Key concepts: power series, radius of convergence, domain of convergence. Power series expansion is a very useful tool for studying analytic functions. But sometimes it is not enough. For instance, if we want to study an analytic function in a neighbourhood of a point where it is not analytic, then power series expansion may be of no use at all. Consider e. In such cases, a Laurent series can be the solution to the problem. These means that we expand the function in powers of both and. In this Section we prove a general theorem about Laurent expansions and also study how to expand a given function in simple cases.
This theory will be important later on for the study of isolated singularities. In this Section we are concerned with the behaviour of an analytic function near a zero or an isolated singularity. In the case of a zero, the behaviour is very similar to that of a polynomial.
In the case of a singularity, the behaviour can be very much like the singularities of rational functions, but it can also be much worse in which case we talk of an essential singularity. The main instrument for studying these singularities is the Laurent expansion of the previous Section. Key concepts: zero, order of a zero, isolated singularity, pole, order of a pole, essential singularity. We know from the Cauchy Theorem that the integral of an analytic function around a closed contour is zero if the function is analytic inside.
If, on the other hand, the function has one or more singularities inside, the integral will in general be non-zero. It turns out to be a very important part of the theory of analytic functions to compute such integrals. The essention notion in this case is that of a "Residue", which is the coefficient in the Laurent expansion around the point corresponding to the term.
The Cauchy Residue Theorem gives an explicit formula for computing such integrals. This will also enable us to compute many real integrals which are much more difficult to compute by real methods. It turns out that very many generalized real variable integrals can very conveniently be computed using Residue Calculus. For instance, integrals over the whole real line can be computed as the limits of integrals around the boundaries larger and larger semi-discs which eventually fill up the whole upper half-plane if the integrand tends to zero sufficiently quickly.
We consider different variations on this theme. We can also use certain contour integrals to estimate the number of zeros and poles inside the contour. This is the content of the Argument Principle. Key concepts: Generalized integral, Principle value, Jordan's Lemma. In this Section, we study how to map one region in the complex plane onto another by using an analytic function. Such maps are called conformal, since they have the property that they preserve angles between curves passing through a point.
We continue the study of conformal maps. To a certain extent, conformsl maps give a way to reduce problems in general simply connected domains to the corresponding problems in the unit disc. We also study the question about how much choice we have when we construct a conformal equivalence between two regions. In particular, this leads to the characterization of the automorphisms of the unit disc.
We also use the Cauchy Formula to solve the Dirichlet Problem, i. In this Section, we consider some applications of conformal mapping theory to the Dirichlet Problem. Although the Poisson Formula gives a general solution, it is often very difficult to compute it explicitly, whereas a clever choice of a conformal map can make the computations much easier. We also consider an example in a non-simply connected domain.
The theory of analytic functions of several variables is a very large and useful part of mathematics.
Here we only consider some very basic facts. But still this is enough to show that the theory contains elements which are very different from the one-variable case. In particular, this is the case with the phenomenon of unrestricted analytic continuation: some domains have the curious property that any analytic function in the domain automatically extends beyond that domain.
Key concepts: open ball, polydisc, Cauchy formula in a polydisc, power series expansion, unrestricted analytic continuation. Weekly outline General. News forum. Grade criteria File. Old exams URL. Notes on analytic functions of several variables File. Important conformal maps File. Leave comments on video lectures Anonymous Feedback. Discussion forum. Complex Numbers. Complex Numbers Material: The first lecture to a large extent deals with prerequests for this course, including elementary properties of the complex numbers, vector and polar forms and the complex exponential function.
Lecture notes 1 File. Lecture notes 2 File. Elementary Functions. Material: In this Section we study elementary examples of analytic function. Text: 3. Lecture notes 3 File. Complex Integration and Cauchy's Integral Theorem.
- Complex Variables.
- MATH20101 Real and Complex Analysis: Complex Analysis learning materials, 2018-19;
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Material: We begin by considering curves in the complex plane and how to integrate along them. Text: 4. Lecture notes 4 File. Cauchy's Theorem, Appendix File. Cauchy's Integral Formula. Material: This Section is devoted to some of the most important consequences of Cauchy's Theorem, in particular the Cauchy Integral Formula which gives the value of an analytic function at any point inside a closed contour in terms of the values of the function on the contour itself. Lecture notes 5 File. Powerseries Expansion of Analytic Functions. Material: In an earlier Section we have seen that analyticity can be characterized either as complex differentiable functions or as functions with locally path-independent integrals.
Advanced Complex Analysis I
Text: 5. Lecture notes 6 File. Laurent series.
Material: Power series expansion is a very useful tool for studying analytic functions. Key concepts: Laurent expansion, annular domain, Text: 5.
- AMATH A: Applied Complex Analysis.
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Video Lecture 7 File. Lecture notes 7 File. Zeros and Isolated Singularities.
Part 1 Proceedings of the Seminar held in Bucharest, June 28 – July 3, 1981
Material: In this Section we are concerned with the behaviour of an analytic function near a zero or an isolated singularity. Lecture notes 8 File. The Residue Theorem with Applications. Material: We know from the Cauchy Theorem that the integral of an analytic function around a closed contour is zero if the function is analytic inside.
Key concepts: Residue, Residue Theorem. Text: 6. Lecture notes 9 File. Generalized Integrals, The Argument Principle. Material: It turns out that very many generalized real variable integrals can very conveniently be computed using Residue Calculus. Lecture notes 10 File.
Advanced Complex Analysis I | CosmoLearning Mathematics
Conformal Mappings. Material: In this Section, we study how to map one region in the complex plane onto another by using an analytic function. Text: 7. Urs Frauenfelder Otto van Koert. Aristide Halanay Vlad Ionescu. Colin MacLachlan Alan W. Alexander Vasil'ev. William Cherry Yang Chung-Chun.
Dang D. Ang Rudolf Gorenflo Vy K. Le Dang D. Jim Agler John McCarthy. John Harnad Alexander R. William T. Ross Harold S. Jose Ignacio Cogolludo-Augustin. Nessim Sibony Jean-Christophe Yoccoz. Rolf S. Sigurdur Helgason. Semigroups in Geometrical Function Theory. Gaston M. Vinogradov Memorial Volume The S. Vinogradov Memorial Volume.
Joseph A. Cima William T. Handbook of Complex Analysis Volume 1. Sundaram Thangavelu. Mirror Symmetry Clay Mathematics Monographs. Lectures on Vanishing Theorems Oberwolfach Seminars. Helene Esnault Eckart Viehweg. Govorov Yu. Piermarco Cannarsa Carlo Sinestrari. Philip L. Bowers Kenneth Stephenson. Peter Kravanja Marc van Barel.
Related Complex Analysis [Lecture notes]
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