Graphing complex functions(the image)

[desaila]

I'm really unsure how to go about graphing a complex function. Like, f(z) = z^2, where z = x+iy.

This ISN'T a homework problem, but I'm studying for an exam and that's an example in a book I'm reading and it says "the image of this function" and goes on explaining some things relevant to the drawing, but there doesn't seem to be a systematic way to go about doing this.

Thanks.

 

[Kreizhn]

Find the real and imaginary images of the mapping.

Since $$z= x+iy$$ then $$z^2 = x^2-y^2+2ixy$$.

Thus we have $$\Re(z^2) = x^2-y^2 \text{ and } \Im(z^2) = 2xy$$

Now think of this as a mapping from $$\mathbb{R}^2 \rightarrow \mathbb{R}^2$$ under the function $$f(x,y) = (x^2-y^2, 2xy)$$

 

[HallsofIvy]

Which means, of course, that you would need a four-dimensional graph!

What is often done is to take u(x,y)+ iv(x,y)= f(z)= x+ iy. Draw some lines in an xy-plane and show what those are mapped into in the uv-plane.
For example, with f(z)= f(x+iy)= (x²-y²+ i(2xy), the
horizontal line y= 0 is mapped into u= x², v= 0 which is just the vertical line v= 0. The horizontal line y= 1 is mapped into u= x²-1, v= 2x so u= v²/4, a parabola.