I have a formula which is analytic in its argument $x$. In it, there is a square root of a variable as in $\sqrt{x}$. Although meaningful results are obtained when positive roots are taken for for positive values of $x$, if I insert negative values of $x$, the meaningful result obtains from the negative imaginary root.
Is there a standardized notation to notate this kind of square root?
$$ \sqrt{x}\enspace\text{"="}\, \begin{cases} |x|^{1/2},&x\geq0\\ -i\,|x|^{1/2},&x<0 \end{cases} $$
So that I may insert it into my formula without having to refer to the ugly case written above?
Somebody suggested that I can use $$ \sqrt{x-i\epsilon^+} $$ in the formula, with the limit $\epsilon \rightarrow 0$ from positive values either implied or written explicitly. Would there be any ambiguities associated with this kind of notation?
I would say that in general the function $\sqrt{x}$ is used to denote the principal square root of a number $x$. For poxitive $x$ it is generally understood that this is the positive one of the two square roots (i.e. $\sqrt9=3$ and not $-3$ by convention.) In the complex case we can't speak of positive or negative. However we do have someting like a principal value of a complex number $z$. It is determined by the princeple value of its argument: $\theta$. If we write $z=re^{\theta i}$ with $\theta=\operatorname{Arg}(z)$ we can usually define the principle value of $\sqrt z$ as:
This way you automatically and unambiguously talk about one particular square root.
N.B It all comes down to conventions. Make shure the reader knows what your conventions are.