## Section

• ANALYTIC FUNCTIONS I fall- 2014

Welcome to the course in Analytic Functions! The course begins on the second of September, 2014. The course is scheduled to run during the first half of the semester with a final exam on the 25 October 2014.

There will be 15 lectures altogether. To each lecture there corresponds a certain amount of text in the textbook. There are also Lecture Notes (pdf-format) and a Video Lecture. These are mainly to help you to get an idea about the main results in the Section, but they do not cover everything in the course.

In addition, there will be 4 WebWork problem sets with 5 problems in each, which can give up to 3 bonus credits on the exam. At least 6 correctly solved problems gives 1 credit point, at least 12 correctly solved problems gives 2 credit points and at least 17 correctly solved problems gives 3 credit points, if you submit your answers before the corresponding dead-lines.

The problems are corrected by our computer system WebWork. If you are not familiar with the use of WebWork, look here.

During the course I will also send letters to the whole group to keep you informed about changes and the general progress of the course. Please make sure that you are activated on this coursepage. If you don't receive any mail, please check the news forum below. If there are messages there which you haven't received, something is wrong with your profile settings. You are also invited to discuss all kinds of course issues with your teachers and fellow students in our Discussion forum.

Martin Tamm, Teacher

email: matamm@math.su.se

Content

We discuss analytic functions, integration and expansions of analytic functions in power series, residues, conformal maps, harmonic functions, physical applications and some elementarty facts about analytic functions of several variables. This material can be applied to the modeling of processes within different areas of natural sciences, i.e. in physics.

Lectures
08.30-10.15 (Martin Tamm) in room 32, Building 5, Kräftriket

Exercises
10.30-11.15 (Jens Forsgård) in room 32, Building 5, Kräftriket.

Textbook and other sources

• Saff and Snider: Fundamentals of complex analysis, Prentice-Hall.
• Tamm: Elementary Properties of Analytic Functions of Several Variables (pdf).
• Lecture notes day4 contains material which is not in the book.
• Earlier examinations with solutions.
• #### Open allClose all

• Instructions: Clicking on the section name will show / hide the section.

• 1
Topic
1

## 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. But in addition we also study the extended complex numbers or, as they are also called, the Riemann sphere. This topic has no immediate application in the beginning of the course, but will be very useful later on when we consider conformal mappings.

The Riemann sphere can by stereographic projection be identified with the complex plane plus one additional point, the point of infinity. And the main geometric property of interest here is that lines and circles in the ordinary complex plane both correspond to circles on the Riemann sphere (i.e. intersections of the sphere with planes in three-space. This property makes the Riemann sphere a very natural arena for studying so called Möbius transformations later on, since these have the property that they can map lines to lines or circles and circles to circles or lines, but always preserve the set of circles AND lines. Note that example 4 in Section 2.1 is also relevant for this lecture.

Key concepts which you are expected to be familiar with after having studied this Section: Complex numbers, a domain in $\mathbb{C}$, rectangular and polar coordinates, the complex exponential function, the Riemann sphere, stereographic projection. It may also be useful to have some idea about what a Möbius transformation is.

• Text: 1.1 - 1.7 (7.3)

Exercises

• 1.1: 8, 10, 25.
• 1.2: 5, 7, 16.
• 1.3: 7(b, d, f, g, h), 13, 16, 17.
• 1.4: 11, 12.
• 1.5: 5, 7(b), 10.
• 1.6: 2, 3, 4, 5, 6, 7 ,8, 11, 12, 16, 17.
• 1.7: 1, 2, 5.

WeBWorK

• WW-Problem Set 1, The test below will open on the 2014-09-02.
• 2
Topic
2

## Analytic and Harmonic Functions, Cauchy-Riemann Equations

Material:

In this part we study the Cauchy-Riemann equations (CRE). These equations are probably familiar to most students, but we start by introducing them in two independent ways. The first way comes from vector analysis. As it turns out the CRE correspond precisely to the condition when we consider line-integrals of complex valued functions. Thus, the CRE are equivalent to the property that integration is locally independent of the path of integration.

The second approach comes from investigating when a function in is complex differentialble. It is a remarkable fact that the two approaches above both lead to the same CRE. Functions which satisfy the CRE are called analytic.

Finally, we also consider harmonic functions. Their importance in this theory comes from the fact that both the real and imaginary parts of an analytic function are harmonic. In addition, this opens up for many applications since harmonic functions occur freaquently in applications, e.g. in potential theory.

Key concepts: independence of the path of integration, the Cauchy-Riemann equations, complex differentiability, harmonic function, harmonic conjugate.

• Text: 2.1 - 2.5

Exercises

• 2.2: 18 (16 in the second edition).
• 2.3: 5, 6, 11(a, c, f, g), 13(a, c, e, g).
• 2.4: 1, 2, 3, 7, 8, 10, 11.
• 2.5: 2, 3(a, b, ce), 6.
• 3
Topic
3

## Elementary Functions.

Material:

In this Section we study elementary examples of analytic function. As it turns out, most of the wellknown elementary functions from calculus, will reappear here in complex form.

In some cases, like with the exponential function and the trigonometric functions, the generalization is quite forward. In other cases, like with the logarithm and the inverse trigonometric functions, we run into difficulties because of the fact that these functions do not in general have a unique extension to the complex plane: either we have to choose a particular extension or we may consider the function as "multivalued", i.e. we consider all possible valkues at once.

Key concepts: complex exponential, complex logarithm, complex powers, complex trigonometric functions and their inverses, multivalued functions, branches principle branch.

• Text: 3.2 -- 3.3, 3.5 (3.1 -- 3.3 in the second edition)

Exercises:

• 3.2 (3.1 in the second edition): 5(a, c, e), 6, 7, 9(b, d, f), 11, 13(a), 15, 17(a, c).
• 3.3 (3.2 in the second edition): 1, 3, 4, 5(b), 8, 11, 19.
• 3.5 (3.3 in the second edition): 1(a,c,d), 3, 7, 8, 11.
• 4
Topic
4

## Complex Integration and Cauchy's Integral Theorem.

Material:

We begin by considering curves in the complex plane and how to integrate along them. This is very similar to how you integrate along curves in Vector Analysis, so this should hopefully not be to difficult. But the rest of this Section is concerned with the most important (and most difficult) Theorem in this course, namely the Cauchy Theorem, i.e. that the integral of an analytic function along a closed contour is zero. This is one of the places where the presentation in our book is not sufficiently rigorous, so I have taken care to write detailed lecture notes on this part which you should consider as an integral part of the course.

The proof consists of two steps: first we prove the Cauchy Theorem in the case of a triangle. Then we prove the general case as a consequence of the first step. The second step requires some (not too difficult) results from advanced calculus. Since advanced calculus is not a prerequest for this course, I have also included an appendix (see below) with elementary proofs of these results.

Key concepts: contour integral, Cauchy's Integral Theorem, homotopy, simply connected domain.

• Text: 4.1 -- 4.4

Exercises:

• 4.1: 1, 2, 4, 8, 9, 11.
• 4.2: 3(b, c), 6(a), 7, 913, 14(b, c), 15.
• 4.3: 1, 5, 6, 7.
• 4.4: 2, 3, 5, 9, 10(b, d), 12, 13, 15, 17.
• 5
Topic
5

## 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.

Important consequences of the integral formula include the Cauchy estimates of the derivatives of an analytic function and the fact that analytic functions are infinitely differentiable. Other consequences include the Meanvalue Property, the Maximum Principle and the Liouville Theorem. From the Liouville Theorem and the Cauchy estimates, we can give a complete self-contained proof of the Fundamental Theprem of Algebra.

Key concepts: smooth curve, contour, closed contour, contour integral, Cauchy's Integral Formula, Cauchy estimates, Meanvalue Property, Maximum Principle.

• Text: 4.5 -- 4.6

Exercises:

• 4.5: 3(a, c, e), 4, 6, 7, 15.
• 4.6: 1, 3, 4, 11.
• 6
Topic
6

## 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. In this Section, we encounter a third way of characterizing analytic functions, namely as functions which can locally be represented by convergent power series expansions. The proof of this fact depends on the Cauchy Integral Formula in the previous Section. In particular, we reobtain all the familiar Taylor Series for the elementary functions which now also make sense for complex arguments.

As a particular consequence of the theory of power series, we deduce an explicit formula for the Fibonacci numbers.

Key concepts: power series, radius of convergence, domain of convergence.

• Text: 5.1 -- 5.3

Exercises:

• 5.1: 2(a, c), 4, 7(a, c, e), 11.
• 5.2: 1(a, c, d, e), 4, 5(c, d, e), 11, 13, 15.
• 5.3: 3(a, b, d, f), 4, 5, 6, 7 ,8, 11, 14, 16.
• 7
Topic
7

## Laurent series.

Material:

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.g. the function $f(z)=e^{1/z}$ in a neighbourhood of the origin.) In such cases, a Laurent series can be the solution to the problem. These means that we expand the function in powers of both $z$ and $1/z$.

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.

Key concepts: Laurent expansion, annular domain,

• Text: 5.4 -- 5.5

Exercises:

• 5.4: 1(a, c), 3(a, b, c, d), 5(a).
• 5.5: 1, 4, 5, 7(a, b), 9.
• 8
Topic
8

## Zeros and Isolated Singularities.

Material:

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.

• Text: 5.6 -- 5.8

Exercises:

• 5.6: 1(a, d, f, g), 2, 3(a, c), 4, 6, 8, 10.
• 5.8: 1, 4, 5.
• 9
Topic
9

## 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. 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 $z_0$ corresponding to the term $1/(z-z_0)$.

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.

Key concepts: Residue, Residue Theorem.

• Text: 6.1 -- 6.3

Exercises:

• 6.1: 1, 3(a, c, f, g), 5, 6.
• 6.2: 1, 3, 4, 5.
• 6.3: 1, 3, 5, 6, 10, 11.
• 10
Topic
10

## Generalized Integrals, The Argument Principle.

Material:

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. As a consequence, we also see have the number of zeros inside a given contour of a complicated function can be estimated by comparing with a simpler function (Rouché's Theorem).

Key concepts: Generalized integral, Principle value, Jordan's Lemma.

• Text: 6.4 -- 6.7

Exercises:

• 6.4: 3, 4, 6.
• 6.5: 1(b, c, d), 2, 4, 7.
• 6.6: 1, 2, 6.
• 6.7: 1, 2, 4, 6, 8, 10.
• 11
Topic
11

## Conformal Mappings.

Material:

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. A particularly important class of such maps is furnished by the Möbius maps which we encountered already in the first Section.

Key concepts: conformal map, Möbius map, cross ratio.

• Text: 7.2 -- 7.4

Exercises:

• 7.1: 3, 7.
• 7.2: 3, 6, 8, 9.
• 7.3: 1, 2, 3, 4, 7, 10.
• 12
Topic
12

## More about Conformal mappings and Harmonic Functions.

Material:

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.e. the problem of constructing a harmonic function with given boundary values.

Key concepts: conformal equivalence, automorphism, Poisson formula.

• Text: 4.7, 7.1, 7.4 (also including $w=z^{\alpha}$, $w=e^z$, $w=\log z$, $w=\sin z$, $w=\frac12 (z+1/z)$ from table B2).

Exercises:

• 7.4: 1, 2, 3, 5, 6, 7, 9, 15 (14 in the second edition), 17  (15 in the second edition), 20 (18 in the second edition).
• 13
Topic
13

## Applications of Conformal Mappings.

Material:

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 methods in this Section can be very useful in various applications.

Key concepts: General Dirichlet problem, potential theory.

• Text: 7.6--7.7.

Exercises:

• 7.6: 1, 6, 9,
• 7.7: 6.
• 14
Topic
14

## Elementary Theory of Analytic Functions in Several Variables.

Material:

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.

• Text: Notes on analytic functions of several complex variables.

Exercises:

• Problems 1-6 in the Notes on analytic functions of several complex variables.
• 15
Topic
15

### Day 15, October 21 - Toggle

Repetion and preparation for the exam.

• 16
Topic
16

### EXAM, October 25 - Toggle

Examination: 09:00-14:00, October 25

Re-examination: 09:00-14:00, December 18