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