## Spring-2015. Math H185 (ccn 54251): Honors Introduction to Complex Analysis

Instructor: Alexander Givental
Lectures:
TuTh 9:30 - 11:00, room: 9 Evans
• Office hours: Wed 1:00-3:00 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: Tue, March 3 in class (Chapters I and II).

#### HOMEWORK

It is recommended to skip no exercises from the book. However, the numbers listed in the homework are those for 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 29: Read Sections 1, 2 from Ch. I of the book, solve Exercises 1,3,4,6 from Ch. I.

HW 2, due Th, Feb. 5: Read Sections 3, 4 from Ch. I, solve Exercises 8, 11, 13, 15 from Ch. I.

HW3, due Th, Feb. 12: Read Ch II, Section 1; Solve:
Exercise 16(i,ii) from Ch. I, Exercise 6 from Ch. II, 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. 19: Read Chapter II. Solve: Exercises 7, 8 from Chapter II, and the following two problems
1. 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.
2. Pove that functions u, v satisfying the Cauchy-Riemann equations are harmonic (i.e. satidfy the Laplace equation f_xx+f_yy =0).

HW5, due Th, Feb. 26: Read Chapter III, sections 1,2. Solve Exercises from Chapter II: 10, 11, 12, 14.

HW6, due Th, Mar. 5: Read Chapter III, sections 1-4; solve Exercises 4,5,6 from Chapter III, and the following
Problem. Expand the function 1/(z-1)(z-2) explicitly into the Laurent series in the regions: (a) |z|<1, (b) 1<|z|<2, (c) 2<|z|.

HW7, due Th, Mar. 12: Read Section 5 of Chapter III; solve Exercises 7, 10, 17(ii), 18.

HW8, due Th, Mar. 19: Read Section ^ of Chapter III. Solve Exerises 13, 19, and the following two problems:
1. Prove that the equation x^2+y^2=1 has infinitely many solutions (x,y) over Q, the field of rational numbers.
2. Is there a meromorphic function, periodic with the periods 1 and i, all of whose poles have the 1st order and are located at the points m+ni (m,n are arbitrary integers)?

HW9, due Th, Apr. 2: Read Section 6 of Chapter III and solve exercises 20 and 23. Start reading Chapter V.

HW10, due Th, Apr. 9: Read Sections 1 and 2 of Chapter V.
Solve Exrcises 1 and 9 from Ch. V, and prove identities (4.1) and (4.2) on page 153.

HW11, due Th, Apr. 16: Read Sections 3 and 4, and solve Exercises 7(both parts i and ii), 8, 10 from Chapter V.

HW12, due Th, Apr. 24: Read Chapter VI, 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. Find explicitly isomorphisms between the following regions:
a. \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);
b. \H (the upper half-plane) and \C - (-\infty, -1] - [+1,+\infty) (complex plane cut along the two rays of the real axis);
c. Half-a-disk (|z|<1, Im z>0) and the upper half-plane;
d. Half-a-strip (0< Im z < \pi, Re z <0) abd half-a-disk.

HW13, due Th, Apr. 30: Read Section 3 of Ch. VI. Solve:
Exercise 2 from Ch. VI of the text, and the following three algebraic problems (Hopefully they will help you understand the material of the next week - not from the textbook - about moduli space of elliptic curves, and Picard's Little Theorem ):
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.