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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, Root

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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 Theorem

Cite this as:

Weisstein, Eric W. "Hurwitz's Root Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HurwitzsRootTheorem.html

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