Hurwitz's Root Theorem

Let  be a sequence of analytic functions regular in a region , and let this sequence be uniformly convergent in every closed subset of . If the analytic function

does not vanish identically, then if  is a zero of  of order , a neighborhood  of  and a number  exist such that if ,  has exactly  zeros in .

REFERENCES:

Krantz, S. G. "Hurwitz's Theorem." §5.3.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 76, 1999.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 22, 1975.