Winter/Spring Semester, 2012 Happy New Year Everyone. To be discussed: Final Exam or Project? (The latter would be that each of you would choose a topic related to complex analysis, for example a topic in your field of concentration--almost every subject contains a topic related to CA, or a new topic (covered not at all or only lightly in class) that you want to explore. Then you would write a report on your investigation (it need not be long but it should be clearly written) and then give a brief (10min) exposition of your project during the last class days of term. ======================================================= Due Friday, Jan. 27 p. 166, ex 1 (Hint: Use Thm 20) p. 171, ex 1 (Hint: You can do this directly or indirectly. Ex: set w=S(z)=i(z-i)/(z+i): UHP \to D. U(0) = \int_0^{2\pi} U(w) dw/iw. U(i)=\int_{-\infty}^{\infty}U(S(x))dx/(1+x^2). In general, send iy instead of i to w=0. p. 171, ex 2, 3 (don't need to generalize), 4,5 =============================================== Due Frday, Feb 3 p. 173, ex 1-4 p, 178, ex 1-3 (hint: try summation by parts, p. 42) Thanks to Eyerusalem for bring #3 pointing out that `summation by parts' is a MISLEADING hint. Better hint: NOte that for s positive, the alternating series converges. Rearrange a bit, write the series 1 - {\sum_2^\infty (-1)^n/n^s } = 1 -{\sum (1/(2n)^s -1/(2n+1)^s)}. Then 1/(2n)^s -1/(2n+1)^s = s\int_{2n}^{2n+1} r^{-s-1} dr := I Now |s|=\sqrt{sigma^2 + t^2} < C sigma if |t| \le T is bdd, C a const. Also |r^{-s-1}| = r^{\sigma -1} So |I| < C\sigma (1/(2m)^\sigma -(2n+1)^\sigma) So now the abs value of the original sum up to (2n-1) is bdd by the alternating series starting with (2n) and going to \infty. This converges since \sigma >0. As n\to infty, the remainder after 2n-1 goes to zero. So the original series uniformly converges for all \sigma >0 and bdd t. So the series in analytic in s in the right halfplane. (For more info on general 'Dirichlet series' see Gamelin, p. 378). ============================================== Midterm Exam: Friday, March 2 ============================================== Due Friday, Feb 10 p. 184, ex 1, 2, 5 p. 186, ex 3, 4, 5 (There seems to be a misprint in the series in ex 4: B_k should be B_{2k}). Hint for ex 5: cot z = i + 2i/(e^{2iz} -1) tan z = cot z - 2cot(2z) =================================================== Due Friday, Feb 17 p. 190, ex 1-5. Hint for #2: 3z^2/(z^3-n^3) = 1/(z-n) + e^{2\pi i/3}/(ze^{2\pi i/3} -n) + e^{4\pi i/3}/(ze^{4\pi i/3} -n). Remember 1+w+w^2 =0, the sum of the cube roots of 1. is Then think about cot(\pi z), etc. NOTE: The original hint forgot two z's in the denominator. (Thanks to Heidi for pointing this out.) Hint for #4: -2ia/[(z+n)^2 + a^2] = 1/[(z+n +ia] - 1/[z+n -ia]. p. 193, ex 1. ========================================================== Due Friday, Feb 24 p. 193, ex 2, 3. p. 197, ex 1, 2. #1 is hard. Here is a hint from my old notes. Assume |a_1|\le |a_2| \le .... Let K_n be the ball \{z:|z|\le |a_n|/2. Set M_n = sup_{z\in K_n} |( A_n g(z))/ (g'(a_n)(z-a_n)) |. Choose \gamma_n so 2^{-\gamma_n} M_n \le 2^{-n}. Then \sum_1^\infty (A_n/g'(a_n))[g(z)/(z-a_n)](z/a_n)^\gamma_n converges to an entire function. Note what happens if z\to some a_k$. #2 typo: sin(\pi \alpha) should appear before the infinite product. ======================================================= Content of Midterm, March 2: A problem on removable singularity of harmonic functions. A Laurent series problem. A basic fact about infinite product convergence. A Poison integral problem (memorize the integral) A cool infinite product formula with hint. ======================================================= Due Friday, March 9. p. 200, ex 1,2. Hint for #1. Replace z by nz. Use the Lemma: sin(\pi/n) sin(2\pi/n) ...sin{(n-1)\pi/n} = n/[2^(n-1)] which you can prove by setting \omega = exp[(\pi i)/n] Then take log-derivatives. p. 166, ex 2. (Hint: formula (61)) ====================================================== Due Friday, March 23 p. 206, ex 3 (first part only...not the Fresnel formulas we derived last semester when we studied residues ) p. 212, ex 2 Exercise 3: Show that -(zeta'(s)/zeta(s)) = \sum(log p)/p^s + \sum(log p)/p^{2s}) + ... where the sums are over the primes p, and real part of s >1. Exercise 4 (from Gamelin, p 379) The Mobius \mu function is defined on positive numbers by \mu(1) = 1, \mu(n) = (-1)^r if n is a product of r distinct primes, otherwise \mu(n) =0. Prove: 1/\zeta(s) = \sum_{n=1}^\infty \mu(n)/n^s, real s > 1. =========================================================== Due Friday, March 30 p.227, ex 2-4. Exercise 4 (Gamelin p.346, ex 3) Apply Runge's theorem to solve the following interpolation formula: Let \Omega be an open set in the plane, Let {z_j} be a discrete, infinite sequence of distinct points in \Omega (that necessarily accumulates to \partial\Omega), Let {w_j} be an infinite sequence of complex numbers (without and further assumptions on this sequence} Prove: There exists an analytic function f in \Omega with f(z_j) = w_j , for all indices j. Gamelin's hint: Exhaust $\Omega$ by suitable compact sets K_n. Construct {h_n(z)} and {f_n(z)} by induction so that *h_n(z_j) =0 , for 1\le j 0 such that the image of \D under any f in $F_2$ contains the disk of radius k : {z:|z|