Cauchy-Riemann Equations

Suppose is a complex-valued function of a complex variable. If the derivatives are the same for the 2 paths —real and imaginary axes— then is analytic.

Suppose for the theorems below.

The equations

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

Cartesian form

Polar form

Here the partial derivatives are about .

Complex form

Theorem 1

If is differentiable at , then

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

Note

Contrapositive is useful when proving is not differentiable at .

Theorem 2

If:

• All partial derivatives exist and
• They satisfy Cauchy-Riemann equations and
• They are 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 .