plot graph of function $f(z)=\frac{1+z}{1-z}$

Question

I am not able to plot graph of function $f(z)=\frac{1+z}{1-z}$. can anyone tell me how to do this without using any software?

Answer 1

This type of map is called a Mobius map, and it is helpful with these to find out where the unit disk goes. Plug in $z=e^{i\theta}$ and see what it simplifies to. The image of the unit disk should split the complex plane into two regions. Then see where the origin goes, and you will know which region $|z|<1$ maps to. The other region is where $|z|>1$ gets mapped to.

Answer 2

It's not clear why you want to do this without a computer. But the traditional route would involve finding certain key points, trends, gradients and such like.

For example, what happens as z tends towards 1? If it approaches from above it tends to negative infinity. If it approaches from below it tends to positive infinity.

What happens as z tends towards infinity? It tends toward -1 (both directions).

Are there any zeros? How does the gradient change?

Actually, just with that you can get a good idea if the shape, but the gradient will help confirm your intuition: you can figure out if there are any zeros, and where it's positive or negative.