What does it mean to perform calculus upon functions of complex values? Complex numbers exist in a plane. This would lead me to believe that calculus views them as multivariate, but I am not real sure. How would one define a rate of change for a complex number valued function, or the area underneath it. Could someone explain this in simple terms?
 A: At a very simple level, any complex number $z$ can be expressed as $x + i y.$
and $f(z) = u(x,y) + i v (x,y)$
The definition of derivative is the same definition.
$f'(z) = \lim_\limits{z\to z_0} \frac {f(z) - f(z_0)}{z-z_0}$
$\frac {\partial f}{\partial x} = \frac {\partial u}{\partial x} + i \frac{\partial u}{\partial y}\\
\frac {\partial f}{\partial y} = - \frac {\partial v}{\partial y} + i\frac {\partial v}{\partial x} $
if $\frac {\partial f}{\partial x} = \frac {\partial f}{\partial y}$ then the function is said to be "holomorphic."
A: When you compute the rate of change of a real valued function via a derivative, you are finding a very local linear approximation ( a line in 1 d, plane for scalar maps of two variables , and hyperplanes as you go up). 
In complex analysis, it works a bit differently, at least visually. You can view infinitesimal changes of the complex component and real component. While this can be thought of as the partial derivatives of a multi variable function from the 2d reals with this $i$ symbol and this measures the rate of change in one of the two directions on the complex plane with respect to the real or imaginary input, thanks to Euler's formula, this manifests itself visually as a local infinitesimal rotation and translation, rather than say a tangent plane.
