Approximation of $\sqrt{1+wi}$

Question

How can $\sqrt{1+wi}$ be approximated? where $-\infty<w<\infty$;

My aim here is getting rid of the square root.

I've tried binomial, Maclurin and Taylor series around various points. but they dont work for large $w$ .

Answer 1

The value is of the square root is

$$\sqrt{1+iw}=\sqrt{\frac{\sqrt{1+w^2}+1}2}+i\frac{w}{\sqrt{2(\sqrt{1+w^2}+1)}}$$

however, this has even more square roots, only that now they are of positive real numbers.