Notater om forelesninger som skjedde for mer enn en uke siden.
Uke 34 er mest viet kapittel 1.
2006-08-23: Onsdagen snakket jeg løst og fast om komplekse tall og regning med disse: Konjugering, real- og imaginærdel, absoluttverdi, trekantulikheten er nøkkelord (men det er et par ulikheter i ruten på side 17 jeg ikke omtalte). Jeg har også snakket om polar form (avsnitt 1.3), med det tilhørende faktum at multiplikasjon med et komplekst tall er en sammensetning av en dilatasjon (selv om jeg ikke brukte det ordet) og en rotasjon. Og så har jeg omtalt DeMoivres formel og bruken av den til å beregne komplekse n-terøtter, og i den forbindelse klarte jeg ikke dy meg, men bare måtte nevne den komplekse eksponensialfunksjonen. Mer om den i morgen.
2006-08-24: Jeg brukte noe tid på å diskutere noen enkle avbildninger i det komplekse plan, og en ikke fullt så enkel en (en Möbiusavbildning). Deretter innførte jeg eksponensialfunksjonen og argumenterte iherdig for identiteten ew+z=ewez. Det ble knapt tid til å begynne på logaritmen, så den får vi ta grundigere neste uke.
Uke 35 avsluttet vi kapittel 1 og komme godt inn i kapittel 2: Områder i det komplekse plan, grenser og kontinuitet, deriverbarhet og analytiske funksjoner. Kanskje blir det også litt tid til Cauchy–Riemann-ligningene.
2006-08-30: Onsdag fortsatte jeg med logaritmen. Vi feide innom de trigonometriske og hyperbolske funksjonene, men brukte ikke mye tid på dem. De komplekse trigonometriske funksjonene oppfyller alle de samme regnereglene som de reelle funksjonene med samme navn, og beviset er så godt som alltid manipulasjon med eksponensialfunksjonen. (De omvendte trigonometriske funksjonene er det verre med: De uttrykkes ved logaritmen og lider av samme problem som logaritmen med multiple verdier, bare et lite hakk verre.) Vi tok også for oss komplekse potenser, derunder n-terøtter. Deretter startet vi på kapittel 2, med en gjennomgang av omegner, randpunkter og åpne mengder. Jeg ser at jeg glemte å definere hva en lukket mengde er: En grei definisjon er at en mengde er lukket dersom den inneholder hele sin rand.
2006-08-31: Vi definerte et område (region) som en åpen, ikketom og sammenhengende mengde. Så gikk vi løs på grensebegrepet: Vi avsluttet dermed omtrent midtveis i avsnitt 2.2. Men jeg rakk å påpeke en feil i læreboken: Teorem 1 side 82: Den siste regelen (6) er feil, om vi ikke legger til en ekstra hypotese f(w0)=A, altså kontinuiteten av f i w0. (Den hypotesen er for øvrig den andre likheten i konklusjonen (6).) Problemet er at beregningen av grenseverdien i (6) kan komme til å trenge verdien av f(w0), men grenseverdien av f avhenger eksplisitt ikke av denne funksjonsverdien.
Week 36 Was spent finishing Chapter 2.
2006-09-06: We have some exchange students in the class, so we changed to English. I finished up the limit concept, and said some words about infinity: We deal with just one infinity in complex analysis, as opposed to the two (plus and minus infinity) that we are used to in real analsys. Something going to infinity means just its absolute value going to inifinity, no matter what phase (or direction) it is.
Continuity was next: A function is called continuous at a point if its function value at that point equals its limit at the point. And it is called continuous in a set if it is continuous at every point in the set.
Finally, I introduced differentiability: A function f is differentiable at a point z0 if we can write f(z)=f(z0)+f′(z0)(z−z0)+ε(z)(z−z0) where ε(z)→0 as z→z0. If you split that up a bit, it says that f(z) is the sum of an affine function f(z0)+f′(z0)(z−z0) and something that is smaller than any nonzero affine function near z0, namely ε(z)(z−z0). (An affine function is just a sum of a constant and a linear function. Very commonly, people simply call them linear, even if they don't map zero to zero.) It is quite easy to see that then f is continuous at z0.
(Web pages are not really good for writing mathematics it seems: In the above, it is very hard to see the primes on the defivative f′. Oh well.)
2006-09-07: We shall call a function analytic in an open set if it is differentiable at every point in the set. We shall also call it analytic at a point if it is analytic in a neighbourhood of that point. Note that we are departing a bit from the terminology in the book here: The book requires that the derivative shall be continuous for the function to be analytic. But we shall see that if f is analytic (my definition), then so is f′ – and then by induction, f has derivatives of all orders. So the difference in terminology is not substantial. I showed that the exponential function is analytic (with the expected derivative, namely itself), and so is the logarithm – with the expected derivative too, no matter which branch of the logarithm you choose (which should not be surprising since the difference between branches is a constant, namely a multiple of 2πi.) and will spend a little time on the Cauchy–Riemann equations.
Week 37 (2006-09-13 and 2006-09-14) saw the start of integration in chapter 3. We have defined the integral, and worked with it a bit. I plan a different path through this material than the one followed by the book.
What is the path integral, and what do we want it for? I spent spend some time talking about paths, path lengths, and the sums that define the integral (like Riemann sums, but different). If you think paths are simple objects, you may need to think again: Take a look, if you wish, at this article (in Norwegian) on simple curves that fill out positive areas in the plane. The article begins by explaining arc length, which is useful anyway. I proved that when the path is (piecewise) smooth, the integral becomes as defined in the book. The integral exists more generally when the integrand is continuous and the path has finite length, but I have not proved this. An “almost proof” can be found in my notes.
Week 38 (2006-09-20 and 2006-09-21) was spent introducing and working with homotopies, and culminating with a proof of the (homotopy version of) Cauchy's integral theorem, what I called Cauchy–Goursat in my notes. The reason for this name is that, while Cauchy required the continuity of the derivative, Goursat showed that you can do without this assumption. If you learn only one thing from the proof, it should be this: The differentiability of a function f at a point z means that f(ζ) is almost an affine function of ζ in a neighbourhood of z. And the integral of an affine function around a closed path is zero: This is shown by a direct calculation. So the integral of f around a closed path in this neighbourhood should be very small. The proof consists of “amplifying” this result from small neighbourhoods to larger regions; for this it is vital to quantify precisely what is meant by “very small” above.
Week 39 (2006-09-27 and 2006-09-28): Peter Lindqvist lectured. He covered the Cauchy integral formula, and its generalization (from my notes). We are saving the last section of the note (the global Cauchy formula) for later (if we have time for it). He will also covered most of section 3.7 in the book, with the exception of Schwarz's lemma.
Week 40: First, Schwarz's lemma. Then we got started on Chapter 4 (series). In order to motivate, I did this by jumping straight into the Maclaurin series (section 4.4). Please note that if you understand Maclaurin series (Taylor series centered at 0), then you understand general Taylor series as well: You get the Taylor series of f around z0 by substituting z−z0 for z in the Maclaurin series for g(z)=f(z0+z). The Maclaurin series comes straight out of Cauchy's formula by substituting the geometric series for (ζ-z)−1 into Cauchy's formula and then interchanging the order of integration and summation. Much of the material of sections 4.1–4.3 consists of showing that such manipulation is justifiable.
After doing that, I returned to section 4.1 (sequences and series). Much of this theory is exactly like the corresponding theory for real sequences, so I see little point in repeating all of it. I expect you to read much of this on your own.
Week 41: There are a few bits of section 4.1 that needed commenting on: Mainly absolute convergence, rearrangements, and the Cauchy product. I tried to keep it brief and not too detailed.
Then section 4.2: I had already stated the definition of uniform convergence and mentioned its main consequence, namely the interchange of limit and integration. Next, we note that for analytic functions, the derivative can be expressed via the Cauchy integral formula, so that uniform convergence can be interchanged with the derivative as well! This is extremely different from the real case, where such manipulation typically leads to wrong results.
Week 42: I am taking a break from complex function theory and started on the theory of differential equations, first doing the basics on Fourier series. (sections 7.1–7.4). Later on, we will spend roughly one lecture a week doing complex function theory and one on differential equations, but we may vary this depending on progress.
2006-10-18: Introduced the Fourier series, both real and complex form (the book does the complex form later). I showed that if a function f is the limit of a uniformly convergent Fourier series, then the Fourier coefficients can be computed via a simple integral. Next, I showed a special case of convergence for the special case x=0 where f(0)=0. It remains to show that this special case actually implies the general case. Oh, and I relied on the Riemann–Lebesgue lemma, which we have not proved yet.
2006-10-23: Showed that the special case I dealt with in the previous lecture implies the general case by the simple expedient of adding a constant to the function, and shifting it sideways. Computed the Fourier series of two simple functions. I have still not proved the Riemann–Lebesgue lemma.
Week 43: Only one lecture this week; the midterm test used up the lecture hour on Thursday.
2006-10-25: I finally proved the Riemann–Lebesgue lemma, for the case of a function that is bounded, Riemann* integrable and periodic. I first proved it for a simple step (this is a trivial calculation), then for a step function (which is a linear combination of simple steps) and finally I approximated an integrable function by a step function, and we're done. With that, our proof of the pointwise convergence for Fourier series is complete. Next, I sort of skimmed through some material in section 7.3 and 7.4, dealing with functions of other periods than 2π (it's all just a question of rescaling) and so-called half-range expansions, where a function defined on an interval is extended first by making it even or odd, then extending it periodically, with the end result being a pure cosine or sine series. Finally, I computed the cosine series of the function (π−x)x on the interval [0,π]. Evaluating the series at x=0, we end up with a proof of the famous formula &sum n−2=π2/6. (There are several other ways to get the same formula, but the vast majority of them relies on Fourier series somehow.)
*Riemann integrable just means that the integral exists, as a limit of Riemann sums – the way you learned it in calculus. More general integrals exist, such as the Lebesgue integral.
Week 44: Starting this week, and continuing until I feel a need to make different arrangements, I will lecture on Fourier series and differential equations on Wednesdays, and on complex functions on Thursdays. More details for this week's lecture will be forthcoming.
2006-11-01: I concentrated on section 8.2, showing how a wave equation is solved by separation of variables and matching of a Fourier series with the initial data. I rescaled the equation so we only needed to consider the case where the length of the string is π, and c=1. (We will come back to convolutions and the Parseval identity (in section 7.5) at a later point.)
2006-11-02: I have a rather ambitious plan for the day: Section 4.3 on power series: Basically just the radius of convergence and the fact that we may integrate and differentiate these series termwise. I have already done most of section 4.4, while trying to motivate all this series stuff. But I'll give a quick refresher. Then section 4.5 on the Laurent series. The existence proof for the Laurent series uses a extension of the proof for the Taylor series.
Week 45:
2006-11-08 – extra lecture 10:15–12:00: This lecture covered section 4.6 on zeros and singularities. (I wrote a short note summarizing the section in just four pages: See under “Syllabus”.) Assume f is analytic in a neighbourhood of a point z0, possibly except at z0 itself. We use the Laurent series around this point to note that there are but three possibilities:
(i) Either f is analytic at z0, or it has a removable singularity there, which means that it will be analytic once we redefine f(z0); or
(ii) f has a pole at z0, which means that its absolute value goes to infinity there; or
(iii) f has an essential singularity at z0, which means it takes every value, with one possible exception, in every neighbourhood of z0. (This latter bit is known as Picard's great theorem. I did not prove it, but settled for a weaker version, the Casorati–Weierstraß theorem, saying that every complex number can be approximated arbitrarily close by function values f(z) for z in any neighbourhood of z0.)
2006-11-08 – regular lecture: Continued the subject of separation of variables with a look at some heat conduction problems from section 8.3 and 8.4.
2006-11-09: This lecture saw the start of residue theory: Basically, section 5.1–5.2. In a sense there is nothing new there, but we will systematize what we already know about integration and Laurent series to better work with path integrals. The basic idea is very simple: By the Cauchy theorem, a path integral around a number of singular points can be expressed as the sum of integrals along smaller paths surrounding just one singularity each. And each of those can be worked out in a variety of ways, including looking at the Laurent series at that point.
Week 46: We are getting near the end of the curriculum.
2006-11-15: Rounded off with an example from 8.6: The Laplace equation (steady state heat equation) in a rectangle.
2006-11-16: Sections 5.3 and 5.4, with the exceptions noted in the syllabus. So from 5.3 we considered only integrals along the entire real line, of a rational function whose denominator has degree at least 2 higher than the numerator. So long as there are no poles on the real axis, the integral is 2πi times the sum of residues of poles in the upper half plane. (Or the lower halfplane, if you multiply by −1. In fact I showed that the sum of all residues of such a polynomial is zero.)
Section 5.4 generalizes this to the situation where the rational function is multiplied by a trigonometric function. I replaced Jordan's lemma by an estimate that is a bit easier to prove, using a rectangular path instead of a semicircular one.
Week 47: These final lectures were used mainly for summarizing.
2006-11-22: Summarized Fourier series and separation of variables.
I had not covered the convolution product before, so I did now, with the important consequence Parseval's identity.
2006-11-23: Summarized complex function theory.
I spent a little extra time on the exponential function and the logarithm, since these important functions are a bit easier to understand with all the theory we now have available.