Rouché,s Theorem
المؤلف:
Challener, D. and Rubel, L.
المصدر:
"A Converse to Rouché,s Theorem." Amer. Math. Monthly 89
الجزء والصفحة:
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11-3-2019
3666
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.
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