Undergraduate Course: Further Complex Variables (MATH11233)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Year 5 Undergraduate)
||Availability||Available to all students
|Summary||This course investigates the theory of complex variables beyond our third year Honours Complex Variables course. In this course we will develop a modern approach to complex analysis, taking a ¿several complex variables¿ perspective in the treatment of classical results such as Runge's approximation and Weierstrass' factorisation theorems. It will also naturally lead us to the sophisticated and important monodromy theorem.
Complex variables is one of the most elegant, beautiful and powerful theories we encounter in our undergraduate curriculum. We already got a nice taste of this topic in Honours Complex Variables which revealed surprising properties of complex differentiable functions; for example, once differentiable implies infinitely differentiable and more! This in turn cracked open the door and led us to Cauchy¿s miraculous theorem, allowing us to evaluate complex integrals over any closed curve.
In this course we will throw the door wide open and unmask the many sparkling gems which underpin the subject. We will investigate the theory of functions which are complex differentiable everywhere (we call such functions entire) and we will find that entire functions behave very much like complex polynomials. We got a glimpse of this from Honours Complex Variables where we saw that every entire function can be represented by its Taylor series expansion on the whole complex plane (one of the sparkling gems in that course). So we can think of entire functions as infinite polynomials. But the analogy with polynomials is far-reaching. Complex polynomials are determined by their zeros. This is the Fundamental Theorem of Algebra. Every complex polynomial can be represented uniquely as a product of linear factors determined by its zeros. We will see that an analgous statement holds for general entire functions. This is the Weierstrass Factorisation Theorem.
This factorisation theorem will lead us to a deep connection between the asymptotic number of zeros of an entire function with its growth rate at infinity. As an incredible consequence, we will prove that the famous Riemann zeta function has infinitely many zeros in the critical strip. This is a significant step towards the Riemann Hypothesis, arguably the most important unsolved problem in mathematics.
The Weierstrass factorisation theorem will follow from Runge's theorem which is a powerful result articulating when we can uniformly approximate complex differentiable functions on a compact set by complex differentiable functions on a given open set containing it. Runge's theorem can also be used to prove the important Mittag-Leffler theorem as well as giving us homological and cohomological forms of Cauchy's theorem. We will give a modern proof of Runge's theorem which is based on solving the inhomogeneous Cauchy-Riemann equation on a general open set. Recall that the homogeneous Cauchy-Riemann equation characterise when a function is holomorphic/complex differentiable.
Honours Complex Variables is an essential pre-requisite for Further Complex Variables. Facility with the general ideas in Honours Analysis and Metric Spaces are important, however the relevant results from these courses will be reviewed and discussed in lecture.
Entry Requirements (not applicable to Visiting Students)
|| Students MUST have passed:
Honours Complex Variables (MATH10067)
||Other requirements|| None
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of each prerequisite course before enrolling.
|High Demand Course?
Course Delivery Information
|Not being delivered|
On completion of this course, the student will be able to:
- Be able to work with Complex Variables by proving unseen results using the methods of the course.
- Correctly state the main definitions and theorems in the course.
- Produce examples and counterexamples illustrating the mathematical concepts presented in the course.
- Explain their reasoning about rigorous Complex Variables clearly and precisely, using appropriate technical language.
|The course does not follow any particular textbook. Notes will be provided. Some good reference books are:|
Complex Analysis in One Variable by R. Narashimhan (Birkhauser, 1985)
Complex Analysis by E.M. Stein and R. Shakarchi, (Princeton Lectures in Analysis, Princeton University Press, 2003)
Complex Analysis by L.V. Ahlfors, (second ed., McGraw-Hill Book Company, 1966)
Functions of one complex variable by J.B. Conway (second ed., GraduateTexts in Mathematics, vol. 11, Springer-Verlag, New York, 1978)
Real and Complex Analysis by W. Ruden (third ed., McGraw-Hill Book Company, 1987)
The Theory of Functions by E.C. Titchmarsh (second ed., Oxford University Press, 1976).
|Graduate Attributes and Skills
|Course organiser||Prof Jim Wright
Tel: (0131 6)50 8570
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427