^xcos y, v = e^xsin y. Then u_x = e^xcos y = v_y, and u_y = −e^xsin y = −v_x. The partials are continuous everywhere, so e^z is entire. Its derivative: f′ = u_x + iv_x = e^xcos y + ie^xsin y = e^z, recovering d/dz e^z = e^z. Numerically at 0 with h = 0.001: (e^{0.001} − 1)/0.001 ≈ 1.0005, within 0.001 of f′(0) = 1.

Common Misconceptions

• "The equations at one point prove differentiability." The theorem needs the equations plus continuous partials in a neighborhood; isolated-point counterexamples exist.
• "No extra conditions needed for sufficiency." Continuity of the partials is the standard extra condition.
• "f′(z) = u_x + iu_y." The correct formula is u_x + iv_x; swapping partials is the classic slip.

Real-World Applications

• Fluid dynamics: analytic potentials give velocity fields that are irrotational and incompressible.
• Electrostatics: u and v are potential and flux functions, field lines orthogonal to equipotentials.
• Given harmonic u, integrating Cauchy–Riemann builds its harmonic conjugate v.

The Cauchy–Riemann equations compress complex differentiability into two PDEs on u and v. They test analyticity, compute derivatives, and link complex analysis to field physics. Whenever a new complex function appears, checking these two equations is the fastest first test of whether it is genuinely analytic.