Cauchy-Riemann Equations

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

Suppose $f(z)=u(x,y)+iv(x,y)$ for the theorems below. $$

The equations

The set of equations mentioned below are the Cauchy Riemann Equations, where $u,v$ are functions of $x,y$.

Cartesian form

Polar form

Here the partial derivatives are about $r, \theta$. $$

Complex form

Theorem 1

If $f$ is differentiable at $z_0$, then

• All partial derivatives $u_x,u_y,v_x,v_y$ exist and
• They satisfy the Cauchy Riemann equations

Theorem 2

If:

• All partial derivatives $u_x,u_y,v_x,v_y$ exist and
• They satisfy Cauchy-Riemann equations and
• They are continuous at $z_0$

Then:

• $f$ is differentiable at $z_0$ and

Theorem 3

If $f$ is analytic at $z_0$, then its first-order partial derivatives are continuous in a neighbourhood of $z_0$.