Details
| Original language | English |
|---|---|
| Pages (from-to) | 801-813 |
| Number of pages | 13 |
| Journal | PRIMUS |
| Volume | 27 |
| Issue number | 8-9 |
| Publication status | Published - 13 Jan 2017 |
Abstract
Rouché’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouché’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouché’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.
Keywords
- Control systems, Laplace transform, Nyquist stability criterion, Rouché’s Theorem, winding number
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Social Sciences(all)
- Education
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In: PRIMUS, Vol. 27, No. 8-9, 13.01.2017, p. 801-813.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Unpacking Rouché’s Theorem
AU - Howell, Russell W.
AU - Schrohe, Elmar
N1 - Publisher Copyright: ©, Copyright © Taylor & Francis Group, LLC. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2017/1/13
Y1 - 2017/1/13
N2 - Rouché’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouché’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouché’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.
AB - Rouché’s Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The winding number provides a geometric interpretation relating to the conclusion of Rouché’s Theorem, but most undergraduate texts give no geometric insights that lead to an understanding of why Rouché’s Theorem holds. In addition, most texts do not inform students that a stronger version of the theorem exists. In this paper we present a simplified proof of the stronger version, which is a suitable topic for students to pursue as a short project, and provide a geometric argument for the weaker version. Finally, as a project for advanced students, we unpack a standard application of this theorem as used in control systems: the Nyquist stability criterion.
KW - Control systems
KW - Laplace transform
KW - Nyquist stability criterion
KW - Rouché’s Theorem
KW - winding number
UR - http://www.scopus.com/inward/record.url?scp=85010703085&partnerID=8YFLogxK
U2 - 10.1080/10511970.2016.1235646
DO - 10.1080/10511970.2016.1235646
M3 - Article
AN - SCOPUS:85010703085
VL - 27
SP - 801
EP - 813
JO - PRIMUS
JF - PRIMUS
SN - 1051-1970
IS - 8-9
ER -