HOMEWORK

click here for comments / solutions to the HW

click here for a summary of the key points of the readings

• Week 13: November 30 to December 4: No class on FRIDAY
  • 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 Monday December 7:  (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 23 to November 27, 2013:
  • Reading: Pages  30 and 31 of pdf http://web.williams.edu/Mathematics/sjmiller/public_html/209/209LectureNotes.pdf (page 22 and 23 on the bottom) on the Laplace Transform.
  • HW: Due Monday December 7:  (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 16 to 20
  • No written HW: Read probability notes. Suggested problem: Finish the computation of the moments of the log-normal and the lognormal plus the sinusoidal term, and show the moments only agree when b=0 and a is a half-integer.

   

• Week 10: November 9 to 13 (no class on Wed Nov 11: will be to make up for the colloquium on  Mon Nov 23 by Blake Mackall on the Uncertainty Principle)
  • Read: Skim Appendix A (concentrate on the section on Laplace's Method),  Read Notes on Central Limit Theorem.
  • HW: Due Monday, November 16: (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 \({\rm Re}(s) > 0\)). (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: November 2 to 6, 2015:
  • 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
  • Due Monday, November 9: 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 26 to 30, 2015: NO  CLASS ON FRIDAY OCT 30
  • Read: Chapter 8 and my online notes.
  • HW Due Monday, November 2: (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:
  • Read: Chapter 5: Sections 5.1, 5.3, 5.4 and my online notes. Read Chapter 8 and my online notes.
  • Due Monday, October 26: 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 14, 2015:
  • Read: Chapter 5: Sections 5.1, 5.3, 5.4 and my online notes. Read Chapter 8 and my online notes.
  • Due Monday, October 26: 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 5, 2015:
  • 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 Monday, October 12 (but no penalty if you hand it in on Wednesday October 14): 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 28 to October 2, 2015:
  • 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 Monday, October 5: 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 21 to 25:
  • Read Chapter 2
  • Remember to glance at the reading highpoints if you want a quick summary of key items of the sections.
  • HW: Due Monday, September 28: 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 14 - 18, 2015:

  • 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: Due by Monday September 21: 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 11, 2015
  • Read: Chapter 1 and my online notes, review your real analysis. Useful resources are the online real analysis book
  • HW: Due by 10am Friday, September ??: 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.