Rouché's Theorem

Given two functions  and  analytic in  with  a simple loop homotopic to a point in , if  for all  on , then  and  have the same number of roots inside .

A stronger version has been proved by Estermann (1962). The strong version also has a converse, as shown by Challener and Rubel (1982).

REFERENCES:

Challener, D. and Rubel, L. "A Converse to Rouché's Theorem." Amer. Math. Monthly 89, 302-305, 1982.

Estermann, T. Complex Numbers and Functions. London: Oxford University Press, p. 156, 1962.

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, p. 111, 1996.

Krantz, S. G. "Rouché's Theorem." §5.3.1 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 74, 1999.

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