Additional comments related to material from the class. If anyone wants to convert this to a blog, let me know. These additional remarks are for your enjoyment, and will not be on homeworks or exams. These are just meant to suggest additional topics worth considering, and I am happy to discuss any of these further.

• Thursday, December 9:
  • For one complex variable, we can consider f(z) = 1/z, which has an isolated singularity (a pole) at z=0. What about two complex variables? If we try f(z1,z2) = 1/z1 then it has poles in the entire plane z1=0, z2 arbitrary. What else could we try? In real variables we have 1/r^2 in 2 or 3 dimensions, such as 1/(x^2+y^2). If we try that here we get f(z1,z2) = 1/(z1^2+z2^2). The problem is that this has a pole in the plane z2 = i z1. There is no isolated singularity in several complex variables -- that's the key takeaway of the Domain of Holomorphy arguments. In several variables, it is even more restrictive to be holomorphic.
  • See the comments from Tuesday, December 7th for more on fractal geometry and iteration of complex functions. We didn't talk about the 3x+1 Problem, but it's a great example of what happens when you iterate a simple function. Very complicated behavior emerges, with some fascinating symmetries and patterns. See the article by Jeff Lagarias for more information.
  • Finally, a few points about optimal algorithms / efficient algorithms.
    • Horner's algorithm to evaluate a polynomial quickly: 4x^3 + 5x^2 - 3x + 8 = ((4x+5)x - 3)x + 8 (saves a few multiplications!). Saving multiplications is very important; one application of evaluating polynomials quickly is in constructing Mandelbrot sets.
    • Telescoping sums: often re-arranging the algebra leads to a significantly easier computation.
    • Fast matrix multiplication: Naively we expect it takes N^3 multiplications to find all N^2 entries of A^2 or AB when A and B are NxN matrices. The Strassen algorithm (see also the Mathworld entry here, which I think is a bit more readable) does it in about N^(log_2 7); the reason for this savings is that they can multiply two 2x2 matrices with seven and not 8 multiplications (3 = log_2 8). The best known algorithm is the Coopersmith-Winograd algorithm, which is of the order of N^2.376 multiplications. See also this paper for some comparison analysis, or email me if you want to see some of these papers.
      • Some important facts. The Strassen algorithm has some issues with numerical stability.
      • One can ask similar questions about one dimension matrices, ie, how many bit operations does it take to multiply two N digit numbers. It can be done in less than N^2 bit operations (again, very surprising!). One way to do this is with the Karatsuba algorithm (see also the Mathworld entry for the Karatsuba algorithm).
  • We then ended the day by a quick tour through the subject. The power of the Cauchy Integral Formula is striking; again and again it appears (analytic is holomorphic, the Cauchy inequalities which lead to Liouville's theorem, the Riemann Mapping Theorem, ...). This integral representation is entirely different than what happens in real analysis, and is responsible for much of what we see.
  • Here is my solution to the look-up table question posed by my brother Jeff from MathWorks (formerly The Mathworks). There are many related questions that are of interest -- if anyone would like to pursue these applications further, please let me know.
  • It's hard to choose the final theorem of the day, so I'll just provide the following link and let you choose.

• Tuesday, December 7:
  • Several Complex Variables is a beautiful subject, of which we can only skim the surface. There are many similarities between several and one complex variable, as well as some important differences. While Cauchy's Integral Formula still holds, an enormous difference is the notion of a domain of holomorphy. An open set is a domain of holomorphy if there is a holomorphic function on U that we cannot extend across the boundary. Note that the punctured unit disk is a domain of holomorphy in one complex variable (simply take the function 1/z); however, the n-dimensional analogue (when n > 1) is not a domain of holomorphy! This is a bit similar to some of the issues in applying Green's theorem in the plane versus Stokes' theorem in 3-space, as a point singularity is not so bad for Stokes as we can deform the curve in the higher dimension.
  • It's interesting finding a function that cannot be extended. We saw one way to do this is to take a dense subset of the boundary and look at Sum_n (1/4^n) 1/(z-z_n). Of course, one needs to do some analysis to prove this provides an example; the key fact (at least for circles and annuli) is that any point in the open set is at least a certain distance from the boundary).
  • See Hartog's theorem for (yet another!) difference between several real and complex variables.
  • Key reading for Fractal Geometry and Iterations of Functions: Julia and Fatou sets, Mandelbrot set and a nice zoom here (see the Mandelbrot explorer and the Chaos game, which you can play here), the Newton fractal, and Chaos theory. A good program to investigate fractals is XaoS.
  • First use of fractal geometry in a movie: Star Trek II: The Wrath of Khan: The Genesis Effect (also known as the `good' Star Trek movie). As long as we're doing Star Trek II scenes, great ones include the battle with Khan (and how things work on a starship) and the final battle, the Kobayashi Maru test, though sadly cannot find quickly the 'I don't like to lose' clip and the 'hours instead of days now we have minutes instead of hours'.
  • For more on efficient algorithms, see Chapter 1 of my book An Invitation to Modern Number Theory (Section 1.2).
  • The theorem of the day? Well, since we're doing Star Trek let's do Fermat's Last Theorem. The Star Trek connection? There's Picard talking about how he likes to try to prove unsolved math problems like Fermat's last theorem (I don't believe he ever says he's related to the complex analyst Picard) as well as Dax (start around 3:25 on the clip) who talks about a previous host having the most original approach since Wiles' successful proof.

• Thursday, December 2:
  • In our application of Laplace's method to proving Stirling's formula for n!, we adjusted the function Phi so that it vanished at the critical point. There is of course no need to do so; some authors prefer to renormalize like this, some don't. One of the homework problems builds on this by looking at the moments of the standard normal, and we're able to get nice formulas for (2m-1)!!. Of course, one could also derive this from (2m)! / (2^m m!). Looking at it like this, it is clear why all the sqrt(pi)'s vanish.
  • Our sketch of the proof of the Prime Number Theorem shows how complex analysis results come into play, and the importance of the location of the zeros of zeta(s). A very important fact is that zeta(s) is non-zero for Re(s) = 1. See for example Chapter 3 of my book An Invitation to Modern Number Theory (see Exercise 3.2.19). The proof there is a very clever application of what has been hailed as the most important inequality in mathematics, namely that for real x, x^2 >= 0.
  • A big technique in studying sums in number theory is partial summation (aka, integration by parts: the discrete version). The idea is that frequently certain sums, though not exactly what we want to study, are easier to handle. It is thus a worthwhile trade to work with these sums first and then modify them to return to what you want. Another instance where this occurs is in attacking problems through the Circle Method, such as the Goldbach Problem.
  • Note that in two problems this week (Stirling's formula and the Prime Number Theorem) we saw the importance of replacing a simple, discrete object with an integral. This occurs throughout higher mathematics. There are numerous advantages to integral representations. For example, in the analysis for the Prime Number Theorem we computed a weighted integral of the logarithmic derivative of zeta(s) two different ways. We want to compute contour integrals of this quantity -- complex analysis is screaming at us to do so; however, if we are computing it as an integral then we need to work on the `other side' with an integral as well.
  • There were questions on whether or not our arguments on pi(x^{1- epsilon}) < x^{1-epsilon} were good. It's not too hard to show that A x / log(x) < pi(x) < B x / log x for some A, B with 0 < A < 1 < B < oo. Chebyshev was the first to do so, around 1850. For more details, see here as well as the nice article here (on Erdos' proof).
  • If you want to know more about the functional equation of the zeta function, as well as proofs of why it isn't zero if Re(s) > 1, I'm happy to meet and chat. You can also look at Chapter 3 of my book An Invitation to Modern Number Theory, where we spend a lot of time trying to emphasize the role complex analysis plays in studying the primes. One of the most interesting items is that it is more natural to weight the primes than to count them equally. Further, it's easier to count all prime powers and not just primes.
  • For the Theorem of the Day: (1) Euler's Partition Identity (I think something very close to this, if not this itself, has surfaced in some discussions with people in the class). It's related to generating functions and expansions. (2) The footballer's theorem. This is a nice article by Harold Boas on H. Bohr (N. Bohr's brother) and some of his results on the boundary of complex analysis and number theory. It discusses many of the concepts we've looked at recently, including analytic continuation and the differences between uniform, absolute and conditional convergence.

• Tuesday Nov 30:
  • We've talked a few times about analytic continuation. It's always worth asking what the source of the analytic continuation is. For the geometric series, it's the geometric series formula. For the Gamma function, it's integration by parts. For the Riemann zeta function, it's the Poisson summation formula, which relates sums of a nice function at integer arguments to sums of its Fourier transform at integer arguments. There are many proofs of this result. In my book on number theory, I prove it by considering the periodic function F(x) = Sum_{n = -oo to oo} f(x+n). This function is clearly periodic with period 1 (if f decays nicely). Assuming f, f' and f'' have reasonable decay, the result now follows from facts about pointwise convergence of Fourier series. There are other proofs; in chapter 4 of our textbook we find a nice proof based on the residue theorem -- it's worth reading this to see yet another example of the power of the residue formula.
  • Laplace's method is a terrific way to approximate the values of integrals as a certain parameter tends to infinity. From it we can derive Stirling's formula for n!. Not surprisingly, there are other ways to prove Stirling's formula. What I like about this proof is that it highlights the need for studying the analytic continuation of the Gamma function. We need the nice integral representation which gives us n! for any positive n. This is not the only application of the method -- see the appendix in the book for other examples.
  • Laplace's method is the first, but not only, approach for estimating certain integrals when a specified parameter tends to infinity. Frequently we want to approximate complex integrals such as Int_{x = -oo to oo} Exp[i s f(x)] g(x) dx; this leads to the Stationary Phase Method.
  • For more on the connections between complex analysis and probability (especially when a sequence of moments uniquely determines a probability distribution, which sadly is not always the case), see my notes from Math 341 (Probability), available here.
  • A very important part of our analysis was showing when an integral is well-defined. This frequently involves understanding the decay of the integrand as |x| --> oo as well as |x| --> 0. A frequent tool in such an analysis is L'Hopital's Rule. Key identities are x^r log x --> 0 as x --> 0 for any r > 0, and x^r exp(-x) --> 0 as x --> oo for any r.
  • It is very useful to be able to express a quantity as an integral; this was a key input in using Laplace's method to derive Stirling's formula. Another situation where this arises is differential equations, where we can often write the solution as a fixed point of an integral operator. See Picard's Iteration Method, as well as my notes from Math 209: Differential Equations.
  • The value Gamma(1/2) is extremely important. It is related to the normalization constant for the normal distribution (and thus, due to the Central Limit Theorem, occurs throughout probability). Here is a link to some notes from a book I'm writing; these notes include information on the Gamma function and sketches of proofs of Stirling's formula (as well as bounds towards Stirling's formula) -- comments welcome!
  • The theorem of the day is the Asymptotic (Half) Liar Formula.

• Tuesday Nov 23:
  • The notion of analytic continuation is one of the most important in complex analysis, allowing us to extend the definition of a function. The Riemann zeta function is defined by zeta(s) = Sum_{n = 1 to oo} 1/n^s for Re(s) > 1, and by unique factorization also equals Prod_{p prime} (1 - 1/p^s)^{-1}. We can analytically continue the zeta function to a meromorphic function on the complex plane, with only one pole, a simple pole with residue 1 at s=1.
  • One of the most famous and important problems in mathematics is the Riemann Hypothesis (this is in fact one of the seven Clay millenial prize problems; click here for the rules). The full problem list is below; familiarizing yourself with these seven problems is a great way to see what issues are driving current mathematical research. Note that often Professor Morgan runs a senior seminar on `The Big Questions', where these and other topics are covered.
    • Birch and Swinnerton-Dyer conjecture
    • Hodge conjecture
    • Navier-Stokes equation
    • P vs NP
    • Poincare conjecture (proved!)
    • Riemann hypothesis (this is not considered a proof!)
      • We do know a bit about the zeros. Hardy proved infinitely many lie on the critical line. Selberg was the first to show a positive percentage lie on the critical line. What this means is if you look at zeros in a box 0 <= Re(s) <= 1 and -T <= Im(s) <= T, then a positive percent have Re(s) = 1/2, and Levinson later extended this to one-third (which in some sense means a majority of zeros are on the line -- see me for the grammatical hair splitting). We know now that about 40% are on the line (the current world record is a t least 41.05%). See the wikipedia entry for more information.
    • Yang-Mills theory
  • In class we defined pi(x) to be the number of primes at most x. We discussed Euclid's argument which shows that pi(x) tends to infinity with x, and mentioned that with some work one can show Euclid's argument implies pi(x) >> log log x. As a nice exercise (for fun), prove this fact. This leads to an interesting sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357....   This sequence is generated as follows. Let a_1 = 2, the first prime. We apply Euclid's argument and consider 2+1; this is the prime 3 so we set a_2 = 3. We apply Euclid's argument and now have 2*3+1 = 7, which is prime, and set a_3 = 7. We apply Euclid's argument again and have 2*3*7+1 = 43, which is prime and set a_4 = 43. Now things get interesting: we apply Euclid's argument and obtain 2*3*7*43 + 1 = 1807 = 13*139, and set a_5 = 13. Thus a_n is the smallest prime not on our list genereated by Euclid's argument at the nth stage. There are a plethora of (I believe) unknown questions about this sequence, the biggest of course being whether or not it contains every prime. This is a great sequence to think about, but it is a computational nightmare to enumerate! I downloaded these terms from the Online Encyclopedia of Integer Sequences (homepage is http://www.research.att.com/~njas/sequences/  and the page for our sequence is  http://www.research.att.com/~njas/sequences/A000945 ). You can enter the first few terms of an integer sequence, and it will list whatever sequences it knows that start this way, provide history, generating functions, connections to parts of mathematics, .... This is a GREAT website to know if you want to continue in mathematics. There have been several times I've computed the first few terms of a problem, looked up what the future terms could be (and thus had a formula to start the induction). One last comment: we also talked about the infinitude of primes from zeta(2) = pi^2/6. While at first this doesn't seem to say anything about how rapidly pi(x) grows, one can isolate a growth rate from knowing how well pi^2 can be approximated by rationals (see http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.2184v3.pdf  for details; unfortunately the growth rate is quite weak, and the only way I know to prove the needed results on how well pi^2 is approximable by rationals involves knowing the Prime Number Theorem!).
  • Key inputs in our study of the zeta function were unique factorization and special values, such as the sum of the reciprocals of the squares equaling pi^2/6 and the harmonic sum diverges. We were able to use the harmonic sum to get information about the sum of the reciprocals of the primes, `proving' there are enough so that this sum diverges. Calculating Brun's constant (the sum of the reciprocals of twin primes) led Nicely to discover the Pentium bug; a nice description of the discovery of the bug is given at http://www.trnicely.net/pentbug/pentbug.html. 
  • The purpose of today's lecture, rushed in places as it was, was to give you a sense of the need for rigor. Innocent looking statements can fail; interchanging limits and integrals can fail. If we're not careful, we can be led to believe we have a proof of the Riemann Hypothesis! I find it fascinating that the analytic continuation of a limit is not necessarily the limit of the analytic continuation, and in fact there is at least one situation where the two differ! This is a very nice result which we proved today. I'll write more extensive notes on this for those who wish.

• Thursday Nov 18:
  • We saw how much easier the pacman contour was than the branch cut method. For the integral of 1/(x^3+1) from 0 to oo, we tried one last approach, namely splitting the integral into two parts, from 0 to 1 and then from 1 to oo, and then using the geometric series formula. Such a splitting of an integral is quite common in number theory, allowing us to exploit the different behavior of the function in the two regions. In fact, this is one of the key steps in the most common proof of the functional equation of the Riemann zeta function.
  • Functional equations allow us to take a function that is initially defined in one region and extend the definition elsewhere. In addition to zeta and L-functions, another common (and very important example in probability, statistics and number theory) is the Gamma function, which generalizes the factorial function.
    • The Riemann zeta function is defined by zeta(s) = Sum_{n = 1 to oo} 1/n^s for Re(s) > 1, and by unique factorization also equals Prod_{p prime} (1 - 1/p^s)^{-1}. An interesting consequence of its analytic continuation is the `identity' that 1 + 2 + 3 + 4 + ... = -1/12; amazingly, `facts' like this arise in modern physics (string theory)!
    • Another nice identity, arising from the continuation of the geometric series, is that 1 + 2 + 4 + 8 + ... = -1. Interestingly, this is true 2-adically.
    • A very important one in probability / statistics is (-1/2)! = sqrt(pi); this comes from the Gamma function, which interpolates the factorial function. Bohr and Mollerup proved that if you assume a continuation of the factorial function has three reasonable properties than it must be the Gamma function.
  • In some sense, these functional equations are similar to the Cauchy integral equation. There we saw that knowing a holomorphic function on the boundary was enough to uniquely determine it everywhere.
  • There are numerous identities / series expansions for simple functions of pi; the difficulty is knowing how to manipulate what one has to get one of these desired relations (I wrote a short note on how to prove Wallis' formula from the Student t-distribution). There are lots of interesting algorithms to calculate digits of pi efficiently. If you're interested in `experimental' mathematics (like this or more standard topics), check out this site.
  • We talked a bit about cellular automata today. A big early example is Conway's Game of Life. Stephen Wolfram has also done a lot of work in the field, see especially his book A New Kind Of Science.
  • In the spirit of when certain values determine a function, the theorem of the day is the Cauchy - Kovalevskaya theorem.

• Tuesday Nov 16: 
  • Today we returned to contour integration, one of the most useful parts of complex analysis. We first considered trigonometric integrals; see my lecture notes on examples of contour integrals. Our first example was integrating Sin[theta]^2n, and uses what I call the `bring it over' method. Here we have some integral I, and by integrating by parts we find I = K + cI, where K is something known. Solving, we find (1 - c)I = K, or I = K/(1-c); so long as c is not 1 (which would force K to be 0) we can solve for I. This is one of my favorite integration techniques, where we write the quantity we want in terms of itself!
    • Another instance of the `Bring it over' method is in generating functions and finding explicit formulas for solutions to recurrence relations. See for example the generating function of the Fibonacci numbers, which leads to Binet's formula for a closed form expression of the Fibonacci numbers.
  • We saw that we could simplify the recurrence relation and get the simple, closed form expression, but we had to make some semi-clever choices of how to multiply by 1 to express our answer in terms of common symbols. It's not a horrible idea to multiply by (2n)!!, as that `completes' the factorials in the numerator. The trick is then pulling out the powers of 2 in the denominator. The best way to see that this is a good thing to do is to note that once we have (2n)! in the numerator, we want to have k! (2n-k)! in the denominator for some k, as this will give us a binomial coefficient.
    • The double factorial is a very important combinatorial function. It arises in many places, perhaps most importantly as the moments of the standard normal, where the 2m-th moment is (2m-1)!!; recall the double factorial means take every other term until you reach 2 (if you start at an even number) or 1 (if you start at an odd number). One interpretation of (2m)!! is that this is the number of ways to pair off 2m objects into m pairs of 2. Here is a derivation: there are (2m choose 2) ways to choose two elements to be our first pair, (2m-2 choose 2) ways to choose the next pair of elements, down to (2 choose 2) ways to choose the last pair. We then have to divide by m!, as we don't care which pair label two elements are given. We thus have (2m) (2m-1)/2 * (2m-2) (2m-4)/2 * ... * (2)(1)/2 / m! = (2m)! / 2^m m! = (2m)! / (2m (2m-2) * ... * 2) = (2m-1) (2m - 3) * ... * 1.
    • Here's another way to see the above formula, this time by induction. The base case is clear. For the inductive step, assume the number of ways to pair 2m objects in m pairs of 2 is (2m-1)!!. If we now have 2m+2 objects, note object 1 must be paired with something -- there are 2m+1 ways to choose its match. We are now left with 2m objects, which by induction can be paired in (2m-1)!! ways. Multiplying gives (2m+1)!! as desired.
  • We converted trigonometric integrals to contour integrals, and saw the advantages of this. This method of integration is quite useful in numerous problems, especially in Circle Method problems in number theory. If you want to read more about the Circle Method, I can pass along the relevant chapter from my book.
  • Our next approach involved branch cuts. Here we show that we can actually use the multivaluedness of the complex logarithm for good (or at least to evaluate some integrals). Not surprisingly, for many problems we have choices as to how to proceed. For the integral from 0 to oo of 1/(x<sup>3</sup> + 1) we have two options. One is to introduce the logarithm function and exploit the branch, while the other is to choose a giant pie contour. In general, it's frequently a trade-off between choosing an easy function and an easy contour. We have to make the clever choice of introducing the logarithm in the branch method, but the resulting algebra was a bit easier to simplify.
    • Note how painful it was to attack the integral via the branch cut. We had to do an enormous amount of book-keeping to show that as epsilon --> 0 and R --> oo that everything converged to what we wanted. There are countless special cases to look at, we found that epsilon log(2R) had to tend to 0, .... Note the write-up in the Bristol notes skips all these arguments, and jumps to the answer in just a few lines. As you do more and more of these integrals, you can `see' the cancellation / convergence. For example, the integral over the semi-circle of radius epsilon is no problem as epsilon log(epsilon) --> 0 as epsilon --> 0. The remaining pieces are `standard' (once you've done enough such problems!).
  • For the theorem of the day, I though it would be nice to have something connecting combinatorics with complex analysis. Here's a nice one relating certain permutations and e.

• Thursday Nov 11: TBD
  • We finally made it to the proof of the Riemann Mapping Theorem! The proof begins with a powerful simplification. We first show that we may may our simply connected proper open subset of the complex plane conformally into the unit disk. Technically this is nice, as we now are working in a compact space. The key to this is the existence of the complex logarithm, and we see now why simply connected is so important -- without this we would not be able to start the proof!
    • One of course should be careful about saying that it is impossible to prove a result without resorting to using specific facts, even though those facts might seem quite obviously necessary to use. A terrific example is the elementary proof of the Prime Number Theorem (which says that as x --> oo, the number of primes at most x is asymptotic to x/log x). It turns out that this statement is equivalent to the fact that the Riemann zeta function zeta(s) = Sum_{n = 1 to oo} 1/n^s (or actually its meromorphic continuation) has no zero on the lien Re(s) = 1. This is quite clearly a complex analytic statement. It was thought that there could be no `elementary' proof of this (elementary doesn't mean easy; it just means without using complex analysis), but if there were one, boy would it open our eyes! Both statements are false. See this article by Dorian Goldfeld for the history of the proof of the Prime Number Theorem (and the priority dispute).
    • It's not inappropriate to talk about the Prime Number Theorem on the day we do the Riemann Mapping Theorem, as a plan of attack was sketched by Riemann in his classic (and only!) paper in number theory! An English translation is available here; the original (in German) is available here.
  • It's a fascinating subject as to what we can say about the boundary under the conformal equivalence. In some cases, such as polygons, we have formulas available for extensions. This is the Schwarz - Christoffel formula. In general, it is hard to discuss what happens at the boundary. If the boundary is nice, however, a bit more can be said.
  • The final step of the proof of the Riemann Mapping Theorem involves a useful technique. We take an object with a certain maximal Property I; if it doesn't have another desired Property II, we show we can find another valid object with even larger value of Property I, a contradiction. I see this as a variant of the following argument. We prove all positive integers exceeding 1 have a prime factorization. If not, let x be the smallest integer not having a prime factorization. Clearly x is not prime, so x = ab with a, b > 1. But as x was the smallest integer without a prime decomposition, a and b have a prime decomposition, which gives one for x. The argument here is a bit more involved, but similar in spirit.
  • A theme running through our proof was replacing one complicated function with a composition of simpler ones. This way we only need to understand the simpler maps and how they piece together. You may have seen this in other classes, such as elementary matrices in linear algebra (which is used for proving results ranging from Gaussian elimination to the Change of Variable theorem). It's frequently a good trade to do lots of simple problems rather than one hard one; thus we do all these maps of our region Omega to get it contained in the unit disk.
    • This is related to the extra credit problem on look-up tables (due December 3). The Babylonians used base 60. If they wanted to multiply x and y, they'd need a 60 x 60 table! Well, we can do a bit better as multiplication is commutative and see we only need to store 60 x 59 / 2 + 60 = 60 x 61 / 2 = 1830. Now, carrying around enough stone tablets to record over 1800 entries is not fun! I've heard that the way they multiplied was to note x y = ( (x+y)^2 - x^2 - y^2 ) / 2. At first this seems foolish, as we now have three multiplications and one division and two subtractions; however, this is a great exchange! The reason is that we need only have a table of squares, and that's just 60 or maybe 120 entries. This is a lot less to lug. In many modern engineering applications, it is essential to find efficient ways to do operations.
  • The theorem of the day seems clear: The Prime Number Theorem!

• Tuesday Nov 9: TBD, including   
  • The Arzela - Ascoli Theorem did not depend on complex analysis (the first part of the proof of Montel's theorem heavily used Cauchy's integral formula, another wonderful example of the utility of that result). I find it nice that Cauchy's integral formula makes another appearance; we had used it a lot earlier in the semester, but not as much lately.
    • Related to holomorphic functions are harmonic functions. These two have the wonderful property that their value at a point can be given by an integral (here it is the mean value property). Not surprisingly, a lot of results for harmonic functions are similar to holomorphic functions.
  • A central theme of the day was doing the analysis on a `nice' set and then showing that sufficed. Specifically, it is convenient to work on a compact set, as compact sets are made for uniform arguments. Remember a key property is that given any compact set, if we have an open cover then there is a finite subcover (see also the Heine-Borel theorem). There's lots of fun covering lemmas, for instance, the Vitali covering lemma. In today's lecture we showed we could do the analysis on a compact sets, and then using the method of exhaustion we write our open set as an increasing union of compact sets. You might recall seeing something similar to this in a geometry class, namely Archimedes determination of pi.
  • Some of you may have seen the Arzela - Ascoli Theorem in a course in Functional Analysis. There are numerous uniformity statements, such as the Uniform Boundedness Principle (also known as the Banach - Steinhaus theorem).
  • A huge part of our analysis today needed some set theoretic and/or topological results. We needed to be able to find a dense subset of any open set in the complex plane, and we also needed to generalize Cantor's diagonal argument to our situation. The diagonal argument is used to show the existence of transcendental numbers without actually constructing them.
    • It's a bit amusing to ponder that Cantor proved almost every number is transcendental, but his method could not give a specific transcendental number, not even one! Fortunately we already knoew of such numbers. Liouville in 1844 constructed some, using nice number theoretic properties related to how well irrational numbers may be approximated by rationals. We have a few interesting proofs for special numbers like e and pi. See the Gelfond-Schneider Theorem for more. Roth won a fields medal for his results on proving how well algebraic numbers may be approximated by rationals (Roth's theorem).
  • As long as we're mentioning Cantor, I'll add his Continuum Hypothesis to the list of comments, and note that it was resolved by my mathematical grandfather, Paul Cohen, using his theory of forcing.Cohen's mathematical father was Zygmund, who is very famous for work in trigonometric (Fourier) series. You can look up people's geneologies online here.
  • The comments above suggest a natural choice for the Theorem of the Day: Cantor's uncountability theorem.

• Thursday Nov 4:   
  • The analysis pre-reqs for the proof of the Riemann Mapping Theorem are significant, but provide a great way to review many of the concepts you've seen in previous classes. One of the first items we need is that of uniform continuity, followed (or perhaps in parallel with) that of a compact set. It's frequently an interesting (but delicate!) matter to determine what type of convergence we have. Big theorems in the subject are Weierstrass' polynomial approximation theorem and Fejer's theorem (which gives us uniform approximation of continuous functions by finite trigonometric polynomials, and Taylor expanding these gives Weierstrass' theorem). Fejer's theorem tells us a weighted Fourier series can converge better than the original Fourier series.
    • For the convergence of Fourier series, one of the best results is Carleson's theorem, which gives us almost everywhere pointwise convergence of the Fourier series of an L2 function to the function. If the function is not continuous it's easy to construct a Fourier series diverging at that point; a fun fact is how much of an overshoot there is (this is called the Gibbs overshoot, after J. W. Gibbs, one of the first American scientists of note on the world stage).
    • Other interesting pathological functions include the Devil's staircase and Weierstrass' continuous but nowhere differentiable function!
    • A big issue in approximation theory is how well a Fourier series converges, and whether or not it converges to the original function. Related to the Fourier series is the Fejer series. Both arise from convolving our function f with a kernel; the difference is that the Fejer kernel has much nicer convergence properties than the Dirichlet kernel (which gives the standard Fourier series). The Weierstrass and Fejer theorems involve the concept of an approximation to the identity (sequences of functions tending towards the Dirac delta functional). For the Weierstrass result, there are many proofs. One of my favorites leads to an explicit construction, using the Landau kernel functions (you have to scroll down on the page). This involves finding non-negative functions that integrate to 1 on [-1, 1] and have most of their mass near 0. This choice is K_n(x) = c_n (1 - x^2)^n. It's a fun exercise to determine c_n. I think the easiest way involves a great technique, the `Bring it over' method of integration. We proceed by induction. Let I_n = Int_{-1 to 1} (1 - x^2)^n dx. Then I_{n+1} = Int_{-1 to 1} (1 - x^2)^{n+1} dx = Int_{-1 to 1} (1 - x^2)^n (1 - x^2) dx = Int_{-1 to 1} (1 - x^2)^n  dx - Int_{-1 to 1} x (1 - x^2) x dx, where we wrote the last integral like this to facilitate integrating by parts. Recalling the definition of I_n gives I_{n+1} = I_n - Int_{-1 to 1} x (1 - x^2)^n x dx. Letting u = x and dv = (1 - x^2)^n dx, we see du = 1 and v = -(1 - x^2)^{n+1} / (2n+2). The boundary term vanishes, and we find I_{n+1} = I_n - 1/(2n+2) Int_{-1 to 1} (1 - x^2)^{n+1} dx = I_n - I_{n+1}/(2n+2). Bringing the unknown I_{n+1} over to the left, we find ((2n+3)/(2n+2)) I_{n+1} = I_n or, assuming I haven't made an algebra error yet, I_{n+1} = (2n+2) I_n / (2n+3). We now have a nice recurrence relation. I find this method somewhat miraculous -- we get the expression we don't know on both sides, and then solve for it!
      • I originally had I_{n+1} = (n+1) I_n / n, which I knew couldn't be right. Clearly I_{n+1} < I_n, and this has I_{n+1} > I_n. The mistake I made was that I forgot the minus sign in going from dv to v.
    • Related to approximations to the identity are mollifiers, which play a very important role in understanding many properties of number theoretic functions.
  • Convergence of sequences of functions is a `central' question in mathematics -- just think about the Central Limit Theorem from probability!
  • One of the most important questions to ask in a given situation is how the quantities in play depend on the various quantities. For this reason, it's often a good idea to have your parameters subscripted. For example, in Central Limit Theorem we have that the (appropriately normalized) sum of independent random variables converges to a standard normal. What's fascinating is that such a statement seems to be independent of the fine structure of the probability distribution. Where the structure comes into play is in the rate of convergence. See for instance the Berry-Esseen Theorem.
  • If anyone is interested in pursuing these topics in greater detail, let me know and I'll make the Fourier analysis chapter of my book available.
  • The Theorem of the Day is the Heine-Borel Theorem on the compactness of [0,1], which ties in nicely with much of our conversations.

• Tuesday Nov 2:   
  • The Schwarz lemma plays a central role in the analysis of many problems in complex analysis. There was a very nice exposition on its history and generalizations that just appeared in the Mathematical Monthly; click here for the article.
  • The result is a bit surprising, namely that we cannot have the first derivative (in absolute value) of an automorphism of the unit disk fixing the origin exceeding 1 in absolute value. Such a statement fails in the real case; see the Mathematica notebook I wrote here. There we considered the function f_a(x) = (a+1) x / (1 + a x²) for a non-negative. If a is at most 1 then this is an automorphism of the interval [-1,1], but it ceases to be for a exceeding 1. Note the derivative at the origin is 1+a, which can be as large as 2. Is this the best we can do? Can we have a larger derivative here?
  • There are many great choices for additional topics here. In addition to the article on the Schwarz lemma, another great choice is the Bieberbach conjecture (proved by de Branges).
  • You should do a lot of the algebra to understand the map psi_alpha(z) = (alpha - z) / (1 - alphabar z). Show that you can rotate and make alpha real, and then try to understand this case (perhaps at the cost of having a rotation outside).
  • There are many applications of conformal mappings, especially automorphisms of the upper half plane. Mobius (or fractional linear transformations) play a central role in the subject, with many applications in number theory and other fields (the cross ratio and geometry is a fun one). Important subgroups are SL(2,R) and SL(2,Z) (also known as the modular group). The latter is an important ingredient in building modular forms (important examples of which arise from elliptic curves). The j-invariant has many wonderful properties, and is even connected (through `moonshine') to the monster group mentioned in 10/28.
  • Why are these maps so important? The modular forms they lead to are generalizations of periodic functions, and we know how important the periodic functions are! A central question becomes the determination of the proper region to study these generalizations. Just as it suffices to study a periodic function of period 1 in any interval of length 1 (such as [0,1) or [-1/2, 1/2)), we look for a basic region to study these modular forms. This set is called the Fundamental Domain.
  • As a final (academic) note, I thought it might be fun to end each day's comments with a link from the site `Theorem of the Day'. I'll start today with their first choice, the Four Color Theorem. It's not a bad idea to go to this site and familiarize yourself with the theorems and statements.
  • For a non-academic final note, the promised well-wishing from Spaceballs: May the Schwartz be with you. My favorite scenes are `then is now', `they've gone to plaid', `industrial strength', 'no sir, I didn't see you...'. Okay, it's a great movie!

• Thursday Oct 28:   
  • A great way to build intuition about conformal maps is to look at lots and lots of examples. The wikipedia page on conformal pictures has some nice examples of what various maps do.
  • We looked at sequences of simple maps today to build more complicated ones. This is a common theme in mathematics, namely try to reduce everything to a combination of basic `moves'. Probably the most important example you've seen is in linear algebra, with elementary matrices. This has numerous applications, ranging from Gaussian Elimination to the Change of Variables formula.
  • Our exponential map f(z) = exp(iz) led to some interesting behavior. We found vertical strips were conformally equivalent to the upper half plane, with the map almost extendable to being from the boundary to the boundary. In fact, we could almost get a vertical strip to be conformally equivalent to the complex plane minus the origin. One of the homework or suggested problems will investigate this in greater detail, and will show that a non-simply connected region cannot be conformally equivalent to a simply connected one.
  • One of the maps we looked at was from the circle to the diamond, given by (u,v) --> (u|u|, v|v|). It's a nice calculation to show that this map is differentiable once but not twice.
  • The book `The Only One Club', in my opinion, has an ending that is at odds with Russell's Paradox.
  • The big object we studied at the end of class was the Automorphism Group of an open set. There are lots of examples listed on the wikipedia page. What's nice is that this is one of our first connections between analysis and algebra in this course. One of the major motivations for studying the Riemann Mapping Theorem is that it allows us to pass from knowledge of the automorphism group of the unit disk to that of other regions. We'll prove this on Tuesday.
    • One of the most famous Automorphism Groups is The Monster Group, which is the automorphism group of the Griess algebra.
    • Another famous one is the automorphism group of the Leech Lattice. There's a fascinating story about how John H. Conway discovered this in a tad over 12 hours one Sunday (he had set aside several hours two days a week for months to work on it). The Leech Lattice has important applications in sphere packings, which have applications in coding theory.
  • John H. Conway (one of my professors when I was at Princeton) is a big player in group theory. He's a fascinating character, and some of his work will make a nice way to round out the additional comments for the day.
    • The Doomsday Rule (for calculating the day of the week of any date, any year).
    • The Game of Life; there's a lot of work in this and its generalizations, Cellular Automata (such as Stephen Wolfram's `A New Kind of Science').
    • The Look and Say (or See and Say) Sequence. This can get addictive; there is some really fascinating math that you'll find (my favorite is that one ends up with a 92 x 92 matrix that is not diagonalizable but only Jordanizable). There's the famous Cosmological Theorem (see here for a proof, or see the links here for more information). You might notice the name of one of the authors is Shalosh B. Ekhad; a picture of this `mathematician' is available here.

• Tuesday Oct 26:   
  • We began by trying to find interesting products subject to constraints (problems #7b and #7c from Chapter 5). A good way to look at these is to have a split in definition for the even and odd subscripts; it just cleans up the algebra enormously. For #7b we tried a_{2n} = f(n) and a_{2n+1} = -f(n), and were led to a nice equation for f(n). For #7c, we tried to find a sequence so that (1 + a_{2n}) (1 + a_{2n+1}) = 1 with a_k --> 0. It's very important to be able to find these examples, as they illustrate how essential conditions are in theorems. For example, #7c showed us that we don't need the series Sum a_n to absolutely converge for the product Prod (1 + a_n) to converge; that's a sufficient but not a necessary condition.
    • Here's a fun problem: is there a non-negative function f(x) such that Int_{0 to oo} f(x)^k dx is finite for integer k < 2010 but infinite for k > 2010? Try to think of one before reading. At first it seems absurd; if the integral is finite, surely f(x) --> 0 and thus f(x) >= f(x)^k; however, it is possible that f is exploding on smaller and smaller subsets. For example, let f be a triangular function with spikes near the integers. We have f(n) = n (for n a positive integer) and f decays linearly to 0 at n +- 1/n^2009. Then Int_{0 to oo} f(x)^k dx = Sum_{n = 1 to oo} n^k / n^2009; this converges for k < 2010 but diverges for k >= 2010. With a bit of work, you can replace this f with an infinitely differentiable function! A good function to know for problems like this is f(x) = 0 outside [a,b] and on [a,b] is given by f(x) = exp(1/(a-x) + 1/(x-b)).
    • For another fun example, let f_n(x) be zero everywhere save [1/n - 1/2n, 1/n + 1/2n], f(1/n) = 2n and f linearly decreases to 0 at the edges. Then Int_{0 to 1} f_n(x) dx = 1, lim f_n(x) = 0 for all x, and lim Int f_n is not Int lim f_n. Switching orders of integration and a limit can be tricky, and leads to various uniformity estimates entering problems. Big theorems on when this can be done are the Monotone Convergence Theorem (MCT) and the Dominated Convergence Theorem (DCT). (For completeness, see also Fatou's lemma.)
    • Here's another fun challenge. Find a sequence a_n of real numbers that tends to zero, converges conditionally but not absolutely, but the sum of a_n^3 diverges?
    • There are lots of examples like the above; Uryshon's lemma is another situation. See also the Wikipedia entry on cut-off functions.
  • Our alternate proof of the Open Mapping Theorem used analytic properties of our map. In particular, if we standardize we may assume f(z) = z^n (1 + g(z)) with g holomorphic and g(0) = 0. While this is always an open map, if n=1 more is true -- this Taylor series is invertible! The Wikipedia page briefly talks about the formal inverse power series. While this proof was a lot longer than using Rouche's theorem, it led to some important results which can be used in studying the Riemann Mapping Theorem.
  • Conformal equivalence is, not surprisingly, an equivalence relation. The reflexive property is trivially shown, and transitivity is mostly easy. The difficult part there is showing the composition of holomorphic functions is holomorphic. The easiest way to do that is via the chain rule. It is possible to show this directly by a brute for expansion, namely the composition of convergent power series is a convergent power series, but the algebra is a bit painful. Oh yeah. We are thus in the situation where, after proving the Riemann Mapping Theorem, we can conformally map any open simply connected proper subset of the complex plane to the unit disk. We can do the analysis in the unit disk, and then map back. This is amazing, as we have an infinitely differentiable map!
  • A logical question to ask is what happens on the boundary of our open set. A good answer is provided, at least for many cases, by the Schwarz - Christoffel formula.
  • Finally, there are big theorems from analysis such as the Inverse Function Theorem and the Implicit Function Theorem. What is nice about these results are that we have simple conditions to test as to when we can express some variables in terms of another. These are big results that are used to prove the existence of solutions to some problems.
    • Along these lines, another good method to know is that of iteration (or Picard's iteration). We use this to prove, among other important results, the existence of solutions to certain first order differential equations (see my notes here). What is going on here is the existence of a contraction map, leading to a fixed point theorem. Results such as these are of great importance in economics and game theory (among other subjects).

• Thursday Oct 21:   
  • There are many proofs of the Open Mapping Theorem. The book treats it as a consequence of Rouche's theorem; it is, however, also possible to derive it just from the fact that f is holomorphic and thus analytic. It's worthwhile doing exercises such as this, as this allows us to see how `deep' a theorem or result is. As always, you should be asking: does a similar statement hold for real analytic functions, and if not why not?
  • We discussed an alternative approach to proving the Open Mapping Theorem. Whenever you see a theorem, you should immediately think of the following: (1) Let me try testing it with some special cases to build intuition; (2) Why do they have the conditions they do? What consequences do we get from these conditions?
    • For the first point, we studied the function f(z) = z<sup>n</sup>. We were led to this by looking at the conditions and seeing that we could standardize our general, non-constant holomorphic function to f(z) = z<sup>n</sup> (1 + Σ_{k = 1 to ∞} c<sub>k</sub> z<sup>k</sup>), and of course the simplest case is when the c<sub>k</sub>'s are all zero. We looked at the map z --> z<sup>n</sup> on a disk centered at the origin. While it was easy to see what happens there, this is a little dangerous, as the point z=0 is a special point for this map. We really need to investigate it elsewhere. Fortunately by rescaling it suffices to do one other point, say z=1.
    • The next item was to think about what theorems we have at our disposal once we assume f is holomorphic. One big one is that a holomorphic function is analytic, so perhaps we can use the series expansion to advantage (and this is of course related to writing the function as we did above). The key insight is that we can replace z<sup>n</sup> (1 + Σ_{k = 1 to ∞} c<sub>k</sub> z<sup>k</sup>) with (z h(z))<sup>n</sup> for some nice h(z). This is essentially Newton's Generalized Binomial Theorem, where we have (1 + Σ_{k = 1 to ∞} c<sub>k</sub> z<sup>k</sup>)<sup>1/n</sup>. It turns out to be relatively easy to get a series expansion for (1+x)<sup>1/n</sup> for |x| < 1. We were left with one last point: can we solve z h(z) = c for some fixed c and h(z) = 1 + ...? It seems clear that the answer should be z = c plus some small correction. How do we find that correction? This is essentially the inverse function theorem. Note that if g(z) = z h(z) with h(z) = 1 + ... then g'(0) is not zero! The Inverse Function Theorem now asserts an inverse for z near 0.
  • A key ingredient in our proof of the Open Mapping Theorem was Rouche's Theorem, which follows from the argument principle. The argument principle is very useful in complex analysis and its applications in number theory. One of my favorites is in numerical verifications of the Riemann hypothesis. We can use the argument principle to count how many zeros should be in a big box containing the critical line Re(s) = 1/2 from say -T to T, and `investigate' how many zeros lie on this line. If the two numbers agree, we have verified the Riemann hypothesis in this region. Amazingly, it isn't too `hard' to numerically find zeros of the zeta function on the critical line. This is done by using the Intermediate Value Theorem! We have π^-s/2Γ(s/2)ζ(s) is real valued when Re(s) = 1/2, and has the same zeros as the Riemann zeta function on the critical line Re(s) = 1/2. We now just move up the critical line and record sign changes; every time there is a sign change we've found a zero (this follows from the Intermediate Value Theorem); we can compute the zeros to whatever order we want simply by taking smaller and smaller steps.
    • For more on this, see the wikipedia entry on the Z function.
    • See also Gram points.
    • There are also now and again distributed computing attacks on RH, such as ZetaGrid (for awhile the computer cluster I maintained at Princeton was one of the top 10 contributors in the world).

• Tuesday Oct 19:   
  • Today we discussed the Weierstrass product and hinted at a `better' version, the Hadamard product. The problem is we need to be very careful when dealing with infinite sums or products. The first step is always to determine exactly what sequences have a chance of leading to a nice result. After a little thought, we realized that if there is an accumulation point in the zeros then our function is identically zero. This gives our first result, namely that the sequence of zeros is unbounded and without an accumulation point. The unboundedness is crucial; it allows us to know that from some point onward we have |z/a_n| < 1/2, and thus we can use log(1 - z/a_n). This was important as we wanted to write 1 - z/a_n as exp(log(1 - z/a_n)), as we could then combine exponential factors and convert multiplication to addition. This is a powerful principle -- it's important to think about how to arrange algebra.
  • In the course of proving the existence of an infinite product with prescribed zeros, we used the exponential convergence factors E_k(z) = exp(z + z²/2 + ... + z^k/k). These factors are good for several reasons. First, of course, is that they are never zero. Second, (1-z) E_k(z) = exp(log(1-z) + z + z²/2 + ... + z^k/k) for |z| < 1/2. We see that the sum is canceling the first k terms in the Taylor expansion of the logarithm, and this will give us good convergence.
  • We split our infinite product into two pieces. We fixed z and looked at all a_n with |z/a_n| < 1/2. As our sequence is unbounded, this condition must be true for all n sufficiently large, say all n > N. We analyze the factors for n < N separately for this z, and then use our analysis above for (1 - z/a_n) E_k(z/a_n). We again encounter a situation where we have a free parameter. This time, we have the freedom to choose k. What should k be a function of? We could have k = k(n,a_n,z); however, it would introduce many problems if k depended on z. Amazingly, there is a choice of k(n,a_n) that works for all sequences simultaneously, namely k(n,a_n) = n. This, however, is quite wasteful. If our series a_n is such that the sum of 1/a_n^r converges for some r, we could actually take k bounded (any integer k exceeding r would work). It is not surprising that there is interest in finding the simplest choices for k(n) possible. If our zeros a_n have some growth, we can bound k. This happens if our function is of finite order. See also some lecture notes of Noam Elkies from Harvard. Fortunately many important functions in number theory, such as the Riemann zeta function, are of finite order; the zeta function has a very simple Hadamard product.
  • We keep using the wonderful fact that an entire function which is never zero is the exponential of an entire function. A fascinating question is just how many values a function can miss and still be entire. The little Picard theorem says just one!
  • In our study of zeros and accumulation points we talked a bit about countable and uncountable sets; we'll need to revisit these subjects again later when we prove the Riemann Mapping Theorem. If you would like to see more about these, let me know and I'll send you Chapter 5 of my number theory book. An interesting fact is that if you have an uncountable sum of non-negative terms, then it must be zero. The proof is simple: break the summands into those in intervals (1/2ⁿ⁺¹, 1/2ⁿ]. At least one of these intervals has infinitely many summands (else you would have countably many terms).
  • We then moved into some topology, talking about simply connected regions. It was essential in defining the complex logarithm that we take a simply connected set, as a key ingredient was the Cauchy integral formula, which requires our function to be holomorphic in the interior of the curve. It's interesting to consider what logarithms we can define. This leads to branch cuts
  • As always, a great topic to study is the Jordan Plane Curve Theorem.
  • One of the key insights in our study of representing functions as products over their zeros was to write (1-z/a_n) E_k(z/a_n) as exp(log(1-z/a_n) + (z/a_n) + ... + (z/a_n)^k/k). Related to this, consider the exponential of a matrix. We set exp(A) = I + A + A²/2! + A³/3! + .... It can be shown that this converges for any matrix. A natural question is does exp(A) exp(B) equal exp(A+B). Amazingly, no (though it does if A and B commute, which a little inspection shows is not unreasonable). The formula for what exp(A) exp(B) equals is the wonderful Baker - Campbell - Hausdorf formula.

• Thursday Oct 14:   
  • It's very useful to write a function as a product of its zeros, though obviously this cannot tell the entire story. For example, we can always multiply by an arbitrary constant and get a new function with the same zeros. For finite polynomials, we have formulas relating the coefficients of the polynomial to the roots. Unfortunately, the other direction is much harder; namely, while it is easy to get the coefficients of the polynomial from the roots, it is hard to go the other way and find the roots from the polynomial's coefficients in general (once the degree is at least five). We do have beautiful formulas such as Newton's identities.
  • The Weierstrass Factorization Theorem allows us to write many meromorphic functions as products over their zeros, so long as we know its value at one place where it does not vanish. A very nice application of the infinite product for sin(πz)/π are expressions for Sum_{n = 1 to ∞} 1 / n^2k. Particularly useful ones are k=2 (which gives π²/6) and k=4 (which gives π⁴/90). These are also ζ(2) and ζ(4), where ζ(s) = Sum_{n = 1 to ∞} 1 / n^s is the Riemann zeta function (which converges absolutely for Re(s) > 1). This is one of the most important functions in number theory, due to the fact that it also equals Prod_{p prime} (1 - 1/p^s)^-1 (this follows from unique factorization of integers). Interestingly, the fact that the sum of the reciprocal of the squares equals π²/6 implies that there are infinitely many primes (if there were only finitely many primes, taking s=2 in the product formula implies that ζ(2) is rational)! This is due to the fact that π² is irrational (which follows from the transcendence of π, but can also be proved simply on its own). There are many proofs of the irrationality and transcendence of π.
    • The above argument for the infinitude of primes is known as a special value proof, and occurs frequently in the higher arithmetic (ie, number theory). There are lots of proofs of the infinitude of primes, going all the way back to Euclid. There's lots of clever ones; see the wikipedia page for more information (one of my favorites is Furstenberg's topological proof; I was fortunate enough to meet him when I was a postdoc at Ohio State). A good source for great proofs such as this (as well as proofs on other subjects) is the book "Proofs from THE Book".
    • We know that the Riemann zeta function at 2k is a rational multiple of π^2k; the actual value involves the Bernoulli numbers. We know very little about the zeta function at odd positive integers. Apery succeeded in proving ζ(3) is irrational (see also this nice article), but we don't know too much more. We have theorems that say "at least so many of the zeta function at these odd integers are irrational", but that's about it.
    • Returning to proving the infinitude of primes (an important question, as the primes are the building blocks of numbers we should know how many there are!), there's still a lot of `fun' exploring that you can do with Euclid's proof (if there were only finitely many primes, say p1, ..., pn, then consider p1 * ... * pn + 1; either it is prime or it is divisible by a prime not in our list as each prime in our list leaves remainder 1 when we divide). A fascinating question is to go through Euclid's proof and write down what primes we get at each stage. This leads to an interesting sequence: 2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357....   Explicitly, this sequence is generated as follows. Let a_1 = 2, the first prime. We apply Euclid's argument and consider 2+1; this is the prime 3 so we set a_2 = 3. We apply Euclid's argument and now have 2*3+1 = 7, which is prime, and set a_3 = 7. We apply Euclid's argument again and have 2*3*7+1 = 43, which is prime and set a_4 = 43. Now things get interesting: we apply Euclid's argument and obtain 2*3*7*43 + 1 = 1807 = 13*139, and set a_5 = 13. Thus a_n is the smallest prime not on our list generated by Euclid's argument at the nth stage. There are a plethora of (I believe) unknown questions about this sequence, the biggest of course being whether or not it contains every prime. This is a great sequence to think about, but it is a computational nightmare to enumerate! I downloaded these terms from the Online Encyclopedia of Integer Sequences (homepage is http://www.research.att.com/~njas/sequences/  and the page for our sequence is  http://www.research.att.com/~njas/sequences/A000945 ). You can enter the first few terms of an integer sequence, and it will list whatever sequences it knows that start this way, provide history, generating functions, connections to parts of mathematics, .... This is a GREAT website to know if you want to continue in mathematics. There have been several times I've computed the first few terms of a problem, looked up what the future terms could be (and thus had a formula to start the induction). One last comment: we also talked about the infinitude of primes from zeta(2) = pi^2/6. While at first this doesn't seem to say anything about how rapidly the number of primes grows, one can isolate a growth rate from knowing how well pi^2 can be approximated by rationals (see http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.2184v3.pdf  for details; unfortunately the growth rate is quite weak, and the only way I know to prove the needed results on how well pi^2 is approximable by rationals involves knowing the Prime Number Theorem!).
  • There are many important generalizations of Weierstrass' theorem, including Hadamard's factorization theorem. These representations are very useful in studying questions in number theory. One of my favorite applications is work by several colleagues of mine (Gonek, Hughes and Keating: A hybrid Euler-Hadamard product formula for the Riemann zeta function). There they combine the two different product expansions of the zeta function to great effect.
  • I'm currently working with a team of mathematicians and engineers on applications of Benford's law to detecting image fraud (I'm also editing the first book on the subject -- if you're interested in participating on that project, let me know). The reason I became involved is the need to understand the size of certain infinite products arising from Fourier analysis. The quick analysis I did a few years back is available here; if you want to improve my bounds and earn a mention, let me know!
  • Finally, we spent a bit of time today trying to figure out what `good' should be in studying the error in the Taylor expansion of log(1+z). Learning how to argue along these lines is one of the most important skills you can develop. The actual nature of `good' (boy can I see a philosophy class having fun with this!) will clearly depend on your problem. You need to take some time and think what the key features are, how your quantities should depend on the parameters of the problem, et cetera.
    • A terrific example of this is curve fitting in probability or statistics. There are three very natural ways to measure error: the difference, the absolute value of the difference, the square of the difference. The third is the `best'. The first has the drawback of being signed, so errors can cancel. The second involves the absolute value function and is not differentiable. The third does have the drawback of giving larger errors more weight, but it is differentiable and thus the tools of calculus are applicable. Click here for a handout I wrote on the Method of Least Squares, describing why the third method is best and how it allows us to use linear algebra and calculus to write down a closed form solution. So, even though it does weight things unequally (unlike the absolute values in the second method), it gives a closed form solution for the best fit curve; the second method cannot do that, and it's worthwhile getting a closed form solution!
  • Okay, one more tidbit. I'm a big fan of Ohio State football (so is Cam -- he loves Brutus). I've done a lot of sabermetrics (applying math to baseball) projects with students at Williams. I also can't stand it when people quantify things poorly and then insist that the quantification is significant because, after all, there's a number! There's a nice article today about how bad the BCS rankings are; it's worth reading (and very well written).
    • There are numerous approaches to trying to rank college football teams, many using graph theory. It's a very hard subject as there are so few data points. (1) One nice article is here. (2) Here's another article. (3) Here's another. You get the point -- very active area of research, and the BCS system forbids researchers from using certain key information in their rankings.
    • What's the solution? So glad you asked. One idea is a playoff series. If you don't want to do that (too much time away from academics), here's a simple solution. All teams who want to compete and are considered legitimate contenders put their names into the hat and agree to a good non-conference game (or perhaps two) with another contender. Some teams, like Ohio State, do this already. Other teams (Penn State, I think, has often gone the safe route) try to win their conference and advance through that (though they did agree to play Alabama this year). Sadly, other teams (Boise State) are willing to play out of conference but other teams won't play them. Such a system will allow us to compare more easily the relative strengths of each conference, and cut down on the number of teams with good records. Is there really a need to have a contender play Appalachia State?

• Thursday Oct 7:   
  • The argument principle shows that the integral of the logarithmic derivative of a nice function along a closed curve is equal to the number of zeros minus the number of poles of the function in the interior. This is a key ingredient in the proof of three big results (discussed below). Instead of integrating f'(z)/f(z) against 1, we could integrate it against a test function g(z). This leads to what is known as explicit formulas; these weighted versions of the argument principle appear throughout number theory (see for instance the proof sketch on Wikipedia of the Prime Number Theorem). We will discuss that argument at length later in the course.
  • Rouche's theorem is the first of many consequences of the argument principle; in fact, the remaining big theorems are consequences of Rouche's theorem. Rouche's theorem can be used to prove the Fundamental Theorem of Algebra. From the Wikipedia article: One popular, informal way to summarize this argument is as follows: If a person were to walk a dog on a leash around and around a tree, and if the length of the leash is less than the minimum radius of the walk, then the person and the dog go around the tree an equal number of times.
  • The Open Mapping Theorem is the next consequence. Again, the complex situation differs enormously from the real case. The proof in the book follows from Rouche's theorem; it is also possible to prove this by analyzing the power series expansion of our holomorphic function. We didn't prove the Open Mapping Theorem, but we'll give a proof later using Rouche's Theorem. Is it possible to prove it another way? As soon as you're told you have a holomorphic function, you should think analytic, you should think power series. Of course, the complex numbers must come into play. The function f(x) = x^2 maps (-1,1) to [0,1), and thus is NOT an open map. Hmm. So what's so special about the complex numbers...?
  • The Maximum Modulus Principle is the biggest implication of the Open Mapping Theorem. This states that a holomorphic function attains its maximum on the boundary. Applications of this include the Schwarz lemma (which is a key ingredient in proving the Riemann Mapping Theorem, which allows us to reduce the study of simply connected open sets not all of the complex plane to studying the unit disk), and the Phragmen - Lindelof Principle, which is very useful in bounding quantities in number theory. One such application is in proving convexity bounds for the Riemann zeta function (and more generally L-functions); see for instance Heath-Brown's note.  It's a very good exercise to work through some similar examples for real valued functions and see what goes wrong. Specifically, look at f(x) = 1-x^2 on (-1,1). Note f((-1,1)) = (0,1]. Note the maximum is attained in the interior, not on the boundary.
  • We talked about including all details in a proof. A great example is when you can construct a regular n-gon with straight edge and compass. Gauss completely resolved this question (well, okay, he reduced it to a determination of which primes are Fermat primes). Johann Hermes has done the 65537-gon (the link is to a website where you can dowload the Mathematica file, as opposed to traveling to his university to view the 200 pages).
  • Another issue we discussed was the danger of infinities in calculations. The Fubini-Tonelli Theorem discusses when we may interchange sums / integrals; a key stumbling block is when the integral of the absolute value is infinite. One way this can happen is to have a bounded function on an infinite domain. See the comments from Sept 14 for more details.
     

• Tuesday Oct 5:  
  • Today we sketched the proof of the Residue Theorem and briefly discussed three applications: Rouche's Theorem, the Open Mapping Theorem, and the Maximum Modulus Principle.
  • The Residue Theorem is an incredibly powerful tool. Even if you only care about integrals of functions of a real variable, it is frequently useful to extend to the complex plane. The reason is that, in general, it is not possible to write down anti-derivatives; integration is hard! (There is an interesting algorithm (due to Risch) to find anti-derivatives involving elementary functions. The linked article has a nice example here, where changing the constant term by 1 leads to the method failing; this is related to a change in the Galois group.) There are several steps to using the Residue Theorem:
    • Step one: determine the function. Frequently it is easy: given f(x) try f(z). Sometimes, though, it's a bit harder. If you have a cos(x) term you could try cos(z) = (exp(iz) + exp(-iz))/2, or you might try taking just exp(iz) and taking the real part.
    • Step two and three are related: choose a contour and find the poles and residues. Often the location of the poles affects what contour you take. You DO NOT want a pole on the contour (I've had to do this a few times in my research, and it is not fun). Sometimes you have to split the integrand up into different pieces, and do one part with one closed curve and another part with another. A big factor in determining contours is how the function decays. Remember that decay is a bit trickier in complex numbers; for instance, let's assume |z| > 2000; then 1/|1+z²| <= 1/(R² - 1); if we restricted to |x| > 2000 then 1/(1+x²) <= 1/(R² + 1). The issue is that we have a phase.
    • Step four: repeat earlier steps as needed. For example, the HW problem (Chapter 3, #1) involved integrating exp(-z²) over a specific contour. There was a lot of difficulty in bounding the contribution over the arc. We solved this by splitting the arc into two pieces; however, it was not clear initially where we should make the split. We had a good reason for wanting the split to occur as close to π/4 as possible, and another reason to want it further from there! The best way to figure out where to put the break point is to introduce a free parameter. In this problem, we make a split at 1/R^a for some a. We then analyze the first piece and see what a we need to take to get a contribution tending to 0 as R tends to infinity. We then analyze the second arc and see what a we need to take for that contribution to tend to 0, and then hope that there is a choice of a that works for both simultaneously. It is very hard to just look at this problems (without lots of experience) and know exactly where to make the cut. Thus, it helps to have a free parameter. This gives you flexibility. For some problems, you often want to equalize the contribution from both pieces -- in some sense, this gives you the smallest possible error (or at least the smallest attainable by this method). For us, that would mean essentially solving something like R^5(a-1) = exp(R^2-a). The actual `best' answer will involve a as a function of R.
  • We did a very important example, finding the normalization constant for integrating 1/(1+x<sup>2</sup>) over the real line. This leads to the Cauchy distribution, which is very important in probability.
    • One of my favorite applications of the Cauchy distribution is by Mandelbrot in his economics research. See the wikipedia entry on Kurtosis risk.
    • The standard random walk hypothesis seems to have lost most of its supporters, though there are variants (and I'm not familiar with all); see also the efficient market hypothesis and technical analysis, and all the links there. (There are also many good links on the wikipedia page on Eugene Fama). Two famous books (with different conclusions) are Malkiel's A random walk down wall street and Mandelbrot-Hudson's The (mis)behavior of markets (a fractal view of risk, ruin and reward). Some interesting papers if you want to read more:
      • Mandelbrot: Variation on certain speculative prices (a must read!)
      • Fama: Mandelbrot and Stable Paretian Hypothesis
      • Fama: Random Walks Stock Prices
      • For more on randomness, check out The Black Swan by Taleb (amazon.com page here, wikipedia page here). Several members of my probability class have recommended this book highly, and from reading excerpts on the web I understand why.
      • For more on fractal geometry, click here. One of the most common is the Koch snowflake; another popular one is the Cantor set. See here for fractal dimensions. To actually compute pictures of items like the Mandelbrot set, one needs to iterate polynomials. This can lead to the fascinating subject of efficient algorithms; when I wrote such programs years ago on what would now be considered `slow' computer, I had to use Horner's algorithm to get things to run in a reasonable time.
  • We ended by finding the poles and residues of 1/1+z²⁰¹⁰. Key is Euler's formula: exp(ix) = cos(x) + i sin(x). Remember that if we want to solve z²⁰¹⁰ = -1 = exp(iπ), we could also write -1 as exp(iπ + 2πin) for any integer n. There will be 2010 distinct solutions (half in the upper half plane, half in the lower half plane).
  • We didn't talk too much about the applications (Rouche's Theorem the Open Mapping Theorem and the Maximum Modulus Principle), but we will. The last two are (yet again) instances where complex analysis is quite different than real analysis.

     

• Thursday Sept 30:  
  • The generalization of Cauchy's Integral Formula, the Residue Theorem,  is phenomenal, and allows us to evaluate many difficult integrals with ease by converting them to algebra problems. The wikipedia page on contour integration has a lot of great examples. There are two main steps to applying Cauchy's Integral Formula. The first is choosing the contours (which is influenced by the decay properties of the function) and the second is computing the residues.
    • Computing residues: We've seen that using the geometric series is a great way to compute residues. For example, if we want a residue at say 3 we replace z everywhere with z-3 + 3; the first factor z-3 is then small. Another useful approach is through differentiation. Say we have f(z) = g(z) / (z-3)¹⁰ with g(z) holomorphic at z=3. To calculate the residue at z=3 we need to find the (z-3)⁹ term of g(z). We could do this with our trick, or we could compute 9 derivatives.
    • Figuring out which paths to take to exploit the decay is very difficult, and becomes easier with practice. It helps to know growth rates of different functions. For this <<< notation is very useful. We write f(x) <<< g(x) if |f(x)/g(x)| tends to 0 as x tends to infinity. A very useful relation is log(x) <<< x^r <<< exp(x) for any r.
  • There are two Picard theorems, Big Picard and Little Picard. Yet again, we see complex functions have a wildly different behavior than real valued ones. We find that a non-constant entire function can miss at most one complex value; thus if we have an entire function that misses two complex numbers it must be constant! This theorem is sharp, as f(z) = exp(z) shows.
    • Speaking of Picard, the reference to Fermat's Last Theorem is from a poor episode, The Royale; start at 1 minute in and watch for a minute online at youtube.
    • Fortunately history in Star Trek is a bit malleable; they correct this in the episode Facets in Deep Space 9; start at 3misn25secs in the youtube video.
  • The Riemann Mapping Theorem is another amazing fact of complex analysis, yet again showing how different the subject is from real variables (see the nice section in the Wikipedia article on why this result is so amazing). The analytic isomorphism between any simply connected open set not all of the complex plane and the unit disk can sometimes be given explicitly; see the Schwarz-Christoffel formula.

   

• Tuesday Sept 28: 
  • It's amazing how the Fundamental Theorem of Algebra falls from Liouville's Theorem. I strongly urge you to look at the list of Fundamental Theorems, and read a few that you haven't heard of to get a sense of what's out there in mathematics. There are a multitude of applications to the Fundamental Theorem of Calculus. One of my favorites is Partial Fraction Decomposition. This is extremely useful in determining integrals of reciprocals of polynomials (and explains why the logarithm often arises). It also makes generating functions. These lead to lots of wonderful results about Fibonacci numbers (and other sequences).
    • There are formulas for quadratic equations, cubic equations (what's amazing is that a real polynomial with real roots can require you to work with imaginary numbers to get those roots!), and quartic equations. There is no general formula for the quintic and higher. This was first proved by Abel, and then in a more general setting by Galois. Lagrange had a wonderful approach to lower order polynomials (quartics and smaller) using resolvents; unfortunately this method breaks down for quintics (it's good to read the article to see why).
    • It is very hard in general to write down roots of a polynomial in explicit form. This is but one of many questions of the following nature: given a fixed tool set, what can I do? The ancient Greeks asked this about a straightedge and compass. Three famous problems the Greeks tried to do, but couldn't, were squaring the circle, doubling the cube, and trisecting an arbitrary angle. All are impossible; attacks on these inspired much of algebra. Another problem was which regular n-gons can be constructed; this was essentially solved by Gauss, and is one of the reasons he chose to become a mathematician. He showed you cannot do a 7-gon, 9-gon, 11-gon, 13-gon, but can do a 17-gon (see also the Mathworld article on constructable polygons).
    • While we cannot write down closed form expressions for the roots in general, we have algorithms such as Newton's method which often do a great job of approximating them. The convergence (or divergence!) of this method leads to fascinating behavior and questions; see the wikipedia article on Newton fractals, for example.
    • As we've already mentioned Liouville, let's do another of his theorems. He was the first to write down a number (these are the Liouville numbers) that is provably transcendental (namely, not the solution of a finite polynomial with integer coefficients). Cantor proved (almost 40 years later) that almost all real numbers are transcendental, though his diagonalization method cannot give specific examples. Determining which numbers in certain expressions are transcendental is an important area (it's Hilbert's seventh problem); see the Gelfand - Schneider Theorem for some results. Properties of Liouville and other numbers are often related to arithmetic dynamics; in fact, how well log(2) / log(10) can be approximated by rationals is a key ingredient in my proof (with Alex Kontorovich) of Benford behavior in the 3x+1 problem (the paper is available here).
  • Liouville's theorem is (yet another) application of Cauchy's formula. It completely classifies all functions that are both holomorphic and entire. There are other situations like this in mathematics. One important one is Harald Bohr (brother of Niels Bohr) and Johannes Mollerup proved that the only function which satisfies the following three properties is the Gamma function: (1) f(1) = 1; (2) f(x+1) = f(x) for x > 0; (3) f is logarithmically convex.
  • Another one of my favorite problems is the Hamburger Moment Problem (and not just because of the name). This is related to when a sequence of moments uniquely determines a probability density (if p(x) is a probability density, the k-th moment is the integral of x^k p(x) dx). Note how similar this is to the Bohr-Mollerup theorem. In fact, this is related to our accumulation point results, and is one of the most important reasons for studying teh subject. Namely, when does a sequence of moments uniquely determine a probability distribution. There are two probability densities that are different but have the same moments. The problem is solved by considering non-integral moments as well, and if there is agreement along a sequence with an accumulation point, then the densities are equal. For full details, see my notes from probability. To rigorously prove many results in probability, such as the Central Limit Theorem, requires many results from complex analysis.

   

• Thursday Sept 23:
  • We finished the contour example, namely integrating (1-cos(x))/x² over the real line. We had to be careful as the complexification (1 - exp(iz))/z² has a pole at z=1, and thus there would be a singularity on the real line. We literally circumvent this by detouring, going along a semi-circle of radius ε about the origin. Fortunately, it turns out that most integrals in complex analysis can be reduced to integrals over circles and semi-circles. Remember when choosing your contour, by Cauchy's formula many will give the same answer, but through different algebra. Make your life as easy as possible. This problem as a 1/x² or a 1/z² in the denominator. It's thus natural to use a large semi-circle of radius R, as there the denominator is exactly R² in absolute value; if we did a rectangle or a triangle, the algebra would be a bit more unpleasant.
  • Cauchy's formula is a true gem of mathematics, and quite surprising at first. We only need to know a holomorphic function on a circle in order to determine its value everywhere in the interior! This is an example of a boundary value problem; the wikipedia article linked here enumerates many of the more famous of these. Note how different the situation is in real analysis, where it is unusual for a function to be uniquely determined by its values on a boundary curve. There are, though, situations where that happens; see for instance harmonic functions (which occur frequently in mathematical physics). The miracle here is that once you specify a holomorphic function on the boundary, you have no choice in how it is defined elsewhere!
  • A big step in our analysis was interchanging a derivative and an integral. The book proves it directly for this special case through series expansions and radii of convergence; however, it's good to be aware of when (in general) this can be done; see for example here.
  • Multiple time today we added zero in a clever way; this is one of the most important skills to master. The first place you might remember seeing this is in Calc I (in proving say the product rule), though if you think back you've seen it even earlier in your career, namely in completing the square to prove the quadratic formula. One of course wants to get a feel for when this should be done. For the proof of Cauchy's integral formula, we have f(ζ) / (ζ-z). As ζ is close to z, it makes sense to add -f(z) + f(z) to the numerator. This gives (f(ζ) - f(z)) / (ζ - z) + f(z) / (ζ - z). The first term tends to f'(z), which by assumption is understood. We actually don't need the full strength of f being holomorphic; all we really need is the quotient (f(ζ) - f(z)) / (ζ - z) is bounded (which then will improve itself to f is holomorphic). If f is Lipschitz, that suffices; Holder continuous does not if the exponent is less than 1.
  • We also saw adding zero when proving a holomorphic function is analytic. To Taylor expand a function about a point z0, we want to have a series Σ a_n (z - z₀)ⁿ where the a_n's are f⁽ⁿ⁾(z0)/n!. Using Cauchy's formula, we can relate these derivatives to integrating along a circle centered at z₀ the function f(ζ) / (ζ-z₀). Unfortunately, we start out knowing the integral expansion for f(z), which involves integrating f(ζ) / (ζ-z) about the same circle. Essentially, we want to replace the z with a z₀. We do this by writing  ζ-z = ζ-z₀ + z₀ - z, which is just (ζ-z₀) - (z - z₀). We now see how well-suited this is to our analysis. We expand with a geometric series (which has nice convergence properties), and are now in the position where we'll be integrating against ζ-z₀. The reason we have good convergence is that we have a geometric series with ratio |z - z₀| / |ζ-z₀| < 1.
  • Finally, the great π-debates continue, namely which is the best: π, 2π, 4π, or 2πi (or maybe even 4π/3)? Each has its proponents. Is the fundamental object the radius of a circle, or its diameter, or its perimeter, or its area, or the surface area of a sphere, or the volume of a sphere.... For example, look and see how the constant is written in Coulomb's law (ie, electricity). Or, even better, in the fine structure constant, one of the most important numbers in mathematical physics (because it is unitless!).

• Tuesday Sept 21:
  • Our first main result is that if a function is holomorphic in a disk then it has a primitive (in other words, an anti-derivative). The big consequence of this is that integrals of our function over closed curves are zero, which implies that the integral along any path depends only on the endpoints, and not the curve taken to connect them. Situations such as this occur all the time in physics; you might remember conservative forces such as gravity and electricity. A conservative force is the gradient of a scalar potential.
  • It is impossible to do path integrals justice in just a few words in class. Feynman used path integral formulations to formulate quantum mechanics! Richard Feynman was an extremely colorful character, and the author of numerous great books, including:
    • The Feynman Lectures on Physics.
    • Surely You're Joking, Mr. Feynman! 
    • What Do You Care What Other People Think?
  • Speaking of good books to read by famous mathematicians / physicists, another classic is Hardy's A Mathematician's Apology. Hardy was one of the top analysts of his time, making seminal contributions to many fields (including applications of complex analysis in number theory). If you are interested in any of these books, I have them in my office.
  • There are lots of ways to do the calculation to show a primitive exists. Instead of using the polygonal path we did one could take the direct line segment. Why did we proceed as in the book? The reason is that there are many times in math that polygonal approximations are useful, and it's often very convenient to move only parallel to the coordinate axes. You might have seen something along these lines (if you'll forgive the pun) when proving Green's theorem. One has to be careful with polygonal approximations, though, as if you are careless you can prove all curves connecting two points have the same length! To see this, take the curve y = x² from x=0 to x=1. This function is strictly increasing, and we can approximate it by very small vertical and horizontal lines. We can make the approximation arbitrarily close, say always withing epsilon pointwise. We would like to then say the length of the curve is approximately the sum of the vertical and horizontal segments; however, all the vertical segments add to the interval [0,1], as do the horizontal segments. Thus this curve has length 2 (as would any other similar curve). What is going horribly wrong? If ds is the infinitesimal distance differential along the curve, the correct formula is ds = sqrt(dx² + dy²), whereas we were using a moment ago ds = dx + dy. It is in general a hard question to find lengths of curves (ie, the arc length).
  • The Joran plane curve theorem tells us when a curve divides the plane into two regions, an inside and an outside.
  • We showed that the integral of a holomorphic function along a closed curve is zero. There is a converse, namely Morera's theorem: if a continuous function integrates to zero along every closed curve then it must be constant.
  • A big part of complex analysis is learning how to choose the function that generalizes the real integrand properly, as well as choosing the right contours. For many examples of doable contour integrals, see the wikipedia page here (some of these won't make complete sense until we've done parts of Chapter 3). The second example from Section 3 is terrific, and illustrates many of the key ideas. We can write sines and cosines in terms of exponentials, and thus replace these real valued functions with complex valued ones. The difficulty is that we must make sure we have rapid decay on some of the sides of integration. Writing cos(z) as (exp(iz) + exp(-iz))/2 is not a bad thing to try, but for z=x+iy, if y>0 then exp(-iz) is rapidly growing in y. We thus need to be careful. One solution is to write 1 - (exp(iz) + exp(-iz))/2 as 1/2 - exp(iz)/2 + 1/2 - exp(-iz)/2, and use contours in the upper half plane for the first and bottom half plane for the second. The other approach is to write 1 - cos(z) as the real part of 1 - exp(iz), and then we can use just one contour in the upper half plane.

• Thursday Sept 16:
  • Stokes' Theorem (or, for us, Green's Theorem as we're in the plane) is a gem of mathematics. We saw how it can be used to compute areas of regions by reducing it to a line integral. The example we did in class today was the area of the ellipse; the special case when it's a circle of radius 2 is done here (it's a shame they do the special case!). Whenever you get an answer in math (in this case, the area of the ellipse is πab), you should check and make sure it agrees with other known, simpler cases. In this instance we check with the case a = b = r, and recover the area of the disk. In general, it is much easier to compute areas than lengths; the length of the ellipse is quite difficult to compute, and involves the elliptic integral of the second kind, which is a special case of the Gauss hypergeometric function. Hypergeometric functions are a very rich family, and arise throughout mathematical physics.
  • The Jordan Plane Curve Theorem is one of the candidates for greatest disparity between expected ease of proof and actual difficulty of proof. The history and further proofs section is a fun read. One of the proos uses the Brouwer fixed point theorem. Fixed point theorems arise all the time, especially in game theory in economics (see for instance the work of Nash).
  • Using polar coordinates to compute the integral of exp(-x²/2) is a wonderful technique, but if feels a bit artificial. There are other ways. One of my favorites is to change variables and observe that we have a special value of the Gamma function (which generalizes the factorial function) and then use some identities of this function. While we can evaluate the integral of this function over the entire real line, there is no nice expression (involving elementary functions) for its anti-derivative. There is an interesting algorithm (due to Risch) to find anti-derivatives involving elementary functions. The linked article has a nice example here, where changing the constant term by 1 leads to the method failing (this is related to a change in the Galois group).
  • We discussed a variety of applications of integration. As a number theorist, one of my favorites is in applications to problems such as Goldbach's problem or Waring's problem. One of the most powerful ways to attack these is the through the Circle Method. If you are interested in learning more about this, let me know and I'll print out the chapter from my book on this. The gist of these ideas is that we have a generating function whose coefficients encode the answer to our problem, and we must somehow pick off these coefficients.
  • It's frequently convenient to work with triangles, as these fit nicely for triangulation purposes (see also the article on triangularizing polygons). One can prove Goursat's theorem (the link here includes the Stokes proof). One very nice consequence is Morera's theorem. Another place where you may have seen triangles is in the proof of the Euler characteristic of a surface in topology.
  • We mentioned CPCTC today, as we needed to show each triangle was similar to the original. There are lots of other useful acronyms (SAS, SSS, ASA and AAS). Speaking of geometry, one of my favorite results is Morley's theorem. I hold the record (I believe) for the longest proof (about 40 - 50 pages, unpublished, see me for the story). John Conway has one of the most beautiful proofs of this fact. If you enjoy geometric arguments like this, you might enjoy some of the non-standard proofs of the irrationality of certain square-roots.
  • There was a nice post on the math arxiv today giving a continuous but nowhere differentiable function.
  • Finally, it's worth reiterating `checking' formulas you can't remember. If you can remember some special cases, you can often correct your guess. We saw how to find where the minus sign is in the Cauchy-Riemann equations by testing our guess with the functions f(z) = z or z². For another example, if you only remember some of the quadratic formula you can try f(x) = x² - 1 or x² - x or x² - 3x + 2, all of which lead to readily computable roots.

• Tuesday Sept 14:
  • The main result today is interchanging a derivative and an infinite sum. Interchanging operations is one of the most important techniques, and sometimes one of the most delicate. The first instance you encounter is probably Fubini's Theorem. Note it is not always permissible to exchange integrals or sums; the difficulty is when infinities are involved. The standard condition is if the integral (or sum) of the absolute value is finite then you are okay. For a problem case, consider the double sequence (m, n >= 0) given by a_{m,n} = 0 for n < m or n > m + 1, a_{m,m} = 1 and a_{m,m+1} = -1. Another important case is differentiating under the integral sign; I strongly urge you to look at the examples here of the integrals this allows you to evaluate. Feynman was a master at this technique. Today we discussed differentiating under the summation sign, which is possible if the series converges absolutely in a disk. The key concepts were the radius of convergence, which involves the notion of the limit superior (or the limsup). Lurking beneath all these proofs is a comparison test with a geometric series.
    • There are many nice ways to prove the geometric series formula. My favorite involves a game of hoops with two players. A always gets a basket with probability p, B with probability q (assume p, q < 1), and first to get a basket wins. Let r = (1-p)(1-q) and x the probability A wins. Then x = p + rp + r² p + r³ p + ...; these add the probabilities of A winning on the first, second, third, ... attempt (as to get to the second attempt both A and B must miss). I claim that x = p + rx, as after both miss the probability A wins from this point onward is just x again! This gives x = p / (1-r), or 1 + r + r² + r³ + ... = 1/(1-r), the geometric series formula! Using the memorylessness nature s a key ingredient in many problems in economics. For more, see the article on martingales.
  • Today was a fast introduction to path integrals, line integrals, and Green's Theorem (which is a special case of the Generalized Stokes' Theorem). If you are continuing in certain parts of math, physics or engineering you will meet these again and again (for example, see Maxwell equations for electricity and magnetism). In fact, one can view all of classical mechanics as path integrals where the trajectory of the particle (its c(t)) minimizes the action; there is also a path integral approach to quantum mechanics.
    • One of the gems of complex analysis is Cauchy's Integral Theorem, A complex differentiable function satisfies what is called the Cauchy-Riemann equations, and these are essentially the combination of partial derivatives one sees in Green's theorem. In other words, the mathematics used for Green's theorem is crucial in understanding functions of a complex variable.
    • For me, I consider it one of the most beautiful gems in mathematics that we can in some sense move the derivative of the function we're integrating to act on the region of integration! This allows us to exchange a double integral for a single integral for Green's theorem (or a triple integral for a double integral in the divergence theorem). As we've seen constantly throughout the year, often one computation is easier than another, and thus many difficult area or volume integrals are reduced to simpler, lower dimensional integrals. This subject is done properly in differential forms.
    • The fact that Int_{t = a to b} grad(f)(c(t)) . c'(t) dt = f(c(b)) - f(c(a)) means that this integral does not depend on the path. If a vector field F = (F1, F2, F3) equals grad(f) for some f, we say F is a conservative force field and f is the potential. The fact that these integrals do not depend on the path has, as you would expect, profound applications.
    • This is a good point to stop and think about the number of spatial dimensions in the universe. Imagine a universe with two point masses under gravity, and assume gravity is proportional to 1/r^{n-1} with r the distance between the masses and n the number of spatial dimensions. If there are three or more dimensions, then the work done in moving a particle from infinity to a fixed, non-zero distance from the other mass is finite, while if there are two dimensions the work is infinite! One should of course ask why the correct generalization to other dimensions is 1/r^{n-1} and not 1/r^2 always. There is a nice geometric justification in terms of flux and surface area; the surface area of a sphere grows like r^2 and thus the only way to have the total flux of force out of it be constant is to assume the force drops like 1/r^2; click here for a bit on the justification of inverse-square laws.
    • Speaking of dimensions, one of my favorite problems from undergraduate days was the Random Walk. In 1-dimension, imagine a person so completely drunk that he/she has a 50% chance at any moment of stepping to the left or the right; what is the probability the drunkard eventually returns home? It turns out that this happens with probability 1. In 2-dimension, we have a 25% chance of moving north, south, east or west, and again the probability of returning is 1. In 3 dimensions, however, the drunkard only returns home with probability about 34%. As my professor Peter Jones said, a three-dimensional universe is the smallest one that could be created that will be interesting for drunks, as they really get to explore! These random walk models are very important, and have been applied to economics (the random walk hypothesis), as well as playing a role in statistical mechanics in physics.
    • We sketched the proof of Green's theorem by saying it suffices to prove it for a rectangle. It might be better to do a triangle as these are a bit easier for triangularizing a region (as the name implies!). Of course, If you can do any rectangle you can do any triangle by using appropriately many rectangles.
  • In real analysis, one develops the theory of Riemann sums to rigorously investigate integrals. To do the subject properly, one should consider an arbitrary partition of the set. The most commonly used ones are into equal pieces, and into 2^n equal pieces. If we divide into n equal pieces, the disadvantage is that at the next step the new subintervals can overlap two previous subintervals; if we do 2ⁿ divisions and then divide again, we have 2n+1 divisions and now each new subinterval is half of a previous subinterval. Usually we want to do one of these two partitions, but not always. In solving differential equations numerically, say via the Euler method or the Runge-Kutta method, one may not want to take equally spaced points. The reason is that if the function is not changing much, it is not a good idea to waste a lot of computational time there; rather, given a finite number of places to evaluate, you want to spend your resources where the function's behavior is wildest. This is a vast topic; see for instance the link here. I did some searching on the web, and found this thesis, which has a nice introduction.
  • In sketching the proof of Green's theorem, we used rectangles and talked about pushing the line integrals to the boundary. Some care is needed here, as otherwise we can prove all curves from (0,0) to (1,1) have the same length! How? Take the polygonal approximation to the given curve; for convenience we assume the curve is convex; in other words, it looks like y = x² or y = x³. Then if we look at the sum of the horizontal polygonal part we get the interval [0,1], which is what we get when we look at the sum of the vertical parts! Thus all curves have the same length, namely 2, which is clearly absurd. What went wrong? We're approximating the infinitesimal distance ds with dx + dy and not sqrt(dx^2 + dy^2). This is all related to how much harder it is in calculus to compute lengths of curves than areas; see the section on arc length in wikipedia.

     

• Thursday Sept 9: (Yes, I know it was really Wednesday the 8th, but Wednesday = Thursday today!). In today's class we discussed some of the basics of complex analysis, from the definition and properties of complex numbers to the definition of the derivative to some differences between real valued and complex valued functions. The comments below are meant to delve more deeply into some of these topics. These are meant for your enjoyment and personal edification; they are entirely optional and are independent of the course work and your grade. That said, reading these will help review some of the concepts we've seen.
  • Complex numbers: complex numbers are of the form z = x + iy, with the complex conjugate z = x - iy with i² = -1. While it is impossible to totally order the complex numbers (unlike the reals, which can), they still have many nice properties. They are associative and commutative. It is possible to generalize the complex numbers further. The first is the four dimensional space of the quaternions, which are commutative but not associative. The next generalization is the octonians, which are eight dimensional (over the reals) and now even associativity is lost! What's next? The sedenions!
  • We showed the definition of the derivative can be rewritten and interpreted as the first order Taylor series does an excellent job approximating the function. Interestingly, a function can have continuous partial derivatives without being differentiable. The big theorem is that if the partial derivatives are continuous then the function is differentiable. It is possible for a function to have a limit along some paths but not others, or to have different limits along different paths. For a function to be differentiable, the quotient must tend to the same value no matter what path you take. In one variable there is essentially just two choices (approach from the left or the right); it's much more interesting in several variables, as Cam and Kayla's artwork show.
    • Here is a nice example: it has the same limit along any straight line or parabola, but a different limit along some cubic paths! Consider f(x,y) = (x⁸ + y⁸)/(x² + y⁸) - (x¹⁰ + y¹⁰)/(x⁴ + y¹⁰) and take the limit as (x,y) approaches (0,0).
  • While the definition of complex differentiable looks innocuously similar to that for one variable, remember that the limit must hold for any path. This implies a variety of strange facts. A good way of viewing this is that functions of a complex variable z can be written as functions of x and y. Using z = x + iy and zbar = x - iy, we have x = (z+zbar)/2 and y = (z-zbar)/2i. Essentially the complex differentiable functions are those which depend on z and not zbar. Thus only special combinations of x and y are allowed. Later we'll see that complex differentiability is equivalent to our function satisfying the Cauchy-Riemann differential equations. One way of interpreting this is that we have a function whose corresponding function of two real variables x and y has zero curl. This means we have restricted ourselves to a very nice subset of real valued functions, in particular ones where Green's Theorem (or Stokes' Theorem) is particularly easy to apply.
    • In comparing results in real analysis to complex analysis, we met some very interesting functions. One is sin(1/x); though this function isn't differentiable, x² sin(1/x) is (but not infinitely often!).
    • Another great example is the function f(x) = exp(-1/x²) for x not zero and 0 otherwise. To take the derivatives requires L'Hopital's rule. This function is extremely important in probability, as it shows that a Taylor series does not uniquely determine a function. In probability this arises as the moments of a probability distribution do not always uniquely determine the probability distribution. This is covered in great depth in some notes I wrote up for Math 341 (probability).
  • We mentioned that there is a way to define a measure on the space of continuous function from the reals to the reals, and in this metric almost all functions are differentiable nowhere! Weierstrass has a wonderful example (see also the article by McCarthy as well as the article by Johnsen).
  • We used dimensional analysis to prove the Pythagorean Theorem. There are many proofs; one particularly nice one is due to James Garfield, a Williams alum (and president of the US). Victor Hill, an emeritus professor of mathematics here, has a very enjoyable article on Garfield and his proof.
  • Recall the exponential function exp is defined by e^z = exp(z) = sum_{n = 0 to oo} z^n/n!. This series converges for all z. The notation suggests that e^z e^w = e^(z+w); this is true, but it needs to be proved. (What we have is an equality of three infinite sums; the proof uses the binomial theorem.) Using the Taylor series expansions for cosine and sine, we find e^(iθ) = cos θ + i sin θ. From this we find |e^(iθ)| = 1; in fact, we can use these ideas to prove all trigonometric identities! For example:
    • Inputs: e^(iθ) = cos θ + i sin θ and e^(iθ) e^(iφ) = e^(i (θ+φ))
    • Identity: from e^(iθ) e^(iφ) = e^(i (θ+φ)) we get, upon substituting in the first identity, that (cos θ + i sin θ) (cos φ + i sin φ) = cos(θ+φ) + i sin(θ+φ). Expanding the left hand side gives (cos θ cos φ - sin θ sin φ) + i (sin θ cos φ + cos θ sin φ) = cos(θ+φ) + i sin(θ+φ). Equating the real parts and the imaginary parts gives the identities
      • cos(θ+φ) = cos θ cos φ - sin θ sin φ
      • sin(θ+φ) = sin θ cos φ + cos θ sin φ
      • Here's a picture that shows the standard proof with a convoluted diagram.
    • One can prove other identities along these lines....
  • Finally, a common theme in mathematics is the need to simplify tedious algebra. Frequently we have claims that can be proven by long and involved computations, but these often leave us without a real understanding of why the claim is true. If you want, let me know and I'll show you my 40-50 page proof of Morley's theorem; Conway has a beautiful proof which you can read here (it's after the irrationality of sqrt(2)). If you like non-standard proofs of the irrationality of sqrt(2), see the article I wrote with a SMALL student (to appear in Mathematics Magazine).