Although I have completed a Bachelor of Science degree, I have never learnt complex analysis in college.
Let $F(a) = a^{q}$ with $q$ non integer. It is known that as $a$ passes over the negative real axis, we need to consider the situation in which $F(a)$ is a multi-valued function.
Thus $(e^{0})^{q}$ is differrent from $(e^{2 \pi i})^{q}$.
Now, is it possible that we calculate $F(a)$ using power series expansion at $a = a_{0}$ with $a_{0}$ very close to and below the negative real axis ?
For instance, the power series of $F(a)$ at $a = a_{0}$ is \[ F(a) = \sum_{n \geq 0} F^{(n)}(a_{0})/n! (a - a_{0})^{n}. \] Now, \[ F^{0}(a_{0}) = a_{0}^{q}, \ldots , F^{(n)}(a_{1}) = q(q - 1)\cdots (q + n - 1)(a_{0})^{q - n}, \] and so the power series of $F(a)$ ought to be valid for $|a - a_{0}| < |a_{0}|$.
That way, could we go over the negative real axis in $a$ claiming that the power series assigns a unique extension to $a^{q}$?
This may be rather a basic question, but I always doubt when multi-valued functions play the role of multi-valuedness.
Thanks in advance.