Is there a geometric projection for every complex function?

Question

I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in the Dimensions movie (part 6) or here. Now, is there is a geometric surface (which you could plot with 3dplot) for every complex function which when projected renders the grid transform? (For $$z \rightarrow\ z + k/z $$ this would probably have to cut or overlap itself.) Also what is the further relationship between geometry and complex numbers? And maybe someone knows what software they used for the animations. Is there any sw along these lines able to visualize your own algebras (vs. functions), i.e. not just $(ac - bd), (ad + bc)$ for multiplication.

Answer 1

It seems you have software available that produces 3D-plots from parametric represenations of surfaces, as $$(\phi,\theta)\mapsto\bigl(\cos \phi\cos \theta,\ \sin \phi\cos \theta,\ \sin \theta\bigr)\qquad\Bigl(0\leq \phi\leq 2\pi, \ -{\pi\over2}\leq \theta\leq {\pi\over2}\Bigr)$$ (the unit sphere). You can use this software to get "grid plots" of complex analytic functions $f$ as well. When such an $f$ is given as an expression $f(z)$ write it out separating real and imaginary parts: $$f(x+iy)=u(x,y)+i v(x,y)\ ,$$ where now $u$ and $v$ are real functions of the real variables $x$ and $y$. Then let your software produce the 3D-plot corresponding to the parametric representation $$(x,y)\mapsto\bigl(u(x,y),\ v(x,y),\ 0\bigr)\ .$$ In the drawing $\bigl($choose the viewpoint at $(0,0,\infty)\bigr)$ you will see the images of the gridlines $x={\rm const.}$ and $y={\rm const.}$ Similarly you could arrange things such that you see the images of circles $|z|={\rm const.}$ and halflines ${\rm arg}(z)={\rm const.}$