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

$$\frac{\partial{u}}{\partial{x}} =u_x = \frac{\partial{v}}{\partial{y}} =v_y \;\;\; \land \;\;\; \frac{\partial{u}}{\partial{y}} =u_y = - \frac{\partial{v}}{\partial{x}} =-v_x$$

Polar form

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

$$u_r = \frac{1}{r}v_\theta \;\;\; \land \;\;\; v_r = -\frac{1}{r}u_\theta$$

Complex form

$$f_x = -if_y$$

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

$$f'(z_0)=u_x(x_0,y_0) + iv_x(x_0,y_0)$$

Note

Contrapositive is useful when proving $f$ is not differentiable at $z_0$.

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

$$f'(z_0)=u_x(x_0,y_0) + iv_x(x_0,y_0)$$

Theorem 3

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