Please spend at least 2 hours a night reading the material/looking at the proofs/making sure you understand the details. Below is a tentative reading list and homework assignments. It is subject to changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day. HW will be due on Mondays, TA sessions on Thursdays and Fridays.

• Week 14: December 6 - 10:
  • Video: Mon:  https://youtu.be/FoKKIMs9wV8  (slides)
   
• Week 13: November 29 to December 3:
  • Please skim the following paper and see how complex analysis enters:
    • The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices (with Gene S. Kopp Murat Koloğlu, Frederick Strauch, Wentao Xiong). Journal of Theoretical Probability  (26 (2013), no. 4, 1020--1060) pdf
  • HW: Due Friday December 3:  (1) Method of Stationary Phase: Use Laplace's Method to estimate \((2m-1)!! = \int_{-\infty}^\infty x^{2m} (1/\sqrt{2 \pi}) \exp(-x^2/2) dx\), the \(2m\)-th moment of the standard normal (recall the double factorial is every other term down to  \(2\) or \(1\), so \(5!! = 5\cdot 3 \cdot 1 = 15\)). DO NOT convert this to a value of a Gamma function and invoke Stirling; the point of this exercise is to go through the Method of Stationary Phase to make sure you know how to use it.  Problem 2: A Poisson random variable \(X_\lambda\) has density \({\rm Prob}(X_\lambda = n) = \lambda^n \exp(-\lambda)/n!\) for \(n\) a non-negative integer and zero otherwise, with \(\lambda > 0\). Calculate the Moment Generating Function of \(X_\lambda\) and of \(Z_\lambda = (X_\lambda - \mu_\lambda)/\sigma_\lambda\) (where \(mu_\lambda, \sigma_\lambda\) are the mean, standard deviation of \(X_\lambda\), and show that as \(\lambda \to \infty\) the moment generating function of \(Z_\lambda\) converges to the moment generating function of the standard normal. What's particularly nice is that if \(X_{\lambda_1}, X_{\lambda_2}\) are two independent Poisson random variables with the obvious parameters then \(X_{\lambda_1} + X_{\lambda_2}\) is a Poisson random variable with parameter \(\lambda_1 + \lambda_2\); thus we can interpret our convergence of \(Z_\lambda\) as what happens when we sum independent identically distributed Poisson random variables and standardize. Problems 3, 4 and 5: Do three (3) of the following five (5) problems at http://web.williams.edu/Mathematics/sjmiller/public_html/209/HW/209HWmay12.pdf
   
• Week 12: November 22
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem.
  • HW: Due Friday December 3:  (1) Method of Stationary Phase: Use Laplace's Method to estimate \((2m-1)!! = \int_{-\infty}^\infty x^{2m} (1/\sqrt{2 \pi}) \exp(-x^2/2) dx\), the \(2m\)-th moment of the standard normal (recall the double factorial is every other term down to  \(2\) or \(1\), so \(5!! = 5\cdot 3 \cdot 1 = 15\)). DO NOT convert this to a value of a Gamma function and invoke Stirling; the point of this exercise is to go through the Method of Stationary Phase to make sure you know how to use it.  Problem 2: A Poisson random variable \(X_\lambda\) has density \({\rm Prob}(X_\lambda = n) = \lambda^n \exp(-\lambda)/n!\) for \(n\) a non-negative integer and zero otherwise, with \(\lambda > 0\). Calculate the Moment Generating Function of \(X_\lambda\) and of \(Z_\lambda = (X_\lambda - \mu_\lambda)/\sigma_\lambda\) (where \(mu_\lambda, \sigma_\lambda\) are the mean, standard deviation of \(X_\lambda\), and show that as \(\lambda \to \infty\) the moment generating function of \(Z_\lambda\) converges to the moment generating function of the standard normal. What's particularly nice is that if \(X_{\lambda_1}, X_{\lambda_2}\) are two independent Poisson random variables with the obvious parameters then \(X_{\lambda_1} + X_{\lambda_2}\) is a Poisson random variable with parameter \(\lambda_1 + \lambda_2\); thus we can interpret our convergence of \(Z_\lambda\) as what happens when we sum independent identically distributed Poisson random variables and standardize. Problems 3, 4 and 5: Do three (3) of the following five (5) problems at http://web.williams.edu/Mathematics/sjmiller/public_html/209/HW/209HWmay12.pdf .
   
• Week 11: November 15 to 19
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem. Skim notes on Fourier Analysis
  • HW: Due Friday, November 19: (1) Let \(G(s) = \int_{0}^{\infty} \exp(-x^2) x^{s-1} dx\). Find a functional equation for \(G(s)\). Hint: there is a nice expression for \(G(s+2)\). (2) Let \(H(z) = 1 + z^2 + z^4 + z^6 + z^8 + \cdots.\) Find an analytic continuation for \(H(z)\). For what \(z\) does your analytic continuation make sense? For what \(z\) is it undefined? What should \(H(2)\) equal? (3) Let \(L(s) = \int_{0}^\infty x^s dx / (x^2+1)\). For what \(s\) does the integral exist? (4) Let \(\zeta_{\rm alt}(s) = \sum_{n = 1}^\infty (-1)^{n-1} / n^s\) (alt for alternating). Prove this series converges for \({\rm Re}(s) > 1\). Show that \(\zeta_{\rm alt}(s) = \zeta(s) - (2/2^s) \zeta(s)\) (hint: group the even and odd terms of \(\zeta_{\rm alt}(s)\) together). From this deduce that \(\zeta(s) = (1 - 2^{1-s})^{-1} \zeta_{\rm alt}(s)\). The importance of this exercise is that, using partial summation, one can show that \(\zeta_{\rm alt}(s)\) is well-defined for all \(s\) with \({\rm Re}(s) > 0\). This furnishes yet another analytic continuation of \(\zeta(s)\) (at least for \({\rm Re}(s) > 0\)). Yes, we did in class but it's very important! (5) Show \(\int_{0}^\infty x^4 dx / (1 + x^8) = (\pi/4) \sqrt{1 - 1/\sqrt{2}}\). Hint: remember if \(f(z) = g(z)/h(z)\) with \(g\), holomorphic and \(h\) having a simple zero at \(z_0\), then the residue of \(f\) at \(z_0\) is \(g(z_0)/h'(z_0)\). (6) Chapter 6, Page 175, \#5: Use the fact that \(\Gamma(s) \Gamma(1-s) = \pi/\sin(\pi s)\) to prove that \(|\Gamma(1/2 + it)| = \sqrt{2\pi/(\exp(\pi t) + \exp(-\pi t))}\) for \(t\) real.
   
• Week 10: November 8 to 12  
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem.
  • HW: Due Monday, November 8: DO ANY FIVE OUT OF THE FOLLOWING SIX: IF YOU DO MORE, THAT'S GOOD BUT ONLY THE FIRST FIVE WILL BE GRADED. (1) Consider the functions \(f_n(x) = n / (1 + n x^2)\) where \(n\) is a positive integer. Prove that each \(f_n\) is uniformly continuous on the real line. Is the family \{\(f_n\): \(n\) a positive integer\} equicontinuous on compact sets? (2) Consider a \(2\times 2\) matrix \(M\) with integer entries and top row \((a,b)\) and bottom row \((c,d)\) such that \(ad-bc = 1\); we denote the set of all such matrices by \({\rm SL}(2,Z)\). Consider the map \(f_M(z) = (az + b)/(cz+d)\) with \(z\) in the upper half plane. Is the family \(\{f_M: M \in {\rm SL}(2,Z)\}\) uniformly bounded on compact sets of the upper half plane? Hint: I think each map is bounded on compact subsets of the upper half plane, but you can find a sequence of matrices such that no bound works simultaneously. (3) Let \(f_n(x) = 1 - nx\) for \(0 \le x \le 1/n\) and 0 otherwise, and let \(F = \{f_n: n\ {\rm a\ positive\ integer}\}\). Prove that \(\lim f_n\) exists and determine it. (4) Consider the family from (3). Prove it is not normal (the problem is that the convergence is not uniform). Specifically, to be normal not only must it converge, but given any epsilon there is an \(N\) such that, for all \(n > N\), \(|f_n(x) - f(x)| < \epsilon\) (or this must hold for a subsequence). (5) Evaluate \(\int_{-\infty}^\infty x^2 dx / (x^4 + x^2 + 1)\). (6) Integrate \(\int_0^{2\pi} d\theta / (a + b \sin \theta)\), where \(a\) and \(b\) are real numbers. What restrictions must we place on \(a\) and \(b\) in order for this to make sense?
  • HW: Due Friday, November 19: (1) Let \(G(s) = \int_{0}^{\infty} \exp(-x^2) x^{s-1} dx\). Find a functional equation for \(G(s)\). Hint: there is a nice expression for \(G(s+2)\). (2) Let \(H(z) = 1 + z^2 + z^4 + z^6 + z^8 + \cdots.\) Find an analytic continuation for \(H(z)\). For what \(z\) does your analytic continuation make sense? For what \(z\) is it undefined? What should \(H(2)\) equal? (3) Let \(L(s) = \int_{0}^\infty x^s dx / (x^2+1)\). For what \(s\) does the integral exist? (4) Let \(\zeta_{\rm alt}(s) = \sum_{n = 1}^\infty (-1)^{n-1} / n^s\) (alt for alternating). Prove this series converges for \({\rm Re}(s) > 1\). Show that \(\zeta_{\rm alt}(s) = \zeta(s) - (2/2^s) \zeta(s)\) (hint: group the even and odd terms of \(\zeta_{\rm alt}(s)\) together). From this deduce that \(\zeta(s) = (1 - 2^{1-s})^{-1} \zeta_{\rm alt}(s)\). The importance of this exercise is that, using partial summation, one can show that \(\zeta_{\rm alt}(s)\) is well-defined for all \(s\) with \({\rm Re}(s) > 0\). This furnishes yet another analytic continuation of \(\zeta(s)\) (at least for \({\rm Re}(s) > 0\)). Yes, we did in class but it's very important! (5) Show \(\int_{0}^\infty x^4 dx / (1 + x^8) = (\pi/4) \sqrt{1 - 1/\sqrt{2}}\). Hint: remember if \(f(z) = g(z)/h(z)\) with \(g\), holomorphic and \(h\) having a simple zero at \(z_0\), then the residue of \(f\) at \(z_0\) is \(g(z_0)/h'(z_0)\). (6) Chapter 6, Page 175, \#5: Use the fact that \(\Gamma(s) \Gamma(1-s) = \pi/\sin(\pi s)\) to prove that \(|\Gamma(1/2 + it)| = \sqrt{2\pi/(\exp(\pi t) + \exp(-\pi t))}\) for \(t\) real.
   
• Week 9: Nov 1 - 5 (class is asynchronous on Friday)
  • Read: If you wish, here is a chapter of my probability book: My handout on Gamma function (including the cosecant identity).
  • Read: Chapter 6 of the textbook
  • Videos: Mon: https://youtu.be/yivEV2yhxgA  (slides)  (Lecture 19: 10/27/17: Riemann Mapping Theorem (Proof) https://youtu.be/x0Yy1Ivn1c4 (2015 Lecture))   Wed:     Fri:
  • HW: Due Monday November 8: DO ANY FIVE OUT OF THE FOLLOWING SIX: IF YOU DO MORE, THAT'S GOOD BUT ONLY THE FIRST FIVE WILL BE GRADED. (1) Consider the functions \(f_n(x) = n / (1 + n x^2)\) where \(n\) is a positive integer. Prove that each \(f_n\) is uniformly continuous on the real line. Is the family \{\(f_n\): \(n\) a positive integer\} equicontinuous on compact sets? (2) Consider a \(2\times 2\) matrix \(M\) with integer entries and top row \((a,b)\) and bottom row \((c,d)\) such that \(ad-bc = 1\); we denote the set of all such matrices by \({\rm SL}(2,Z)\). Consider the map \(f_M(z) = (az + b)/(cz+d)\) with \(z\) in the upper half plane. Is the family \(\{f_M: M \in {\rm SL}(2,Z)\}\) uniformly bounded on compact sets of the upper half plane? Hint: I think each map is bounded on compact subsets of the upper half plane, but you can find a sequence of matrices such that no bound works simultaneously. (3) Let \(f_n(x) = 1 - nx\) for \(0 \le x \le 1/n\) and 0 otherwise, and let \(F = \{f_n: n\ {\rm a\ positive\ integer}\}\). Prove that \(\lim f_n\) exists and determine it. (4) Consider the family from (3). Prove it is not normal (the problem is that the convergence is not uniform). Specifically, to be normal not only must it converge, but given any epsilon there is an \(N\) such that, for all \(n > N\), \(|f_n(x) - f(x)| < \epsilon\) (or this must hold for a subsequence). (5) Evaluate \(\int_{-\infty}^\infty x^2 dx / (x^4 + x^2 + 1)\). (6) Integrate \(\int_0^{2\pi} d\theta / (a + b \sin \theta)\), where \(a\) and \(b\) are real numbers. What restrictions must we place on \(a\) and \(b\) in order for this to make sense?

• Week 8: October 25 to 29
  • Read: Chapter 8 and my online notes.
  • HW Due Monday, November 1: (1) Evaluate \(\int_{-\infty}^{\infty} \cos(4x)dx / (x^4 + 1)\). (2) Let \(U\) be conformally equivalent to \(V\) and \(V\) conformally equivalent to \(W\) with functions \(f: U \to V\) and \(g: V \to U\). Prove \(g \circ f\) (\(g\) composed with \(f\)) is a bijection. (3) The Riemann mapping theorem asserts that if \(U\) and \(V\) are simply connected proper open subsets of the complex plane then they are conformally equivalent. Show that simply connected is essential: find a bounded open set \(U\) that is not simply connected and prove that it cannot be conformally equivalent to the unit disk. (4) Chapter 8, Page 248: \#4. (5) Chapter 8: Page 248: \#5. (6) Chapter 8: Page 251: \#14.
   
• Week 7: October 18 to 22
  • Read: Chapter 5: Sections 5.1, 5.3, 5.4 and my online notes. Read Chapter 8 and my online notes.
  • Due Friday, October 29:  Chapter 5: Page 155: \#6, \#7, \#9 (extra credit: what is the combinatorial significance of this problem?). Chapter 3: Page 104: \#10. Additional Problems: (1) Find all poles of the function \(f(z) = 1 / (1 - z^2)^4\) and find the residues at the poles. (2) Consider the map \(f(z) = (z - i) / (z + i)\). Show that this is a 1-to-1 and onto map from the upper half plane (all \(z = x + iy\) with \(y > 0\)) to the unit disk. (3) Calculate the Weierstrass product for \(\cos(\pi z\)) (this is also problem \#10b in Chapter 5, and the answer is listed there), and for \(\tan(\pi z)\).
   
• Week 6: October 11 - 15 (no class Mon b/c of Reading Period, no class Fri b/c either MtDay or Exam):
  • Read: Chapter 5: Sections 5.1, 5.3, 5.4 and my online notes. Read Chapter 8 and my online notes.
  • Due Friday, October 29: Chapter 5: Page 155: \#6, \#7, \#9 (extra credit: what is the combinatorial significance of this problem?). Chapter 3: Page 104: \#10. Additional Problems: (1) Find all poles of the function \(f(z) = 1 / (1 - z^2)^4\) and find the residues at the poles. (2) Consider the map \(f(z) = (z - i) / (z + i)\). Show that this is a 1-to-1 and onto map from the upper half plane (all \(z = x + iy\) with \(y > 0\)) to the unit disk. (3) Calculate the Weierstrass product for \(\cos(\pi z\)) (this is also problem \#10b in Chapter 5, and the answer is listed there), and for \(\tan(\pi z)\).
   
• Week 5: October 4 - 8:
  • Read Chapter 3 (Sections 3.1, 3.2, 3.3, 3.4, 3.6), Chapter 5 (Sections 5.1, 5.3 and 5.4) and my online notes.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due Friday, October 8 (if Mountain Day happens then it's due on Monday, October 11): Chapter 3, Page 103: #1, #2, #5 (this is related to the Fourier transform of the Cauchy density), #15d, #17a (hard). Additional: Let \(f(z) = \sum_{n = -5}^\infty a_n z^n\) and \(g(z) = \sum_{m = -2}^{\infty} b_m z^m\) be the Laurent expansions for two functions holomorphic everywhere except possibly at \(z=0\). a) Find the residues of \(f(z)\) and \(g(z)\) at \(z=0\); b) Find the residue of \(f(z)+ g(z)\) at \(z=0\); c) Find the residue of \(f(z) g(z)\) at \(z=0\); d) Find the residue of \(f(z)/g(z)\) at \(z=0\).
  • Suggested Problems: Chapter 3: Chapter 3: Do #2 with \(x^{2k}\) instead of \(x^4\), #3, #9, #12 (great problem: also take \(u = 1/2\) and deduce a formula for the sum of \(1/n^2\) -- amazingly, this should be doable from just knowing the sum over the odd integers!), #13, #15c, #19. Also: is there a nice formula for the residue of \(f(g(z))\), where \(f\) and \(g\) are the functions from the Additional problem above?

   

• Week 4: September 27 to October 1: no in-person class on Friday
  • Read Chapter 2 and my online notes for Section 2.4. Just skim Section 5 so you're aware of those results; we won't cover them in detail now. Read Chapter 3.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due Friday, October 1: Chapter 2, Page 64: #1, #8. Also do: Chapter 2: (#1) In the proof of Liouville's theorem we assumed \(f\) was bounded. Is it possible to remove that assumption? In other words, is it enough to assume that \(|f(z)| < g(z)\) for some real-valued, non-decreasing function \(g\)? If yes, how fast can we let \(f\) grow?  (#2) a) Find all \(z\) where the function \(f(z) = 1 / (1+z^4)\) is not holomorphic; b) Let \(a, b, c\) and \(d\) be integers such that \(ad - bc = 1\). Find all \(z\) where the function \(g(z) = (az + b) / (cz + d)\) is not holomorphic. (#3) Compute the power series expansion of \(f(z) = 1 / (1-z)\) about the point \(z = 1/2\) (it might help to do the next problem first, or to write \(1 - z\) as \(1/2 - (z - 1/2))\). (#4) Do Chapter 1, Page 29, #18.
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 2: #2, #3, #5, #6. Also evaluate the integral of \(\sin^2 x/x^2\) for \(-\infty < x < \infty\). Also, the fundamental theorem of algebra says a polynomial of degree n has n complex roots. The roots thus lie in a disk of radius \(R\). Find an upper bound for \(R\) in terms of the coefficients of the polynomial and its degree; in other words, find a computable number \(R\) (as a function of the parameters of the problem) such that all the roots must lie in the disk \(|z| < R\). Finally, read about a fundamental theorem you don't already know.

• Week 3: September 20 to 24:

  • Read Chapter 2, 3

  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.

  • HW: Due Friday, September 24: Chapter 1: Page 24: #16abc, #24, #25ab. Chapter 2: (#1) We proved Goursat's theorem for triangles. Assume instead we know it holds for any rectangle; prove it holds for any triangle. (#2) Let gamma be the closed curve that is the unit circle centered at the origin, oriented counter-clockwise. Compute Integral_gamma f(z) dz where f(z) is complex conjugation (so f(x+iy) = x - iy). Repeat the problem for Integral_gamma f(z)^n dz for any integer n (positive or negative), and compare this answer to the results for Integral_gamma z^n dz; is your answer surprising? (#3) Prove that the four triangles in the subdivision in the proof of Goursat's theorem are all similar to the original triangle. (#4) In the proof of Goursat's theorem we assumed that f was complex differentiable (ie, holomorphic). Would the result still hold if we only assumed f was continuous? If not, where does our proof break down?

  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 1: #5, #7, #16d, #18, #23, #26.

   
• Week 2: September 13-17, 2017:
  • Videos: Mon: https://youtu.be/aJBDcKw7pfE
  • Read : Chapter 1: Section 2.2 (includes if and only if for Cauchy-Riemann), Section 2.3 (work on series), Section 3 (integration); Chapter 2: Section 2.1 (Goursat's Theorem), Section 2.2 (Cauchy in a disk).
  • You can also click here for my notes for Chapter 1 (includes review of real analysis concepts: let me know if any items here are unfamiliar / unclear)
  • You can also click here for my notes for Chapter 2
  • HW#2: Due by Friday September 17: Chapter 1: Page 24: #1abcd, #3, #13. The assignment is deliberately light so you can spend a lot of time reading....
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 1: #5, #7, #16d, #18, #23, #26.
  • Extra credit: see the extra credit list. The list is from the 2010 iteration, but they're still good problems.

• Week 1: September 10, 2021
  • Read: Chapter 1 and my online notes, review your real analysis. Useful resources are the online real analysis book
  • Homework: To be emailed to me by noon on Sunday, September 10: Email me a short note on what you want to get out of this course, and what lesson  you learned from the graduation speech (https://whatrocks.github.io/commencement-db/2006-michael-uslan-indiana-university/). Full credit, 20/20, so long as you answer both questions on time.
  • HW#2: Due by 11am Friday, September 17: Chapter 1: Page 24: #1abcd, #3, #13.
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 1: #5, #7, #16d, #18, #23, #26.
  • Extra credit: see the extra credit list.

FIXING DATES FROM BELOW

• Week 13: November 27 to December 1:
  • Please skim the following paper and see how complex analysis enters:
    • The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices (with Gene S. Kopp Murat Koloğlu, Frederick Strauch, Wentao Xiong). Journal of Theoretical Probability  (26 (2013), no. 4, 1020--1060) pdf
  • HW: Due Friday December 1:  (1) Method of Stationary Phase: Use Laplace's Method to estimate \((2m-1)!! = \int_{-\infty}^\infty x^{2m} (1/\sqrt{2 \pi}) \exp(-x^2/2) dx\), the \(2m\)-th moment of the standard normal (recall the double factorial is every other term down to  \(2\) or \(1\), so \(5!! = 5\cdot 3 \cdot 1 = 15\)). DO NOT convert this to a value of a Gamma function and invoke Stirling; the point of this exercise is to go through the Method of Stationary Phase to make sure you know how to use it.  Problem 2: A Poisson random variable \(X_\lambda\) has density \({\rm Prob}(X_\lambda = n) = \lambda^n \exp(-\lambda)/n!\) for \(n\) a non-negative integer and zero otherwise, with \(\lambda > 0\). Calculate the Moment Generating Function of \(X_\lambda\) and of \(Z_\lambda = (X_\lambda - \mu_\lambda)/\sigma_\lambda\) (where \(mu_\lambda, \sigma_\lambda\) are the mean, standard deviation of \(X_\lambda\), and show that as \(\lambda \to \infty\) the moment generating function of \(Z_\lambda\) converges to the moment generating function of the standard normal. What's particularly nice is that if \(X_{\lambda_1}, X_{\lambda_2}\) are two independent Poisson random variables with the obvious parameters then \(X_{\lambda_1} + X_{\lambda_2}\) is a Poisson random variable with parameter \(\lambda_1 + \lambda_2\); thus we can interpret our convergence of \(Z_\lambda\) as what happens when we sum independent identically distributed Poisson random variables and standardize. Problems 3, 4 and 5: Do three (3) of the following five (5) problems at http://web.williams.edu/Mathematics/sjmiller/public_html/209/HW/209HWmay12.pdf .
   
• Week 12: November 20
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem.
  • HW: Due Friday December 1:  (1) Method of Stationary Phase: Use Laplace's Method to estimate \((2m-1)!! = \int_{-\infty}^\infty x^{2m} (1/\sqrt{2 \pi}) \exp(-x^2/2) dx\), the \(2m\)-th moment of the standard normal (recall the double factorial is every other term down to  \(2\) or \(1\), so \(5!! = 5\cdot 3 \cdot 1 = 15\)). DO NOT convert this to a value of a Gamma function and invoke Stirling; the point of this exercise is to go through the Method of Stationary Phase to make sure you know how to use it.  Problem 2: A Poisson random variable \(X_\lambda\) has density \({\rm Prob}(X_\lambda = n) = \lambda^n \exp(-\lambda)/n!\) for \(n\) a non-negative integer and zero otherwise, with \(\lambda > 0\). Calculate the Moment Generating Function of \(X_\lambda\) and of \(Z_\lambda = (X_\lambda - \mu_\lambda)/\sigma_\lambda\) (where \(mu_\lambda, \sigma_\lambda\) are the mean, standard deviation of \(X_\lambda\), and show that as \(\lambda \to \infty\) the moment generating function of \(Z_\lambda\) converges to the moment generating function of the standard normal. What's particularly nice is that if \(X_{\lambda_1}, X_{\lambda_2}\) are two independent Poisson random variables with the obvious parameters then \(X_{\lambda_1} + X_{\lambda_2}\) is a Poisson random variable with parameter \(\lambda_1 + \lambda_2\); thus we can interpret our convergence of \(Z_\lambda\) as what happens when we sum independent identically distributed Poisson random variables and standardize. Problems 3, 4 and 5: Do three (3) of the following five (5) problems at http://web.williams.edu/Mathematics/sjmiller/public_html/209/HW/209HWmay12.pdf .
   
• Week 11: November 13 to 17 
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem.
  • Video: Mon:
  • Video: Wed:
  • Video: Fri:
  • HW: Due Friday, November 17: (1) Let \(G(s) = \int_{0}^{\infty} \exp(-x^2) x^{s-1} dx\). Find a functional equation for \(G(s)\). Hint: there is a nice expression for \(G(s+2)\). (2) Let \(H(z) = 1 + z^2 + z^4 + z^6 + z^8 + \cdots.\) Find an analytic continuation for \(H(z)\). For what \(z\) does your analytic continuation make sense? For what \(z\) is it undefined? What should \(H(2)\) equal? (3) Let \(L(s) = \int_{0}^\infty x^s dx / (x^2+1)\). For what \(s\) does the integral exist? (4) Let \(\zeta_{\rm alt}(s) = \sum_{n = 1}^\infty (-1)^{n-1} / n^s\) (alt for alternating). Prove this series converges for \({\rm Re}(s) > 1\). Show that \(\zeta_{\rm alt}(s) = \zeta(s) - (2/2^s) \zeta(s)\) (hint: group the even and odd terms of \(\zeta_{\rm alt}(s)\) together). From this deduce that \(\zeta(s) = (1 - 2^{1-s})^{-1} \zeta_{\rm alt}(s)\). The importance of this exercise is that, using partial summation, one can show that \(\zeta_{\rm alt}(s)\) is well-defined for all \(s\) with \({\rm Re}(s) > 0\). This furnishes yet another analytic continuation of \(\zeta(s)\) (at least for \({\rm Re}(s) > 0\)). Yes, we did in class but it's very important! (5) Show \(\int_{0}^\infty x^4 dx / (1 + x^8) = (\pi/4) \sqrt{1 - 1/\sqrt{2}}\). Hint: remember if \(f(z) = g(z)/h(z)\) with \(g\), holomorphic and \(h\) having a simple zero at \(z_0\), then the residue of \(f\) at \(z_0\) is \(g(z_0)/h'(z_0)\). (6) Chapter 6, Page 175, \#5: Use the fact that \(\Gamma(s) \Gamma(1-s) = \pi/\sin(\pi s)\) to prove that \(|\Gamma(1/2 + it)| = \sqrt{2\pi/(\exp(\pi t) + \exp(-\pi t))}\) for \(t\) real.

   

• Week 10: November 6 to 10  
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem.
  • HW: Due Friday, November 10: DO ANY FIVE OUT OF THE FOLLOWING SIX: IF YOU DO MORE, THAT'S GOOD BUT ONLY THE FIRST FIVE WILL BE GRADED. (1) Consider the functions \(f_n(x) = n / (1 + n x^2)\) where \(n\) is a positive integer. Prove that each \(f_n\) is uniformly continuous on the real line. Is the family \{\(f_n\): \(n\) a positive integer\} equicontinuous on compact sets? (2) Consider a \(2\times 2\) matrix \(M\) with integer entries and top row \((a,b)\) and bottom row \((c,d)\) such that \(ad-bc = 1\); we denote the set of all such matrices by \({\rm SL}(2,Z)\). Consider the map \(f_M(z) = (az + b)/(cz+d)\) with \(z\) in the upper half plane. Is the family \(\{f_M: M \in {\rm SL}(2,Z)\}\) uniformly bounded on compact sets of the upper half plane? Hint: I think each map is bounded on compact subsets of the upper half plane, but you can find a sequence of matrices such that no bound works simultaneously. (3) Let \(f_n(x) = 1 - nx\) for \(0 \le x \le 1/n\) and 0 otherwise, and let \(F = \{f_n: n\ {\rm a\ positive\ integer}\}\). Prove that \(\lim f_n\) exists and determine it. (4) Consider the family from (3). Prove it is not normal (the problem is that the convergence is not uniform). Specifically, to be normal not only must it converge, but given any epsilon there is an \(N\) such that, for all \(n > N\), \(|f_n(x) - f(x)| < \epsilon\) (or this must hold for a subsequence). (5) Evaluate \(\int_{-\infty}^\infty x^2 dx / (x^4 + x^2 + 1)\). (6) Integrate \(\int_0^{2\pi} d\theta / (a + b \sin \theta)\), where \(a\) and \(b\) are real numbers. What restrictions must we place on \(a\) and \(b\) in order for this to make sense?
  • HW: Due Friday, November 17: (1) Let \(G(s) = \int_{0}^{\infty} \exp(-x^2) x^{s-1} dx\). Find a functional equation for \(G(s)\). Hint: there is a nice expression for \(G(s+2)\). (2) Let \(H(z) = 1 + z^2 + z^4 + z^6 + z^8 + \cdots.\) Find an analytic continuation for \(H(z)\). For what \(z\) does your analytic continuation make sense? For what \(z\) is it undefined? What should \(H(2)\) equal? (3) Let \(L(s) = \int_{0}^\infty x^s dx / (x^2+1)\). For what \(s\) does the integral exist? (4) Let \(\zeta_{\rm alt}(s) = \sum_{n = 1}^\infty (-1)^{n-1} / n^s\) (alt for alternating). Prove this series converges for \({\rm Re}(s) > 1\). Show that \(\zeta_{\rm alt}(s) = \zeta(s) - (2/2^s) \zeta(s)\) (hint: group the even and odd terms of \(\zeta_{\rm alt}(s)\) together). From this deduce that \(\zeta(s) = (1 - 2^{1-s})^{-1} \zeta_{\rm alt}(s)\). The importance of this exercise is that, using partial summation, one can show that \(\zeta_{\rm alt}(s)\) is well-defined for all \(s\) with \({\rm Re}(s) > 0\). This furnishes yet another analytic continuation of \(\zeta(s)\) (at least for \({\rm Re}(s) > 0\)). Yes, we did in class but it's very important! (5) Show \(\int_{0}^\infty x^4 dx / (1 + x^8) = (\pi/4) \sqrt{1 - 1/\sqrt{2}}\). Hint: remember if \(f(z) = g(z)/h(z)\) with \(g\), holomorphic and \(h\) having a simple zero at \(z_0\), then the residue of \(f\) at \(z_0\) is \(g(z_0)/h'(z_0)\). (6) Chapter 6, Page 175, \#5: Use the fact that \(\Gamma(s) \Gamma(1-s) = \pi/\sin(\pi s)\) to prove that \(|\Gamma(1/2 + it)| = \sqrt{2\pi/(\exp(\pi t) + \exp(-\pi t))}\) for \(t\) real.
   
• Week 9: October 30 - November 3, 2017:
  • Read: If you wish, here is a chapter of my probability book: My handout on Gamma function (including the cosecant identity).
  • Read: Chapter 6 of the textbook
  • HW Due Friday, November 3: (1) Evaluate \(\int_{-\infty}^{\infty} \cos(4x)dx / (x^4 + 1)\). (2) Let \(U\) be conformally equivalent to \(V\) and \(V\) conformally equivalent to \(W\) with functions \(f: U \to V\) and \(g: V \to W\). Prove \(g \circ f\) (\(g\) composed with \(f\)) is a bijection. (3) The Riemann mapping theorem asserts that if \(U\) and \(V\) are simply connected proper open subsets of the complex plane then they are conformally equivalent. Show that simply connected is essential: find a bounded open set \(U\) that is not simply connected and prove that it cannot be conformally equivalent to the unit disk. (4) Chapter 8, Page 248: \#4. (5) Chapter 8: Page 248: \#5. (6) Chapter 8: Page 251: \#14.
  • HW: Due Friday, November 10: DO ANY FIVE OUT OF THE FOLLOWING SIX: IF YOU DO MORE, THAT'S GOOD BUT ONLY THE FIRST FIVE WILL BE GRADED. (1) Consider the functions \(f_n(x) = n / (1 + n x^2)\) where \(n\) is a positive integer. Prove that each \(f_n\) is uniformly continuous on the real line. Is the family \{\(f_n\): \(n\) a positive integer\} equicontinuous on compact sets? (2) Consider a \(2\times 2\) matrix \(M\) with integer entries and top row \((a,b)\) and bottom row \((c,d)\) such that \(ad-bc = 1\); we denote the set of all such matrices by \({\rm SL}(2,Z)\). Consider the map \(f_M(z) = (az + b)/(cz+d)\) with \(z\) in the upper half plane. Is the family \(\{f_M: M \in {\rm SL}(2,Z)\}\) uniformly bounded on compact sets of the upper half plane? Hint: I think each map is bounded on compact subsets of the upper half plane, but you can find a sequence of matrices such that no bound works simultaneously. (3) Let \(f_n(x) = 1 - nx\) for \(0 \le x \le 1/n\) and 0 otherwise, and let \(F = \{f_n: n\ {\rm a\ positive\ integer}\}\). Prove that \(\lim f_n\) exists and determine it. (4) Consider the family from (3). Prove it is not normal (the problem is that the convergence is not uniform). Specifically, to be normal not only must it converge, but given any epsilon there is an \(N\) such that, for all \(n > N\), \(|f_n(x) - f(x)| < \epsilon\) (or this must hold for a subsequence). (5) Evaluate \(\int_{-\infty}^\infty x^2 dx / (x^4 + x^2 + 1)\). (6) Integrate \(\int_0^{2\pi} d\theta / (a + b \sin \theta)\), where \(a\) and \(b\) are real numbers. What restrictions must we place on \(a\) and \(b\) in order for this to make sense?
   
• Week 8: October 23 to 27, 2015:
  • Read: Chapter 8 and my online notes.
  • HW Due Friday, November 3: (1) Evaluate \(\int_{-\infty}^{\infty} \cos(4x)dx / (x^4 + 1)\). (2) Let \(U\) be conformally equivalent to \(V\) and \(V\) conformally equivalent to \(W\) with functions \(f: U \to V\) and \(g: V \to U\). Prove \(g \circ f\) (\(g\) composed with \(f\)) is a bijection. (3) The Riemann mapping theorem asserts that if \(U\) and \(V\) are simply connected proper open subsets of the complex plane then they are conformally equivalent. Show that simply connected is essential: find a bounded open set \(U\) that is not simply connected and prove that it cannot be conformally equivalent to the unit disk. (4) Chapter 8, Page 248: \#4. (5) Chapter 8: Page 248: \#5. (6) Chapter 8: Page 251: \#14.
   
• Week 7: October 19 to 23, 2015: NO  CLASS ON FRIDAY
  • Read: Chapter 5: Sections 5.1, 5.3, 5.4 and my online notes. Read Chapter 8 and my online notes.
  • Due Friday, October 27:  Chapter 5: Page 155: \#6, \#7, \#9 (extra credit: what is the combinatorial significance of this problem?). Chapter 3: Page 104: \#10. Additional Problems: (1) Find all poles of the function \(f(z) = 1 / (1 - z^2)^4\) and find the residues at the poles. (2) Consider the map \(f(z) = (z - i) / (z + i)\). Show that this is a 1-to-1 and onto map from the upper half plane (all \(z = x + iy\) with \(y > 0\)) to the unit disk. (3) Calculate the Weierstrass product for \(\cos(\pi z\)) (this is also problem \#10b in Chapter 5, and the answer is listed there), and for \(\tan(\pi z)\).
   
• Week 6: October 11, 2017: (no class Mon b/c of Reading Period, no class Fri b/c either MtDay or Exam)
  • Read: Chapter 5: Sections 5.1, 5.3, 5.4 and my online notes. Read Chapter 8 and my online notes.
  • Due Friday, October 27: Chapter 5: Page 155: \#6, \#7, \#9 (extra credit: what is the combinatorial significance of this problem?). Chapter 3: Page 104: \#10. Additional Problems: (1) Find all poles of the function \(f(z) = 1 / (1 - z^2)^4\) and find the residues at the poles. (2) Consider the map \(f(z) = (z - i) / (z + i)\). Show that this is a 1-to-1 and onto map from the upper half plane (all \(z = x + iy\) with \(y > 0\)) to the unit disk. (3) Calculate the Weierstrass product for \(\cos(\pi z\)) (this is also problem \#10b in Chapter 5, and the answer is listed there), and for \(\tan(\pi z)\).
   
• Week 5: October 2 - 6:
  • Read Chapter 3 (Sections 3.1, 3.2, 3.3, 3.4, 3.6), Chapter 5 (Sections 5.1, 5.3 and 5.4) and my online notes.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due Friday, October 6 (if Mountain Day happens then it's due on Monday, October 9): Chapter 3, Page 103: #1, #2, #5 (this is related to the Fourier transform of the Cauchy density), #15d, #17a (hard). Additional: Let \(f(z) = \sum_{n = -5}^\infty a_n z^n\) and \(g(z) = \sum_{m = -2}^{\infty} b_m z^m\) be the Laurent expansions for two functions holomorphic everywhere except possibly at \(z=0\). a) Find the residues of \(f(z)\) and \(g(z)\) at \(z=0\); b) Find the residue of \(f(z)+ g(z)\) at \(z=0\); c) Find the residue of \(f(z) g(z)\) at \(z=0\); d) Find the residue of \(f(z)/g(z)\) at \(z=0\).
  • Suggested Problems: Chapter 3: Chapter 3: Do #2 with \(x^{2k}\) instead of \(x^4\), #3, #9, #12 (great problem: also take \(u = 1/2\) and deduce a formula for the sum of \(1/n^2\) -- amazingly, this should be doable from just knowing the sum over the odd integers!), #13, #15c, #19. Also: is there a nice formula for the residue of \(f(g(z))\), where \(f\) and \(g\) are the functions from the Additional problem above?
   
• Week 4: September 25 to September 29:
  • Read Chapter 2 and my online notes for Section 2.4. Just skim Section 5 so you're aware of those results; we won't cover them in detail now. Read Chapter 3.
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due Friday, September 29: Chapter 2, Page 64: #1, #8. Also do: Chapter 2: (#1) In the proof of Liouville's theorem we assumed \(f\) was bounded. Is it possible to remove that assumption? In other words, is it enough to assume that \(|f(z)| < g(z)\) for some real-valued, non-decreasing function \(g\)? If yes, how fast can we let \(f\) grow?  (#2) a) Find all \(z\) where the function \(f(z) = 1 / (1+z^4)\) is not holomorphic; b) Let \(a, b, c\) and \(d\) be integers such that \(ad - bc = 1\). Find all \(z\) where the function \(g(z) = (az + b) / (cz + d)\) is not holomorphic. (#3) Compute the power series expansion of \(f(z) = 1 / (1-z)\) about the point \(z = 1/2\) (it might help to do the next problem first, or to write \(1 - z\) as \(1/2 - (z - 1/2))\). (#4) Do Chapter 1, Page 29, #18.
  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 2: #2, #3, #5, #6. Also evaluate the integral of \(\sin^2 x/x^2\) for \(-\infty < x < \infty\). Also, the fundamental theorem of algebra says a polynomial of degree n has n complex roots. The roots thus lie in a disk of radius \(R\). Find an upper bound for \(R\) in terms of the coefficients of the polynomial and its degree; in other words, find a computable number \(R\) (as a function of the parameters of the problem) such that all the roots must lie in the disk \(|z| < R\). Finally, read about a fundamental theorem you don't already know.
   

• Week 3: September 18 to 22:

  • Read Chapter 2, 3

  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.

  • HW: Due Friday, September 22: Chapter 1: Page 24: #16abc, #24, #25ab. Chapter 2: (#1) We proved Goursat's theorem for triangles. Assume instead we know it holds for any rectangle; prove it holds for any triangle. (#2) Let gamma be the closed curve that is the unit circle centered at the origin, oriented counter-clockwise. Compute Integral_gamma f(z) dz where f(z) is complex conjugation (so f(x+iy) = x - iy). Repeat the problem for Integral_gamma f(z)^n dz for any integer n (positive or negative), and compare this answer to the results for Integral_gamma z^n dz; is your answer surprising? (#3) Prove that the four triangles in the subdivision in the proof of Goursat's theorem are all similar to the original triangle. (#4) In the proof of Goursat's theorem we assumed that f was complex differentiable (ie, holomorphic). Would the result still hold if we only assumed f was continuous? If not, where does our proof break down?

  • Suggested Problems (these are NOT to be turned in, but rather for your own personal edification): Chapter 1: #5, #7, #16d, #18, #23, #26.