Komplekse funksjonar

Hans Georg Schaathun

September 2016

$$f: x \mapsto x^2$$

$$f: \mathbb{R} \to \mathbb{R}$$

$$f: z \mapsto z^2$$

$$f: \mathbb{C} \to \mathbb{C}$$

  1. Kontinuitet: OK
  2. Grenseverdiar: OK
  3. Derivasjon: strenge krav

$$f(x+iy) = u(x,y) + iv(x,y)$$

Cauchy-Riemann-likningane

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \quad\quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}$$

$$f'(x+iy) =\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}$$