Hurwitz's Root Theorem
Let {f_n(x)} be a sequence of analytic functions regular in a region G, and let this sequence be uniformly convergent in every closed subset of G. If the analytic function
| lim_(n->infty)f_n(x)=f(x) |
does not vanish identically, then if x=a is a zero of f(x) of order k, a neighborhood |x-a|<delta of x=a and a number N exist such that if n>N, f_n(x) has exactly k zeros in |x-a|<delta.
See also
Argument Principle, RootExplore with Wolfram|Alpha
WolframAlpha
More things to try:
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.Referenced on Wolfram|Alpha
Hurwitz's Root TheoremCite this as:
Weisstein, Eric W. "Hurwitz's Root Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HurwitzsRootTheorem.html