This function was studied since Newton's time; I suppose the power series was obtained by Newton himself. It is analytic, but not in the whole
$(a,b)$ space since it has singularities, in particular when $a=b$. See, for example
<a href="https://www.math.purdue.edu/~eremenko/Pdf/ellipse.pdf">this file</a>. The answer is an exlpicit function, which is a special case of the hypergeometric function.