ittag-Leffler's Theorem

If a function analytic at the origin has no singularities other than poles for finite , and if we can choose a sequence of contours  about  tending to infinity such that  never exceeds a given quantity  on any of these contours and  is uniformly bounded on them, then

where  is the sum of the principal parts of  at all poles  within . If there is a pole at , then we can replace  by the negative powers and the constant term in the Laurent series of  about .

REFERENCES:

Jeffreys, H. and Jeffreys, B. S. "Mittag-Leffler's Theorem." §12.006 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 383-386, 1988.