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Cauchy-Riemann Equations

Source lecture(s): PHY622 Lec3

Intuition

Differentiability in the complex plane imposes two coupled real equations, far stricter than real differentiability.

Formal Definition

For \(f=u+iv\), \(\partial u/\partial x = \partial v/\partial y\) and \(\partial u/\partial y = -\partial v/\partial x\).

Mathematical Formulation

\[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\]

Derivation

Write \(\Delta z\) along real (\(\Delta x\)) and imaginary (\(i\Delta y\)) axes. The limit must be unique, giving two equations.

Worked Example

For \(f(z)=e^z=e^x(\cos y+i\sin y)\), CR gives \(u_x=v_y=e^x\cos y\) and \(u_y=-v_x=-e^x\sin y\).

Common Mistakes

  • Checking only one of the two equations.
  • Assuming CR guarantees analyticity everywhere (need continuous partials).

Quiz

Q1. Are the Cauchy-Riemann equations sufficient alone?

Answer

No—need continuous partial derivatives.