A topological version of the argument principle and Rouche’s theorem

2007 ◽  
Vol 146 (1) ◽  
pp. 5592-5602
Author(s):  
E. G. Sklyarenko
Keyword(s):  
Argument Principle ◽  
Topological Version ◽  
Rouche’S Theorem ◽  
Rouché's Theorem

A Remark on Rouché's Theorem

1976 ◽  
Vol 83 (3) ◽  
pp. 186-187 ◽  
Author(s):  
I. Glicksberg
Keyword(s):  
Rouche’S Theorem ◽  
Rouché's Theorem

Application of Rouche’s theorem for MP filter design

2009 ◽  
Vol 211 (2) ◽  
pp. 329-335 ◽  
Author(s):  
Gordana Jovanovic Dolecek ◽  
Vlatko Dolecek
Keyword(s):  
Filter Design ◽  
Rouche’S Theorem ◽  
Rouché's Theorem

scholarly journals A local mean value theorem for the complex plane

1969 ◽  
Vol 16 (4) ◽  
pp. 329-331 ◽  
Author(s):  
J. M. Robertson
Keyword(s):  
Complex Plane ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Local Mean ◽  
Rouche’S Theorem ◽  
Image Position ◽  
Rouché's Theorem

The equationneed not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).

Sign in / Sign up

Export Citation Format

Share Document