Rouche’s Theorem in Complex Analysis
Marian C. Whitney
1Khushil Rana
11
Applied Mathematics, University of Connecticut, Storrs
May 7, 2020
Introduction
Eugene Rouch´e was born in 1832 in Southern France and was an influential man of his time. Rouch´e graduated from Saint-Barbe College, located in Paris. He then continued his studies and graduated from Ecole Polytechnique in 1852. A couple years later he taught physics and mathematics at Nantes and Lycee Charlemagne. During his tenure he was an admissions examiner and years later was nominated to become Professor of Descriptive Geometry. He was a father of one. He was published in journals and has written famous mathematics textbooks. His main mathematics topics that he studied were complex functions, descriptive geometry, probability theory and algebra. Trait´e de g´eom´etrie ´el´ementaire was a two volume work published in French and has been revisioned to a 7th edition published in 1900. This two volume work held contents about angles, lines, circumferences, polygons and more. He also published M´emoire sur la s´erie de Lagrange in 1862, Trait´e de g´eom´etrie ´el´ementaire (1864-1866), Sur la discussion des equations du premier degr´e in (1890), and many others. In Sur la discussion des equations du premier degr´e is where he produced results on solving systems of linear equations [1].
One of his most well-known publications was in his alma mater’s institutional mathematics journals. This journal included Rouche’s theorem in complex analysis. A. Austin stated that, “The theorem gives a method for telling when two holomorphic functions have the same number of zeros in a region of the plane bounded by some curve.” In other words, this helps to find the number of roots of a function in a region under certain conditions or parameters.
Proof
To begin the proof on Rouche’s Theorem, it is important to note the following: Suppose that C is a closed contour There are two functions, f(z) and g(z), that are analytic inside of C f(z)g(z) at each point on C. Then, f(z) and f(z) + g(z) will have the same number of zeros, with multiplicity included, inside of C.
Rouche’s Theorem and the Fundamental Theorem of Algebra
Rouch`e’s Theorem can be used to prove the Fundamental Theorem of Algebra. This proof is only true if the polynomial, denoted P(z), has n number of roots with degree n.
On the next page is another example of Rouche’s Theorem and the Fundamental Theorem of Algebra being put to use. This is a much more broken-down variant of the Theorem as it incorporates multiple steps.
Examples
Rouche’s Theorem can be applied to numerous functions with the intent of determining analyticity and roots of various functions. The purpose of this is to narrow down the number of roots in a given function under set conditions. This theorem is a useful tool in complex analysis that aims to split the main function
into two, distinct functions to serve that very purpose. This following example sets to find the number of zeros in a given function:
For this first example, let us consider the function
We can confirm that there are three roots inside the unit circle, given that the highest power is 3. To prove this, we can set
which still has three roots Given the theorem, we can already say that both f and g are analytic on C, and they do not have any roots on the unit circle as well. Thus, we can conclude the following:
In this case, the subtraction of f(z) and g(z) implies that it is equal to the function that has three roots, which would be f(z).
Connection to Coursework
In relation to the course material throughout the semester, Rouche’s Theorem can be applied to concepts discussed during the lectures. One of the most prominent relations that Rouche’s Theorem can be applied
with is the Fundamental Theorem of Algebra. In complex analysis, the Fundamental Theorem of Algebra states that if there is a polynomial, p, with complex numbers of degree n, then the polynomial has exactly n roots. In terms of Rouche’s Theorem, the polynomial is split up into two separate functions, f(z) and g(z). The theorem states that the sum of these two functions would yield the same number of zeros as the degree the polynomial is in. As shown in the example, the sum of f(z) and g(z) have n number of zeros, since the polynomial is in the nth degree. This applies when both functions are analytic inside C and that f(z) ¿ g(z) at every point inside of C. In addition, the theorem can be applied to show that there is an n amount of zeros in a given function. For instance, say there is a function:
inside of C with the constraint that z=2. You want to prove that there are exactly 5 roots in this function. So, the main function can be split as:
Since we mentioned that Rouche’s Theorem would hold true if f(z) is greater than g(z), which is precisely true. In this scenario:
That being said,
which renders the theorem to hold true. This also proves that there are exactly 5 roots in the original function. Rouche’s Theorem is mainly associated with the Fundamental Theorem of Algebra for the purpose of rationalizing zeros in the complex plane.
Applications
The idea of Rouche’s Theorem is to determine how many zeros are present and to analyze the behavior of such zeros if given a constraint. One of the many applications that incorporates Rouche’s Theorem is the Queueing Theory. The Queueing Theory is a branch of mathematics that studies the behavior of waiting in lines, or rather a queue. The use of Rouche’s Theorem becomes relevant in “[proving] the existence of a certain number of zeros in the domain of regularity of a given function.” If the theorem can be successfully integrated, then it can help find the “solution of the functional equation for the generating function of the stationary distribution” [9].
However, that is no easy task. It can be difficult to incorporate the classical Rouche Theorem. Queueing problems run into the issue of determining the analyticity of zeros in the function on the unit disk. To make it convenient, components of the theorem are utilized to assist in For instance, the function:
incorporates the a(z), representing the generating function. If the function A(z) were to be parameterized, then the function changes to
where t parameterized from
The behaviors of the zeros would be analyzed when t approaches, or is less than 1 [9]. Apart from its use in Queueing Theory Problems, Rouche’s Theorem can be applied to enhance dimensions in various design modules. For instance, authors Jose Ricardo Garcia Baez and Gordana Jovanovic Dolecek wanted to incorporate the theorem for the purpose of “[increasing] attenuation in combfolding bands” through the use of a “simple comb zero rotation on the unit circle in the complex z-plane.” Through implementation of Rouche’s Theorem, the zeros on the unit circle can be found by “adding and subtracting a positive constant to a comb-system function” [10]. When implementing this formula, the end result is said to be a new filter that has increased width far more durable than the old filter. Essentially, Rouche’s theorem is applied in this context to “identify a necessary and sufficient condition. . . .[which will] guarantee all zeros of the rotated term are on the unit circle” [10]. The whole premise of the theorem is to analyze and find zeros on the unit circle within the complex plane in order to determine the behavior of specific constants. The authors have created the main polynomial, HR+(z) which is defined by
Since the sum of f(z) and g(z) can be represented as
they have also deduced it will have the same number of zeros in the contour C. And since all of those zeros are present, they automatically lie on the unit circle. As demonstrated in the computations for the filter
design, both authors enhanced the equations by modifying the function to approach a certain point, thus rendering all of the zeros on the unit circle. [3]
Conclusion
To conclude, Rouche’s Theorem “...gives a method for telling when two holomorphic functions have the same number of zeros in a region of the plane bounded by some curve.” In other words, it is used to find the number of roots of a function in a region under certain conditions or parameters. Without the
contributions and dedication from Eugene Rouch´e, this theorem would not be possible. We also know that Rouch`e’s Theorem can be used to prove the Fundamental Theorem of Algebra and can enhance dimensions in various design modules. Rouche’s Theorem can be applied to numerous functions with the intent of determining analyticity and roots of those various functions. It is also used in many applications; one of the more popular applications that incorporates Rouche’s Theorem is the Queueing Theory, as stated before. Even with it being difficult to incorporate, Rouche’s theorem is still applicable in certain areas of research and applied mathematics. It continues to ease the burden of solving complex functions and remains one of the most versatile theorems in complex analysis to this day.
1
References
[1] J. J. O’Connor, E. F. Robertson. Eugene Rouche. School of Mathematics and Statics University of St. Andrews. (2017). Web. http://mathshistory.st-andrews.ac.uk/Biographies/Rouche.html
[2] A. Austin. Rouche’s Theorem. The University of Oxford. (2019).Web. https://www.chebfun.org/examples/complex/RoucheTheorem.html
[3] Jovanovic Dolecek, Gordana and Vlatko Dolecek. ”Application of Rouche’s Theorem for MP Filter Design.” Applied Mathematics and Computation 211, no. 2 (2009): 329-335.
doi:10.1016/j.amc.2009.01.058.
[4] “Rouche’s Theorem.” Mathonline, WikiDot, mathonline.wikidot.com/rouche-s-theorem.
[5] Monard, Francois. “Argument Principle, Rouche’s Theorem and Consequences.” Argument Principle, Rouche’s Theorem and Consequences, University of California Santa Cruz,
people.ucsc.edu/ fmonard/Sp17Math207/lecture6.pdf.
[6] Orloff, Jeremy. “Argument Principle.” Argument Principle, Massachusetts Institute of Technology, math.mit.edu/ jorloff/18.04/notes/topic11.pdf.
Alissa.pdf.
[8] TheMathCoach. “Rouche’s Theorem with Example — Complex Analysis 13,” Youtube, 27 Mar. 2017, https://www.youtube.com/watch?v=3nOAkEp-rFM
[9] Klimenok, V. On the Modification of Rouche’s Theorem for the Queueing Theory Problems. Queueing Systems 38, 431–434 (2001). https://doi.org/10.1023/A:1010999928701
[10] Jose Ricardo Garcia Baez Gordana Jovanovic Dolecek (2017) On simple comb zero rotation: Rouche’s theorem approach, International Journal of Electronics, 104:4, 569-582, DOI:
