I am studying complex analysis and themain thing about complex analysis that disturbs me is the difficulty in visualization.I have studied exponential function and logarithmic function in complex analysis.Now I want to visualize them as maps from $z$-plane to $\omega$-plane.I want to know how these maps act on the complex plane.Can someone help me understand this with suitable diagrams or animations?
What I know about exponential map and logarithmic map is the following:
I know that exponential function defined on $\mathbb C$ is $2\pi i$-periodic and hence is not injective,so inverse does not exist.So,we cannot talk about $\log$ function,but exponential function is injective on each strip $S=(-\infty,\infty)\times (-\pi,\pi)$ which projects onto $\mathbb C-(-\infty,0]$.In fact the map $z\mapsto e^z$ is a covering map from $\mathbb C-\{(2n-1)\pi:n\in \mathbb Z\}$ to $\mathbb C-(-\infty,0]$.So,we can see that on each strip $e^z$ is a homeomorphism,in particular injective,so we can define the logarithmic function $\log:\mathbb C-(-\infty,0]\to S$.
Is this all correct?
Suppose you have some region $D$ in the complex planes, then you can 'mesh' it by putting two family of non parallel curves. In the picture you have shown, we have the curves of constant x and constant y. Once we have this, we can understand the complex function by the mapp of this whole mesh.
This is actually useful because it gives the characteristic property of analytic function: they preserve infinitesimal squares. In more specific words, suppose you make a mesh in which the whole domain is split into tiny squares, then you would find that the square is mapped to another infinitesimal square. For visualizing the mesh, you can check out this stack over flow post which uses python to show maps of the mesh.
On the point of inverses, it is somewhat complicated because there is a lot of aspect such as branch, branch cut, Riemann surfaces. But, the essential philosophy is that the natural inverse for the $e^z$ function is $\ln(z)$ as a multi-function (mf). THis is because the evaluate of $e^z$ repeats everytime we add $2\pi i$ to it's input, when we invert the function, we get all those values which differ by $2\pi i $ as for one input.
One may think we can say that at a point we can take one of the branchs eg: take one of the angle which correspond to that point for evaluating $\ln(z)$ but if we are to try consider a loop around the origin then again we would realize that the mf issue persists.
To truly dissect the mf into a single valued function, you need a branch cut. The branch cut involves removes removing points in the domain so that we can't even do the loop thing I said.
Once we put a branch cut, we can consider an argand plane for each branch. After this, we can stich together the different planes as a Riemann surface which finally makes our function single valued from a Riemann manifold to an argand plane.