Overview
The derivative of complex function is defined to be the given limit
{% f'(z_0) = \lim_{\Delta z \to 0} \frac{f(z_0 + \Delta z) - f(z_0)}{|\Delta z|} %}
when it exists, where {% z_0 %} is a complex number, and {% \Delta z = z-z_0 %}
Properties
The complex derivative exhibits the following properties:
- {% \frac{d f}{dz} = 0 %} when {% f(z) = c %} a constant
- {% \frac{dz^n}{dz} = nz^{n-1} %}
- {% \frac{d}{dz} (f(z) + g(z)) = f'(z) + g'(z) %}
- {% \frac{d}{dz}[\frac{f(z)}{g(z)}] = \frac{g(z)f'(z) - f(z)g'(z)}{g(z)^2} %}
Derivative Paths
The limit defined above can be approached in two dimensions. In the horizontal dimension, only the {% x %} value is varied.
{% f'(z_0) = \lim_{\Delta x \to 0} \frac{u(x_0 + \Delta x, y_0) - u(x_0,y_0)}{\Delta x}
+ i \; \lim_{\Delta x \to 0} \frac{v(x_0 + \Delta x, y_0) - v(c_0,y_0)}{\Delta x}
%}
In the veritcal dimension, only the {% y %} value is varied.
{% f'(z_0) = \lim_{\Delta y \to 0} \frac{u(x_0, y_0 + \Delta y) - u(x_0,y_0)}{\Delta y}
+ i \; \lim_{\Delta y \to 0} \frac{v(x_0 , y_0 + \Delta y) - v(c_0,y_0)}{\Delta y}
%}
Cauchy Riemann Equations
Because both of these equations represent the derivative, the following must hold if the derivative exists:
{% \frac{\partial u(x_0,y_0)}{\partial x} = \frac{\partial v(x_0, y_0)}{\partial y} %}
and
{% \frac{u(x_0,y_0)}{\partial y} = - \frac{v(x_0,y_0)}{\partial x} %}