simple question on how to find complex roots

Question

I have the following: What are the solutions to:

$$e^{z^{2}}=-1$$

in the circle around $z=0$ of radius $R=2$

Apllying the complex $LOG$

I derived: $$z^2 = \pi*i(1+2k)$$

What is the simplest way to continue from here? I tried to write $z=x+iy$ and then derived that $x=+-y$ and then: $2x^2=\pi*i(1+2k)$ but im getting mixed with positive and negetive values and the values of $k$ are unclear...how can I solve this one easily? Thanks

Answer 1

One way would be to put things in polar form: write $z=r e^{i \theta}$, note that $i=e^{i \pi/2}$, and figure out what r and $\theta$ are.

EDIT: Didn't actually read your whole question. This would be how to solve $z^2 = (2 k + 1) \pi i$, but it's overkill for your actual question, which is how to solve $e^{2z} = -1$.