The Cauchy-Riemann Equations

Abstract

Certainly every student of complex analysis learns of the Cauchy-Riemann equations

$$ \begin{gathered} \frac{{\partial u}} {{\partial x}} = \frac{{\partial v}} {{\partial y}}, \hfill \\ \frac{{\partial u}} {{\partial y}} = - \frac{{\partial v}} {{\partial x}}. \hfill \\ \end{gathered} $$

These identities, which follow directly from the definition of complex derivative, give an important connection between the real and complex parts of a holomorphic function. Certainly conformality, harmonicity, and many other fundamental ideas are effectively explored by way of the Cauchy—Riemann equations.