Additional Comments

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.

Friday Dec 6: Today was the Jordan Plane Curve theorem, a rite of passage to modern math.
- Nice article by Hales: http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/hales882.pdf
- Some issues with generalizations: https://people.math.osu.edu/fiedorowicz.1/math655/Jordan.html
- Why is it not obvious: http://math.stackexchange.com/questions/61849/why-is-the-jordan-curve-theorem-not-obvious
- Video: http://www.gogeometry.com/topology/topology_jordan_curve_theorem_video.html
- Related topic: Smale's paradox and some related videos.
  - http://www.gogeometry.com/topology/topology_jordan_curve_theorem_video.html
  - http://www.youtube.com/watch?v=wO61D9x6lNY

Wednesday Dec 4: A great application of complex analysis is to dynamical systems and iterations of functions.

Monday Nov 2: Today we explored some of the consequences of several complex variables.
- 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.
- 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 \(f(z_1,z_2) = 1/z_1\) then it has poles in the entire plane \(z_1=0, z_2\) 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 \(f(z_1,z_2) = 1/(z_1^2+z_2^2)\). The problem is that this has a pole in the plane \(z_2 = i z_1\). 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.
  - Here are some notes for additional reading.
  - Another reference here. Has the nice 1-dimensional example of \(\sum_k z^{k!}\).
  - Very detailed set of notes here.
- 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.
- We also discussed branch cuts and other methods of contour integration.
  - My lecture notes on branch cuts and trigonometric integration.
  - Article on these methods.

Monday Nov 25: Today we finished some loose ends, doing the Weierstrass nowhere differentiable function and a bit more analytic number theory.
- It is possible for a function to be continuous but differentiable nowhere; here is a nice example from Weierstrass. In fact, almost all continuous functions are differentiable nowhere! You'll find such results in a functional analysis class. For more on this, click here and see the references.
- 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.
- Here is a fun integral I learned about in Physics 260 (Professor Firk) right before Thanksgiving my freshman year at Yale. He wanted us to have something to talk about over the holidays. Let \(\int\) be the operator \(\int f\) means \(\int_{t = 0}^x f(t) dt.\) The solution to \( \int f = f - 1 \) (ie, \(\int_{t=0}^x f(t) dt = f(x) - 1\)) is \(f(x) = e^x.\) Let's see why.
  - \(\int f = f - 1\) ==> \(f - \int f = 1.\)
  - Thus \((1 - \int) f = 1.\)
  - Therefore \(f = (1 - \int)^{-1} 1.\)
  - Using the geometric series expansion \((1-r)^{-1} = 1 + r + r^2 + \cdots\) with \( r = \int\) we find \(f = 1 + \int 1 + \int \int 1 + \int \int \int 1 + \cdots.\)
  - Now \(\int 1 = \int_{t=0}^{x} 1 dt = x.\)
  - Now \(\int \int 1 = \int(\int 1) = \int_{t=0}^{x} t dt = x^2/2 = x^2/2!.\)
  - Now \(\int \int \int 1 = \int(\ \int(\int 1)\ ) = \int_{t = 0}^{x} t^2/2 dt = x^3/3!.\)
  - We find \(f = f(x) = 1 + x + x^2/2! + x^3/3! + \cdots = e^x.\)
- The big question is: can this be made rigorous, and if so how? Happy Thanksgiving!

Friday Nov 22: We finished our discussion of analytic number theory, proving the Prime Number Theorem. We then turned to Weierstrass's continuous but nowhere differentiable function.
- The complex analytic proof of the Prime Number Theorem uses several key facts. We need the functional equation of the Riemann zeta function (which follows from Poisson summation and properties of the Gamma function), the Euler product (namely that \(zeta(s)\) is a product over primes), and the important fact that the Riemann zeta function does not have a zero on the line Re(s) = 1! If this happened, then the main term of \(x\) from integrating \(\zeta'(s)/\zeta(s) \ast x^s/s\) arising from the pole of \(\zeta(s)\) at \(s=1\) would be cancelled by the contribution from this zero! Thus it is essential that there be no zero of zeta(s) on Re(s) = 1. There are many proofs of this result. My favorite proof is based on a wonderful trig identity: \(3 + 4 \cos(x) + \cos(2x) = 2 (1 - \cos(x))^2 \ge 0\) (many people have said that \(w^2 \ge 0\) for real \(w\) is the most important inequality in mathematics). If people are interested I'm happy to give this proof in class next week (or see Exercise 3.2.19 in my number theory textbook; this would make a terrific aside if anyone is still looking for a problem). There is an elementary proof of the prime number theorem (ie, one without complex analysis). For those interested in history and some controversy, see this article by Goldfeld for a terrific analysis of the history of the discovery of the elementary proof of the prime number theorem and the priority dispute it created in the mathematics community. We mentioned Riemann computed zeros of zeta(s) but didn't mention his achievement; the method only came to light about 70 years later when Siegel was looking at Riemann's papers. Click here for more on the Riemann-Siegel formula for computing zeros of zeta(s). Finally, terrific advice given to all young mathematicians (and this advice applies to many fields) is to read the greats. In particular, you should read Riemann's original paper. In case your mathematical German is poor, you can click here for the English translation of Riemann's paper. The key passage is on page 4 of the paper: One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.
- 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 \(x \to \infty\), 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 \(\zeta(s) = \sum_{n = 1}^{\infty} 1/n^s\) (or actually its meromorphic continuation) has no zero on the line \({\rm Re}(s) = 1\). 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).

Wednesday Nov 20: Today we returned to the basics, and actually did a contour integral again to find the Fourier transform of a Gaussian. We then saw an application to the analytic continuation of the Riemann zeta function.
- We calculated the Fourier transform of the Gaussian by doing a contour integral as, well, this is a class in complex analysis! There are other ways. Click here for a fun one from the theory of differential equations.
- We needed the Fourier transform of the Gaussian for two purposes. The first was to see when the inequality in the Hesienberg Uncertainty Principle is sharp (it's at a Gaussian). The second was for the analytic continuation of the Riemann zeta function \(\zeta(s)\). The proof we gave reduces the continuation of the completed function \(\xi(s) = \pi^{-s/2} \Gamma(s/2) \zeta(s)\) to the functional equation of \(\theta(u) = \sum_{n=-\infty}^\infty \exp(-\pi n^2 u)\). The proof of analytic continuation often comes down to a functional equation; for the analytic continuation of the Gamma function it was \(\Gamma(s+1) = s \Gamma(s)\). One difficulty in the proof of the analytic continuation of \(\zeta(s)\) is figuring out how to begin. When we have the expression for the completed Riemann zeta function it's natural to replace the Gamma function with its defining integral, and the zeta function with its defining sum. Unfortunately that doesn't allow us to change orders, as there is no intersection of variables in the sum and integral. In times like these, it's natural to try a change of variables, and bring the summands into the integration and then change orders.
- An enormous amount can be said about the Gamma function, which generalizes the standard factorial function. We gave a proof of its functional equation, Γ(s+1) = sΓ(s); this allows us to take the Gamma function (initially defined only when the real part of s is positive) and extend it to be well-defined for all s other than the non-positive integers. We can express all the moments of the standard normal in terms of special values of the Gamma function. There are a lot of nice, special values of the Gamma function. One of the most important is \(\Gamma(1/2)\). We can deduce these values from either the multiplication or reflection formula, but that just begs the question on how these are proved. See our textbook for details.
- Below are a lot of comments on the Gamma function and where it occurs, and some related problems.
  - One nice application of the Gamma function and normalization constants is a proof of Wallis' formula, which says π/2 = (2·2 / 1·3) (4·4 / 3·5) (6·6 / 5·7) ···. I have a proof which is mostly elementary (see my article in the American Mathematical Monthly). Not surprisingly, the proof uses one of my favorite techniques, the theory of normalization constants (caveat: it does have on advanced ingredient from measure theory, namely Lebesgue's Dominated Convergence Theorem).
  - Many functions in mathematical physics initially exist only for some values of the parameters but can be continued elsewhere; my favorite is the Riemann zeta function (and the extension uses the Gamma function). What is amazing (and not initially apparent) is that the following frequently occurs. We have some function and we only care about its values at the real numbers (or maybe even just the integers); nevertheless,it is often easier to study it as a function of a complex variable (z = x + iy), as then we have all the tools and techniques of complex analysis at our disposal. A terrific example is the Prime Number Theorem (which says that, to first order, the number of primes at most x is about x/log x). This is a statement about integers, yet the `easiest' and `best' proofs all use the Riemann zeta function at complex arguments (and, as you may reasonably ask, why should we need to use complex numbers to count integers!). What follows is an aside on an aside -- this is clearly not needed for the course!

Monday Nov 18: Today was a big day for physics, finally showing how the Hesienberg Uncertainty Principle is a consequence of properties of the Fourier transform. This was a very fast paced lecture; in an ideal world we'd have another 15 minutes. It was a tough choice, but I decided it's better to do it in one day while everything is fresh then to split into several. This means that we can't do all the steps in class together. I strongly urge you to go through the final few steps yourself, and to revisit the last few pages of the handout, if you want to see the nuts and the bolts. There are a lot of calculations to do. This is exactly how lectures are in graduate school. All the details are not done; the algebra or standard calculations are often left for you. Thus you need to show \(\langle Qf, f\rangle = \langle f, Qf\rangle\), and similarly for \(P\) (this is equivalent to saying these two operators are Hermitian); the proof involves computing each separately and then integrating one of them by parts. If you haven't done these calculations they're great exercises, and they also explain why we defined \(P\) to be \(-i d/dx\) and not \(d/dx\) (the presence of the \(-i\) makes it Hermitian). The last bit today was doing algebra to look at \(1 = ||f||_2^2 = i\int_{-\infty}^\infty ((PQ - QP)f)(x) \overline{f(x)} dx\), as \(PQ - QP = -iI\). Though we were a bit rushed, if you go through the algebra and use the fact that the operators are Hermitian (so \(\langle Af, f\rangle = \langle f, Af\rangle\) for \(A\) a Hermitian operator), and use Plancherel to note \(||(Pf)||_2 = ||\widehat{(Pf)}||_2\) (up to factors of 2 and \(\pi\)), we get the product of the variances is at least a universal constant.
- We started with the definition of the Fourier transform. We restricted our analysis to functions in the Schwartz space.
- Though we didn't really need it, a big result is the Fourier inversion formula. We did, however, need Plancherel's theorem.
- One of the reasons the Fourier transform is so useful is that it converts differentiation to multiplication, and vice-versa.
- Other ingredients from today include the mean, the variance, the Cauchy-Schwartz inequality, and the Fourier transform of the Gaussian (see entry 206).
- Finally, we also need the position operator and momentum operator

Friday Nov 15: We finished our Fourier unit, getting through the Fourier transform and Poisson summation.
- We started by using Parseval's theorem to evaluate \(\sum 1/n^4\) and then Dirichlet's theorem to get \(\sum 1/n^2\). Interestingly using \(f(x) = 1/2 - |x|\) only gives us the sums over odd integers, but we can pass from that to all integers by noting the evens are a simple multiple of the entire sum, and then using the Bring it over method.
- We then turned to Poisson summation. By assuming decay of our functions the sums converge; we saw it was natural to study \(F(x) = \sum_n f(x+n)\). The decay conditions ensure that the sum exists and is perioidic, continuous and differentiable, and thus we can use all our results from Fourier series! One of the hardest parts of proofs is figuring out what to study; as we want to show \(\sum_n f(n) = \sum_n \widehat{f}(n)\), it is natural to try and convert one side to a function of \(x\) and then evaluate at a special choice of \(x\) (we take \(x=0\)). Once we think to study this function, we're well on our way!
- Finally, we looked at functions \(g(x) \ge 0\) such that \(\int_0^\infty g(x)^k dx < \infty\) for \(k \le 2012\) but is infinity for \(k = 2013\). Interestingly a function whose integral is finite doesn't mean its values tend to 0 as \(x \to \infty\); what can happen is that the function goes to 0 almost everywhere but explodes on an arbitrarily small set. A great example is \(g(x) = n\) for \(|x-n| \le 1/2n^\delta\) with \(\delta = 2014\).
Wednesday Nov 13: The goal of today's lecture was Parseval's theorem, which has a lot of nice applications.
- We started with Bessel's inequality. This followed from one of the most important inequalities in math: if \(x\) is real then \(x^2 \ge 0\).
- Showing Bessel's inequality is an equality is Parseval's theorem. We used a lot of great methods from real analysis. We assumed our function was continuous and then broke the analysis into two parts. First we used Fejer's theorem to show \(f\) was close to \(T_N f\), and then we showed \(T_N f\) is close to \(S_N f\). In general we don't know that our first function is continuous, so we have to go through a standard analysis argument: an \(L^1\) function can be approximated by step functions, which can be approximated by continuous functions, .... After that, we had to do some work to study the sum \(\sum_{n=-N}^N (n^2/N^2) |\widehat{f}(n)|^2\). The idea was to break the sum into small \(n\), say \(|n| \le \sqrt{N}\), and large \(n\). The small case is easily handled as the Fourier coefficients are bounded (by both the \(L^1\) and \(L^2\) norm of \(f\)), and we have an \(N^2\) in the denominator. For the larger values, \(n^2/N^2 \le 1\) and the fact that the unweighted sum of Fourier coefficients converges allows us to show that our sum, which is a slightly decreased version of that tail, converges. These are very common methods in real analysis, and worth knowing.
- I wasn't expecting interest in doing the integration, so we're a few minutes behind; that's perfectly fine, I think it's good to role up the sleeves and do calculations. We computed the Fourier coefficients, but it looks like we're only going to get information about sums of \(1/(2n+1)^4\). This leads to two questions. (1) How do we pass from this to a sum over all integers? (2) We want sums of \(1/n^2\) and not \(1/n^4\); is it possible to get that from what we know? Stay tuned and come to Friday's lecture for the exciting conclusion.

Monday Nov 11: In honor of Veterans Day, we began today's lecture with one of my favorite examples of applications of mathematics, the Battle of Midway; the specific section is on code-breaking in the battle.
- We then turned to Fejer kernels and Dirichlet kernels. The main result we proved today was Fejer's theorem, using a three-epsilon approach.
- We saw a nice consequence, Weierstrass' Approximation Theorem, which can be generalized to the Stone - Weierstrass Theorem.
- A huge industry is devoted to convergence of Fourier series. A great result along these lines is Carleson's theorem.
- Dirichlet's theorem for pointwise convergence using differentiability, but actually all that's needed is the function to be Lipschitz with exponent 1. It is possible for a function to be continuous but differentiable nowhere; here is a nice example from Weierstrass. In fact, almost all continuous functions are differentiable nowhere! You'll find such results in a functional analysis class. For more on this, click here and see the references.

Friday Nov 8: We began our study of Fourier series in earnest today.
- While one can study generalized coordinates, orthonormal vectors are nice! We saw how easy projections are and expanding the function, but we must be careful to make sure we're not missing any directions.
- We showed the family \(\{e_n(x)\}_{n=-\infty}^\infty\) are orthonormal, where \(e_n(x) = e^{2\pi i n x}\); this is often called the Fourier basis. We didn't do the calculation, but the Taylor basis \(1, x, x^2, \dots\) is not.
- For an application of Fourier series, see my Calc III lecture on the mathematics behind streaming video: http://www.youtube.com/watch?v=WjnHNHPhJtU
- We studied the various \(L^p\) spaces, including inclusions. We saw a beautiful trick to get \(L^2(0,1) \subset L^1(0,1)\), noting \(|f| \le |f|^2 + 1\). Identities like this are very powerful, and allow us to propagate information. Of course, this would be useless if instead of a finite interval we had an infinite interval. We also saw the two spaces are not the same as \(x^{-1/2}\) is in \(L^1(0,1)\) but not \(L^2(0,1)\).
- The extra credit challenge problem: find a continuous, non-negative function \(f\) such that \(\int_0^\infty f(x)dx = 1\) but \(\lim_{x\to\infty} f(x) \neq 0\).
- We ended with the Dirac delta function, which is very useful. We can reach it through approximations to the identity.

Wednesday Nov 6: We discussed analytic continuation of the geometric series and \(\zeta(s)\), and sketched the proof of the Prime Number Theorem. In the proof sketch we saw why we need to analytically continue well beyond the real part of \(s\) exceeds zero.
- 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 if you want to extend to all of the complex plane, 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. We'll prove it by considering the periodic function \(F(x) = \sum_{n = -\infty}^{\infty} f(x+n)\). 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.
- 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 \(\zeta(s) = \sum_{n = 1}^{\infty} 1/n^s\) for \({\rm Re}(s) > 1\), and by unique factorization also equals \(\prod_{p\ {\rm prime}} (1 - 1/p^s)^{-1}\). 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 \(0 \le {\rm Re}(s) \le 1\) and \(-T \le {\rm Im}(s) \le T\), then a positive percent have \({\rm Re}(s) = 1/2\), 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
- We gave two proofs of analytic continuation of the Riemann zeta function. The first was through the alternating zeta function, and was fairly straightforward for \(s\) real and greater than zero. I checked and the standard definition agrees with what we did in class: \(\zeta^\ast(s) = \sum_{n=1}^\infty (-1)^{n+1} / n^s\) (although some authors use \((-1)^{n-1}\) instead of \((-1)^{n+1}\). The second was comparing the sum to an integral and doing lots of algebra (which I've left to you); you can find that online in many sources, for example:
  - Page 92: http://www.math.tifr.res.in/~publ/ln/tifr01.pdf (this is a good reference for \(\zeta(s)\)).
  - http://www.math.harvard.edu/~elkies/M259.02/zeta1.pdf
  - Another variant of the calculation is in the textbook, page 173.
- We will do much more with the zeta function and the prime number theorem later. We saw that it's natural to look at the logarithmic derivative of \(\zeta(s)\), as this is \(\zeta'(s)/\zeta(s)\); this is ideally suited for contour integration. The reason this is useful is the Euler product, which is screaming at us to apply a logarithm. Finally, we saw the value in weighting the primes by logarithmic weights; these can be removed easily via partial summation.
Monday Nov 4: After finishing the last ingredient for the proof of the Riemann mapping theorem, we started our number theory tour by looking at the Riemann zeta function.
- 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 result we needed to finish the proof of the Riemann mapping theorem was that the limit of a family of 1-1 holomorphic functions that converge uniformly on compacta is either constant or 1-1. We looked at what happens if we remove the initial functions are 1-1, and got lots of counter-examples (\(f_n(z) = z^2\) is a nice family to look at -- the convergence is clearly uniform as there is no \(n\) dependence, but the resulting limit is neither 1-1 nor constant). Our proof involved looking at the integral of an appropriately chosen function. We spent a lot of time discussing why it was natural to look at \(g_n(z) = f_n(z) - f_n(z_1)\) instead of equaling \(f_n(z) - f(z_1)\); the issue is that we nee to control where the zeros are.
- In class we defined \(\pi(x)\) to be the number of primes at most \(x\). You may have seen Euclid's argument, which shows that \(\pi(x)\) tends to infinity with \(x\); with some work one can show Euclid's argument implies \(\pi(x) \gg \log \log x\). 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 theharmonic 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, and a sense of what is true.

Friday Nov 1: We proved Montel's theorem today, which gives us the final step of the proof of the Riemann Mapping Theorem.
- 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.

Wednesday Oct 30: Our tour of analysis began today!
- 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 \(I_n = \int_{-1}^{1} (1 - x^2)^n dx\). Then \(I_{n+1} = \int_{-1}^{1} (1 - x^2)^{n+1} dx\) \(=\) \(\int_{-1}^{1} (1 - x^2)^n (1 - x^2) dx\) \(=\) \(\int_{-1}^{1} (1 - x^2)^n dx - \int_{-1}^{1} x (1 - x^2) x dx\), where we wrote the last integral like this to facilitate integrating by parts. Recalling the definition of \(I_n\) gives \(I_{n+1} = I_n - \int_{-1}^{1} x (1 - x^2)^n x dx\). 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 \(I_{n+1} = I_n - 1/(2n+2) \int_{-1}^{1} (1 - x^2)^{n+1} dx = I_n - I_{n+1}/(2n+2)\). 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 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 natural choices for the Theorem of the Day: Cantor's uncountability theorem and the Heine-Borel Theorem on the compactness of \([0,1]\), which tie in nicely with much of our conversations.

Monday Oct 28: We sketched the proof of the Riemann Mapping Theorem, leaving the difficult step 2 for later. We 'almost' had time to give the proof of Step III without using
- 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 \(x \to \infty\), 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 \(\zeta(s) = \sum_{n = 1}^{\infty} 1/n^s\) (or actually its meromorphic continuation) has no zero on the line \({\rm Re}(s) = 1\). 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 an extra credit problem on look-up tables (due December 1). 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!

Friday Oct 25: We discussed the Schwarz lemma and (yet again) the differences b/w real and complex.
- 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? Note this value of 2 beats what we did in class, where we used \(f(x) = (4/\pi) {\rm arctan}(x)\) and \(g(x) = \sin(\pi x / 2)\).
  - Here is my paper (with Williams alum David Thompson) where we answer the above question on how large the derivative can be.
- 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).

Wednesday Oct 23: We continued our discussion of conformal maps, looking at a lot of examples and special cases.
- You should do a lot of the algebra to understand the map \(\psi_\alpha(z) = (\alpha - z) / (1 - \overline{\alpha} z)\). 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). It's interesting that \(\psi_\alpha\) is its own inverse!
- 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 occasionally include a link from the site `Theorem of the Day'. I'll choose 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!

Monday Oct 21: The Riemann Mapping Theorem is one of the strangest results in a very strange subject; to reach the proof of it we'll see many useful facts along the way. We started today by building up our intuition for complex maps.
- 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.
- The exponential map f(z) = exp(iz) leads to some interesting behavior, which we'll discuss next time. Vertical strips are conformally equivalent to the upper half plane, with the map almost extendable to being from the boundary to the boundary. In fact, we can 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.
- Consider the map 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 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.
- 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).
- In a previous iteration of the course I spent more time on the Open Mapping Theorem; in order to save time for Quantum Mechanics applications I'm cutting this, but will provide some comments on an alternative approach below.
  - 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?
  - 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, study the function f(z) = zⁿ. We are led to this by looking at the conditions and seeing that we can standardize our general, non-constant holomorphic function to f(z) = zⁿ (1 + Σ_{k = 1 to ∞} c_k z^k), and of course the simplest case is when the c_k's are all zero. We look at the map z --> zⁿ on a disk centered at the origin. While it is 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 is 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 zⁿ (1 + Σ_{k = 1 to ∞} c_k z^k) with (z h(z))ⁿ for some nice h(z). This is essentially Newton's Generalized Binomial Theorem, where we have (1 + Σ_{k = 1 to ∞} c_k z^k)^1/n. It turns out to be relatively easy to get a series expansion for (1+x)^1/n 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 the book's 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).
- In a previous iteration of the course I also spent a bit more time on Weierstrass products. Here are some more things to ponder.
  - 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.

Wednesday Oct 16: We finished our discussions on infinite products and functions that vanish to prescribed orders at prescribed places, and talked about formulas for \(pi\).
- We talked a lot about the book-keeping needed to bound certain sums. Great notation for this is big-Oh notation and little-oh notation.
- Gregory-Leibnitz formula for \(pi\).
- Wallis formula for \(\pi\)
- click here for my paper proving the Wallis' formula from an analysis of the student t-distribution.

Friday Oct 11: We continued our discussions on infinite products and functions that vanish to prescribed orders at prescribed places.
- We started by trying to find a product that vanished only at prescribed zeros \(a_n\), with \(|a_n| \to \infty\). We tried \(\prod_n (z - a_n)\), but this won't converge. We thus tried \(\prod_p (1 - z/a_n)\), which does have better convergence properties, but not enough to ensure that it exists. This led us to introduce exponential factors, as these won't add zeros but can help with the convergence (these were the Weierstrass factors \(E_n(z) = \exp(z + z^2/2 + \cdots + z^n/n)\).
- A big part of all the analysis today was splitting into two cases: where things are well behaved (say \(|a_n| \le 1/2\)) and where things could be poorly behaved (but there are only finitely many of these). This is a powerful idea, and a frequent strategy. Break off the `bad' cases and handle them separately, then study the remainder where you have additional properties to simplify the analysis. We frequently exploited the power of \(|a_n| \le 1/2\); one of my favorites uses was to show \(\sum_n \left(a_n - a_n^2/2 + a_n^3/3 - a_n^4/4 + \cdots\right)\) converges. We have to be careful due to the signs and the fact that we can't immediately relate the sums of \(a_n\) to various powers; however, \(|a_n|^k \le (1/2)^{k-1}|a_n|\), which allows us to replace the sum of the various powers of \(a_n\) with a geometric series which converges.
- We frequently looked at extreme cases to get a sense of what was going on. This is a wonderful method, and we may see it again when we study the Riemann zeta function (more on that below). A great example was studying \(\prod_n (1 + a_n)\). We can sandwich the value by \(\prod_n (1 - |a_n|)\) and \(\prod_n (1 + |a_n|)\), and are now in a situation where we're just taking logarithms of real valued quantities.
- I can't emphasize enough how important it is to do special cases to build intuitions. When trying to see if \(\prod_n (1 + a_n)\) converged we looked at two special cases. The first was \(\prod_n (1 - 1/(n^2-1))\), which was just \(\frac{2\cdot 2}{1\cdot 3} \frac{3 \cdot 3}{2 \cdot 4} \frac{4 \cdot 4}{3 \cdot 5} \cdots\), which converges to 2 by cancellation; the second was \(\prod_n \frac{n+1}{n}\), which was just \(\frac{2}{1} \frac{3}{2} \frac{4}{3} \frac{5}{4} \cdots\), which diverges to infinity. From this we saw we had convergence of the product \(\prod_n (1 + a_n)\) when the sum of \(|a_n|\) converged, and divergence when the sum diverged. This wasn't a proof, but highly suggestive of the answer (which we then proved). There was a great question in class as to whether or not it goes the other way -- I strongly encourage you to explore some interesting possibilities.
- 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 \(\sum_{n = 1}^{\infty} 1 / n^{2k}\). 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) = \(\sum_{n = 1}^{\infty} 1 / n^s\) 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 \(\prod_{p\ {\rm prime}} (1 - 1/p^{-s})^{-1}\) (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 following are some nice comments on primes. I was going to give them a little later, but as you have so many off days before the next class....
  - 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). This was important when we were looking at the Weierstrass factors \(E_n(z)\). 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 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, which I think we're close to getting. 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. 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?

Wednesday Oct 9: We began our discussions on understanding a function from its zeros.
- 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 (and if it's a polynomial, one of the same degree). More generally, we can multiply by the exponential of a function.
- 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.
- If we are given the coefficients of a polynomial then the roots are uniquely determined; however, if the degree is 5 or larger it is not possible in general to write down an explicit, closed form expression for the roots as functions of the coefficients.
  - For the qunitic, see: http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
  - For Galois theory, see: http://en.wikipedia.org/wiki/Galois_theory#Solvable_groups_and_solution_by_radicals
- We spent a lot of time looking at how to try and write down a function with a prescribed set of zeros. If the function has only finitely many zeros it wasn't too bad, and we saw any two such functions differ by the exponential of a holomorphic function. The situation is far more interesting if we have countably many zeros. The example from class, where \(a_n = 1/2^n\), couldn't be the write set to look at as it has an accumulation point. This led to the challenge question: is it possible to write down \(\prod_n (z - a_n)\) and have this converge to a non-zero value for \(z\) in some open set? The point of all these discussions is to get a sense of what it means for products to converge, and to try and build intuition and understand why we define quantities the way we do.
- Finally, earlier in the class we showed that if \(f\) is holomorphic on a simply connected set and never zero then there is a holomorphic \(g\) such that \(f(z) = e^{g(z)}\). Looking at this, it seems like \(g\) is the logarithm of \(f\), so it was natural to integrate \(f'(z)/f(z)\) from some point \(z_0\) to \(z\). As our region is simply connected, the answer is independent of the path (so using this integral as the definition of \(g(z)\) is well-defined. Unfortunately it gives us \(g(z_0) = 0\) (this is the only point to try to evaluate \(g\), as it's the only point we know is in our set! This led us to subtracting a constant from the definition of \(g\) (which is fine and doesn't affect it being a primitive for \(f'/f\). We chose that constant \(c_0\) such that \(g(z_0) = c_0\) with \(e^{c_0} = f(z_0)\). The point of all this, in addition to getting a beautiful result, is to understand the thought process. Why did we do the integral we did. Why did we subtract the constant we did.

Monday Oct 7: We discussed simply connected regions, homotopic curves, and the complex logarithm.
- The key concept lurking behind our toy contours is that they're all simply connected regions. We then showed that homotopic curves in such domains give the same integrals of holomorphic functions. The Wikipedia site, http://en.wikipedia.org/wiki/Homotopic, has a nice illustration
- Here is a great youtube video on turning a sphere inside out (and why you can't do that to a circle!): http://www.youtube.com/watch?v=BVVfs4zKrgk; I strongly recommend taking a class in topology if you want to continue in pure mathematics.
- We then discussed how to define the complex logarithm, and how to make sure our definition extends that of the real logarithm. We started with our knowledge of how to create a primitive by looking at a small circle about a point and then integrating along east-west and north-south paths, and then using the existence of the primitive to show that the resulting definition is independent of the path.
- We've talked several times this semester 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).

Friday Oct 4: The comments from October 2nd still apply; then we stated the results, now we proved it. Below are some items worth considering.
- Be careful about hidden assumptions creeping in. We needed to know there was some circle about \(z_0\) such that \(|f(z) - w_0| > \delta > 0\) on that circle (in order to use Roche's theorem to prove the Open Mapping Theorem). If there were no such circle, we would have a point of accumulation which would lead to our function being constant.
- There are many ways to teach the course / present the material. I prefer to do it with you reading the material ahead of time and then reviewing it in class, and concentrating on the subtle issues and seeing why we make the assumptions we do. We saw the constraint that \(f\) is non-constant naturally emerge in our proof of the Open Mapping Theorem. We also saw that it doesn't matter what is going on inside the region, only on the boundary.
- The hardest part of these proofs is figuring out what to use, when to use them, and how to apply them. Knowing that we want to use Roche's theorem isn't enough; we need to figure out which functions to use it on. I hope our discussion made our choice of \(F\) and \(G\) natural. We start with a function that we know has a zero, and then pass to the function we want to have a zero.

Wednesday Oct 2: We began by talking about the behavior of functions at infinite. This led to stereographic projection. From there we went to the argument principle and its consequences.
- Unfortunately the logarithm of a complex number doesn't have the same nice properties as the logarithm of real numbers; fortunately one key property survives: \((f_1f_2)'/f_1f_2 = f_1'/f_1 + f_2'/f_2\). We used this to prove the argument principle. Well, truth be told, we didn't use it. We derived it again from scratch when we took the logarithmic derivative of \(a z^n (1 + g(z))\); we could have taken \(f_1(z) = a z^n\) and \(f_2(z) = 1 + g(z)\); remember that if \(h(z) = \log f(z)\) then \(h'(z) = f'(z) / f(z)\); you should have a Pavlovian response to do a contour integral of a logarithmic derivative.
- 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 when we talk about proofs of the Prime Number Theorem.
- 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 won't prove the Open Mapping Theorem this way, 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.

Monday Sept 30: We explored the different types of singularities today (removable, pole and essential).
- The first result is Riemann's theorem on removable singularities. We talked at length as to what is the correct constraints to impose. Continuity is too much (as then removable is trivial), while factorization is too hard to use. We settled on locally bounded near the point of interest, and proved Riemann's theorem on removable singularities.
- We then turned to essential singularities. The big result is the Casorati-Weierstrass theorem.
- We spent a lot of time looking at a specific example with an essential singularity, \(e^{1/z}\). We wanted to show that, for any non-zero number \(w\) we could find a \(z_w\) such that \(e^{1/z_w} = w\). Writing \(z_w = x + iy\) we got a system of equations for \(x\) and \(y\). There are typically infinitely many solutions, as we write \(w = r e^{i (\theta + 2\pi n)}\) and we get freedom from the \(n\). As a great exercise, explicitly solve for \(x\) and \(y\) in terms of \(r, \theta\) and \(n\).

Friday Sept 27: We continued working on the Residue Theorem. The difficulty is always exploiting decay, which we saw required splitting the top arc carefully. Of course, if we integrate \(\sin^2 x / x^2\) by parts we get a better decay....
- 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 a few classes ago, finding the normalization constant for integrating 1/(1+x²) 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.
  - T
    he 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
    - 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.

Wednesday Sept 25: We talked about the Residue theorem, and the power of notation.

Monday Sept 23: Cauchy's formula has numerous consequences, which we discussed today, ranging from definitional equality (holomorphic = analytic) to algebraic (the Fundamental Theorem of Algebra) to analytic (the Central Limit Theorem).
- 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.
- 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).
  - 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).
- We talked today about the Fundamental Theorem of Calculus. Remember, the `meat' is not that \(\int_a^b f(x) = F(b) - F(a)\) for some \(F\) with \(F' = f\), but that the integral represents the area under the curve \(y = f(x)\) from \(x = a\) to \(x = b\). If we don't say area, it's just the Fundamental Definition of Calculus. Similarly, the meat in today's lecture is that holomorphic and analytic refer to different concepts that turn out to be equal in certain settings.
- There are not that many fundamental theorems in mathematics -- we do not use the term lightly! Other ones you may have seen are the Fundamental Theorem of Arithmetic and the Fundamental Theorem of Algebra; click here for a list of more fundamental theorems (including the Fundamental Theorem of Poker!).
- Here's a strange, Calc III proof of the Fundamental Theorem of Algebra!
  - Lagrange Multipliers and the Fundamental Theorem of Algebra (Theo de Jong), American Mathematical Monthly, Volume 116, Number 9, November 2009, pp. 828-830(3). This is a beautiful proof mostly using what you've done in Calc III (gradients, Lagrange Multipliers, Level Sets). There is a bit of real analysis terminology and results, but not too much, and I think confined enough so that you can get the general flavor.

Friday Sept 20: We continued integrating, and saw the subtleties that can arise in extending some holomorphic results. In particular, the difficulties in interchanging a derivative and an integral, or a limit and an integral.
- 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.
  - Here's an example that illustrates the dangers in the subject. Let \(f_n(x)\) be the triangular function from \(1/n\) to \(3/n\), starting at 0, rising to n at \(2/n\), and falling to \(0\) at \(3/n\). Note \(\int_0^\infty f_n(x)dx = 1\) for all \(n\), so \(\lim_{n\to\infty} \int_0^\infty f_n(x)dx = 1\); however, \(\int_0^1 \lim_{n\to\infty} f_n(x) dx = 0\). Thus we have to be VERY careful in interchanging operations.
- 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!).

Monday Sept 16: Today we finished the proof that a holomorphic function on a disk has a primitive, and in fact a constructable one. At first it didn't look like we used the holomorphicity, but remember that was needed to reduce to integrating along the straight line. We then talked about integrating over certain closed curves, and started to consider \(\int_{-\infty}^\infty (1-\cos x) dx/x^2\).
- 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:
- 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 \(y = x^2\) 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 within \(\epsilon\) 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 \(ds = \sqrt{dx^2 + dy^2}\), 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.
- 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.

Friday Sept 13: On Wednesday, September 18th there will be no class.
- Stokes' Theorem (or, for us, Green's Theorem as we're in the plane) is a gem of mathematics. It can be used to compute areas of regions by reducing it to a line integral. A great example is to find the area of an 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.
- We talked about how many problems can be reduced to a (very difficult) integral. For example, \(f_N(x) = \int_{x=0}^1 \left(\sum_{p \le N} \exp(2 \pi i p x)\right)^2 \exp(-2\pi 2m x) dx\) gives the number of ways \(2m\) can be written as a sum of pairs of primes where each is at most 10. This is 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.
- What about other important integrals? 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).
- 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 used 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.
Wednesday Sept 11: Today we explored series expansions and integration along curves.
- 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.
    - The geometric series formula only makes sense when \(|r| < 1\), in which case \(1 + r + r^2 + \cdots = 1/(1-r)\); however, the right hand side makes sense for all r other than 1. We say the function \(1/(1-r)\) is a(meromorphic) continuation of \(1+r+r^2+\cdots.\) This means that they are equal when both are defined; however, \(1/(1-r)\) makes sense for additional values of \(r\). Interpreting \(1+2+4+8+\cdots\) as \(-1\) or \(1+2+3+4+5+ \cdots = -1/12\) actually DOES make sense, and arises in modern physics and number theory (the latter is \(\zeta(1)\), where \(zeta(s)\) is the Riemann zeta function)! We'll see \(\zeta(s)\) later in the course, as well as analytic continuation.
  - Another infinity to be aware of is the derivative of a sum is the sum of the derivatives. If we only have finitely many terms in our sum it's true, and follows immediately from the case of just two terms from grouping. We have (f + g + h)' = (f + (g + h))' = f' + (g + h)' (using the rule for the derivative of a sum of two terms) = f' + (g' + h') (again using the rule for the derivative of the sum of two terms) = f' +g' + h' (as we can drop parentheses by associativity). We can continue by induction and get the derivative of a finite sum is the finite sum of the derivatives, but we do not get the derivative of an infinite sum is the sum of the derivatives. This is a great example of how to really milk a result for all it's worth, as well as the dangers of being too greedy.
  - Here is a video from Cameron when he was 2, explaining how to switch sums.
- See yesterday's notes for more on path integrals and similar concepts.
- 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).
- Finally, it's worth reiterating `checking' formulas you can't remember. If you can remember some special cases, you can often correct your guess. A great example is 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.

Monday Sept 9: On Wednesday, September 18th there will be no class; you should watch my lecture on Green's Theorem in a day from Calc III (you can watch it before then if you'd like; this quickly goes over path integrals and explains the calculations we've done). After exploring a bit the definition of complex differentiable (with some examples) and proving a holomorphic function satisfies the Cauchy-Riemann equations, we then saw how the integral of a power series without negative terms around a circle centered at the origin was zero. More generally, using Green's Theorem we saw a reason to believe that \(\oint_{\partial \Omega} f(z) dz = 0\) for \(f\) holomorphic. We looked at \(f(z) dz\) as \((u(x,y) + iv(x,y)) \cdot (dx + idy)\), and interpreted that as \(udx - vdy\) plus \(i(vdx + udy)\), and then applied Green's Theorem to each piece.
- 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.
- In sketching the proof of Green's theorem, the video uses rectangles and talkes 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 \(y = x^2\) or \(y = x^3\). 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.

Friday Sept 6: 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 \(i^2 = -1\). 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 \(\overline{z} = x - iy\), we have \(x = (z+\overline{x})/2\) and \(y = (z-\overline{z})/2i\). Essentially the complex differentiable functions are those which depend on \(z\) and not \(\overline{z}\). 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 \(e^z = \exp(z) = \sum_{n = 0}^{\infty} z^n/n!\). 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).