Consider the complex number $z = \sqrt{X+i\delta}$, with $0< \delta \ll 1$ and $X$ may be negative or positive.
I would like to reexpress this in the form $z= a+ i\delta$ where $a$ is positive. If $X>0$, then $z = \sqrt{X(1+ i\delta/X)} = \sqrt{X} + i \delta$. What about the case $X<0$? In particular, I'd like to understand if the sign of the imaginary part changes there.
To give context, I was considering an equation of the form $$\tilde{x_0} = 1-\sqrt{X+i\delta},$$ and I’d like to write it in the form $$\tilde{x_0} = x_0 \pm i \delta.$$ That is, extract the imaginary part. I need to do this to do some analytic continuation. Here, $x_0$ is an analytic function.
Square root of a complex number is a multi-valued function, you get 2 complex numbers normally (unless you're taking square root of zero).
I don't think it's a matter of what we want but what it is. Your delta is already positive, you cannot change that. So you cannot impose restrictions on the $a$. It is what it is.
Also, if you square this number $\sqrt{X} + i \delta$ I don't think you get the original number $X + i \delta$, do you? So I don't think this is the right way of taking the square root. I usually go through polar coordinates when taking square root of a complex number (using DeMoivre's theorem).
Finally, see this for a general formula which involves
just the algebraic form of the complex number:
Square root of a complex number