Complex Functions of a complex variable

[Solidmozza]

Hi,
Im doing a course in Differential Calculus and we are up to Euler's formula and representing complex functions of a complex variable. I'm finding it really difficult to visualise/sketch these types of functions: z|-->z^2 such that {z=x+2i}, or z|-->e^z = w such that {z=(-1+i)t | t e R, t>_0}
Could someone please explain to me how to sketch these sorts of things, or direct me to an explanation? For instance, in the first of these above equations, I've gone z^2 = x^2 - 4 + 4xi but then I get mixed up with the axes: arn't they meant to be real and imaginary axis, not x and y, so what can I do with this x value?
Any help muchly appreciated!

 

[HallsofIvy]

Yes, of course you are going to have trouble! every complex number, z= x+ iy, requires two dimension so a graph of f(z) would have to be in 4 dimensions- 2 for z itself and 2 for the value of f.
Do you really need to draw graphs in order to do the problems you are given?

 

[Solidmozza]

My fault - I didn't make myself clear. I realize fully that you would need 4 dimensions to draw it on one graph, however what I meant was that they want us to draw two complex planes - the z plane (inputs) and a w plane (outputs). For instance, if we chose a subset A of the complex numbers such that f:C-->C and A={z=x+iy|-pi<y_<pi} what are the set of output values of z maps to w, where w=e^z? Sorry if this doesn't make it clear either :S

 

[HallsofIvy]

Ah, I've never really liked that method! The best you can do is take some simple curves or lines in the z= x+ iy plane and sketch the result in the f(z)= u+ iv plane.

For example to graph f(z)= z², look at the vertical lines z= x. Then f(z)= x^{2 which is just u= x2 (constant for each vertical line and so a horizontal line- but not evenly spaced and above the real axis for both x and -x). The horizontal lines, z= iy, give z2= -y2, again horizontal lines but below the real axis.}