I would like to expand $x^a$ with positive, real-valued $a$, in powers of $x$. That is, I am looking for an expression of the form
$$ x^a = \sum_{n=0}^\infty c_n x^n ~, $$
with explicit expressions for the expansion coefficients $c_n$ in terms of $a$.
No such expansion exists unless $a$ is an integer, in which case it is trivial. If $f(x) = \sum_{n=0}^{\infty} c_n x^n$ is equal to a convergent power series in some interval $I$ around $0$ (i.e. $f$ is analytic), then it is infinitely differentiable. But if $a$ is not an integer, the function $f(x) = x^a$ is singular at $x=0$. See also this answer.
