Spring-2014. Math H185 (ccn 54255): Honors Introduction to Complex Analysis

Instructor: Alexander Givental
Lectures:
TuTh 11:00 - 12:30, room: 35 Evans
  • Office hours: Tu 2:00-4:00 p.m., in 701 Evans
  • Textbook author: Henri Cartan,
  • Textbook title: Elementary Theory of Functions of One and Several Complex Variables, Dover, ISBN 9780486685434
  • Syllabus: We will try to cover Chapters I,II, III, V, and the first 3 sections of Chapter VI.
  • Grading: 40% Homework, 20% Midterm, 40% Final.
  • HW: Weekly homework assignments are posted to this web-page, and your solutions are due on Th in class.
    Typically your homework will be returned to you in a week from the due date with some of the problems graded.
  • Academic honesty policy: All exams are closed books / closed notes. In homework, while you are recommended to work on your own, no form of collaboration is prohibited. So, one can discuss problems with others, read books, use electronic sources, hire tutors, etc. However, any use of outer sources must be acknowledged in the submitted solution. Failure to acknowledge the use of someone else's ideas is commonly known as academic plagiarism.
  • Midterm Exam: Tu, Mar. 4 in class (Chapters I and II). Note the date change!
  • Final Exam: Thursday, May 15, 8-11 a.m.. Solutions

    HOMEWORK

    It is recommended to skip no exercises from the book. However, the numbers listed in the homework are those fro which you are asked to submit written solutions. Most likely, only part of the problems will be graded (say, two problems from each set), but you are not told in advance which (two) ones.

    HW 1, due Th, January 30: Read Sections 1, 2 from Ch. I of the book, solve Exercises 1,3,4,6 from Ch. I.
    For those hwo don't have a book yet, here is the photo of relevant book pages.

    HW 2, due Th, Feb. 6: Read Sections 3, 4 from Ch. I, solve Exercises 8, 10, 11, 13 from Ch. I.
    For those who still don't have the textbook, problems beginning of Exercise 8 can be found on the scan for HW1, and the end of the formulation togather with Exercises 10, 11, 13 - on this scan (but you'll need to right-click on your mouse to rotate it 180 degrees :-)

    HW3, due Th, Feb. 13: Read Ch II, Section 1; solve Exercises from Chapter I: 15, 16(i, ii) (see this scan ) and the following two prolems:
    1. Let f be an analytic function on the complex plane \C, such that for every z there exists a non-negative integer n=n(z) such that the nth derivative of f vanishes at z. Prove that f is a polynomial.
    2. Invariance of Green's formula under changes of variables:
    Let \w = P(x,y)dx+Q(x,y)dy be a differential form (see Section 1 of Chapter II) with continuously differential coefficients P, Q, defined in a bounded closed domainan D of the plain; let x=x(u,v), y=y(u,v) be a two-times continuously differentiable change of variables: D' --> D, and let \W = \P(u,v) du + \Q (u,v) dv be the differential form obtained from \w by this change of variables. Prove that, up to sign, \int\int_D (Q_x-P_y) dx dy = \int\int D' (\Q_u-\P_v) du dv, and find out what the sign in this identity depends on.

    HW4, due Th, Feb. 20: Read Chapter II. Solve: Exercises 2, 6, 7 from Chapter II, and solve this
    Problem. The following properties of a connected domain D are equivalent: (a) D is simply-connected; (b) every contnuous mapping to D of the circle |z|=1 can be extended to a continuous mapping of the disk |z| =< 0 (less or equal); (c) every continuous mapping to D of the boundary of the square [0,1]x[0,1] can be continuously extended to the square; (d) any two paths in D with the same endpoints are homotopic with fixed endpoints.

    HW5, due Th, Feb. 27: Read Chapter II. Solve Exercises: 5, 8, 10, 12.

    HW6, due Th, Mar. 6: Read Chapter III, sections 1,2; solve Exercises 2,4,5,6 from Chapter III.

    HW7, due Th, Mar. 13: Read Section 3,4,5 of Chapter III; solve Exercises 7, 10, 13, as well as the following one: Expand the finction 1/(z-1)(z-2) explicitly into Laurent series in the regions (a) |z|<1, (b) 1<|z|<2, (c) 2<|z|. ( Hint: Decompose the function into elementary fractions.)

    HW8, due Th, Mar. 20: Read carefully Chapter III, section 5; solve 17 (ii), 18, 19, and compute residues at each pole of the differential form (\sin z)^{-5} dz (dz divided by the 5th power of \sin z).

    HW9, due Th, Apr. 3: Read Section 6 of Chapter III and solve exercises 20 and 23. Start reading Chapter V.

    HW10, due Th, Apr. 10: Read Chapter V, Sections 1, 2. Solve:
    1. Exercise 1 from Chapter V;
    2. Prove identities (4.1) and (4.2) on page 153 of the book;
    3. Let $X, d$ be a metric space, and let f: [0,\inft) -> [0,\infty) be a continuous increasing convex-down function such that f(0)=0. Prove that D(x,y)=f(d(x,y)) defines a new metric on X which defines the same topology as d.
    4. Prove that for every real-valued harmonic (i.e. sasisfying the Laplace equation) function u(x,y) on a connected simply connected domain D in the plane, there exists a real-valued harmonic function v(x,y) such that u(x,y)+iv(x,y) is a holomorphic function of $z=x+iy$, and that such a v is unique up to a constant summand. (Assume u continuously differentiable twice.)

    HW11, due Th, Apr. 17: Read Chapter V, Sections 2,3,4. Solve exercises 7, 8, 9, 10 from Chapter V.

    HW12, due Th, Apr. 24: Read Chapter 6, sections 1,2. Solve:
    1. Let E be a dense subset in a compact metric space space K. Prove that a complex-valued function on E extends to a continuous function on K if and only if it is uniformly continuous on E. Is such an extention unique?
    2. For E and K as in the previous problem, prove that an equicontinuous sequence of continuous functions on K, which converges pointwise on E, converges uniformly on K.
    (Remark. To recall the definition, a family of functions is equicontinuous if for every \epsilon > 0 one can find \delta > 0 such that for all functions f in the family and all x, y within the distance \delta from each other, the distance between f(x) and f(y) does not exceed \epsilon.)
    3. Prove that a linear transformation R^n -> R^n which preserves angles between any vectors is the composition of an orthogonal transformation (i.e. a linear transformation preserving dot-products) with the multiplication by a scalar.
    4. Show that Zhukovsky's mapping w=(z+(1/z))/2 transforms concentric circles and radial rays of the z-plane into confocal ellipses and (branches of the) confocal hyperbolas of the w-plane. Which hyperbolas the real and imaginary axes of the z-plane are mapped to?

    HW13, due Th, May 1: Read Chapter 6, Sections 1-3. Find explicitly isomorphisms between the following regions:
    1. \C - [-1,1] (the complex plane with a cut along the interval [-1,1] of the real axis) and \U -{0} (the unit disk without the center);
    2. \H (the upper half-plane) and \C - (-\infty, -1] - [+1,+\infty) (complex plane cut along the two rays of the real axis);
    3. Half-a-disk (|z|<1, Im z>0) and the upper half-plane;
    4. Half-a-strip (0< Im z < \pi, Re z <0) abd half-a-disk.

    HW14, NOT due Th, May 8: Don't read anything. We meet TuTh 11-12:30 in 35 Evans, as usual, to discuss more elliptic curves, their moduli space, the application to Picard's Little Theorem, as well as solutions to past HW problems. Here are some exercises that could help to understand the subject:
    1. On the (p,q)-plane, draw the locus of points, such that polynomial x^3-px-q has double roots.
    2. Prove that the group GL_2(\Z_2) of invertible 2x2-matrices over the field of 2 elements is isomprphic to the permutation group S_3.
    3. Show that S_3 has two generators, a,b, saisfying the relations a^2=b^2=(ab)^3=1.
    In the group SL_2(\Z) consider the normal subgroup G which is the kernel of the homomorphism SL_2(\Z) -> GL_2(\Z_2) (reduction modulo 2). (Thus, G consists of invertible integer 2x2-matrices congruent mod-2 to the identity matrix.) Describe a funcdamental domain of G acting on the upper half-plane.
    4. Prove that the quotient of an elliptic curve \C / \Omega (where \Omega is a lattice) by the central symmetry z \mapsto -z is isomorphic to the Riemann sphere \CP^1 ( Hint: How does the symmetry act on the Weierstrass curve (\P')^2=4\P^3-60 G_2 \P - 140 G_3 ?)
    5. On an elliptic curve y^2=x^3-px-q, examine differential form y^{-1} dx (that is, dx divided by y). Show (by rewriting the differential form in a local coordinate on the curve, first near a typical point, then near the points with y=0, and finally near the point at infinity) that this differential form is holomorphic everywhere (and has no zeros).
    6. Consider the multiple-valued function on an elliptic curve y^2=x^3-px-q, defined by integrating the differential form y^{-1}dx:
    z(x,y) := \int_{\infty}^(x,y) y^{-1} dx
    where the integration is performed along a real path lying on the elliptic curve and connecting the point at infinity with a point, (x,y), on the elliptic curve. Taking in accountnt that the elliptic curve is homeomorphic to a torus, show that the ambiguities of the multiple-valued finction z(x,y) form a lattice \Omega in \C, and that the elliptic curve is thus mapped by (x,y)|->z(x,y) to \C /\Omega.
    ( Remark: This exercise shows how to prove that every curve of the form y^2=x^3-px-q is the quotient of the complex plane by a lattice, provided that the polynomial has distinct roots.
    7. Identify the upper half-plane with the unit disk in such a way that 0,1, \infty are mapped to the vertices of a regular triangle on the boundary of the unit disk. Draw the image in the unit disk of the fundamental domain D of the modular group SL_2(\Z) on the upper half-plane. ( Hint: Fractional-linear transformations map lines and circles to lines or circles.)