Cauchy Riemann Equations

The set of equations mentioned below are the Cauchy Riemann Equations, where are functions of .

Theorem 1

Suppose , and is differentiable at . Then

• All partial derivatives exist
• They satisfy the Cauchy Riemann equations

Note

Contrapositive is useful when proving is not differentiable at .

Theorem 2

Suppose . All partial derivatives exist, and they are all continuous at . Then is differentiable at . And:

Theorem 3

If is analytic at , then its first-order partial derivatives are continuous in a neighbourhood of .