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Alexandre Eremenko
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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 $a=b=0$. See, for example this file. The answer is an explicit function, which is a special case of the hypergeometric function.

Alexandre Eremenko
  • 91.8k
  • 9
  • 260
  • 431