Uses Mittag-Leffler star

an illustration of regions of convergence of mittag-leffler expansion , taylor series expansion around (the regions bounded blue curve , red circle respectively).

any complex-analytic function ƒ defined around point in complex plane can expanded in series of polynomials convergent in entire mittag-leffler star of ƒ at a. each polynomial in series linear combination of first several terms in taylor series expansion of ƒ around a.

such series expansion of ƒ, called mittag-leffler expansion, convergent in larger set taylor series expansion of ƒ @  a. indeed, largest open set on latter series convergent disk centered @ , contained within mittag-leffler star of ƒ at a