1. ## unusual equality

Find the solution of $\displaystyle e^z = z\text{ in }|z|<2$
and $\displaystyle \tan x = x\text{ in }-\frac{3}{2}\pi<x<\frac{3}{2}\pi$

2. Neither of those can be solved analytically. You will need a numerical method, such as graphing on a calculator.

For example, on the first one, you could set $\displaystyle y=e^x-x$ and then find all the zeros graphically on $\displaystyle -2<x<2$.

3. I believe the first one is supposed to be a problem in complex analysis (use of $\displaystyle z$ rather than $\displaystyle x.)$ Perhaps Rouché’s theorem might help.

4. What is Rouche's theorem????
I have tried to solve this but it got fail >_<