Suppose I have a branch of the logarithm, that is, a continuous function $L(z)$ on some region $\Omega$ such that $e^{L(z)} = z$. We see that this defines a branch for the square root function on $\Omega$, via $\sqrt{z} = \exp(1/2 L(z))$, since
$$(\exp(1/2 L(z))^2 = \exp(L(z)) = z$$
I am wondering if a sort of converse of this holds. Suppose on the other hand, we have a branch for the square root, i.e. some continuous function $R(z)$ on $\Omega$ such that $R(z)^2 = z$. Is there some way to get a branch of the logarithm from $R(z)$? If so, does this generalize (i.e. what branches for multi-valued functions will determine a branch of the logarithm)?
Best Answer
While every cut for the square-root function is admissable as a cut for the logarithm in $\mathbb{C}\backslash\{0\}$, the complex plane without the origin, there are only two "layers" of the square-root function and countably many for the logarithm. So it cannot be said that a branch of the square-root function "determines" a branch of the logarithm, unless you are willing to simply impose some choice of branches ad hoc.