## Spring-2021. Math H185 (class # 25022): Honors Introduction to Complex Analysis

Instructor: Alexander Givental
Lectures:
MWF 12:00 - 12:59, zoom:
• Office hours: Sunday 10:00 a.m.-12:00 p.m., on the same zoom as lectures
• Textbook author: Henri Cartan,
• Textbook title: Elementary Theory of Functions of One and Several Complex Variables, Dover, ISBN-13 9780486685434
• Syllabus: We will try to cover Chapters I,II, III, V, and the first 3 sections of Chapter VI.
• HW: Weekly homework assignments will be posted to this web-page. Your solutions are due on Monday before the class, in PDF format. You may pdflatex your solutions, or simply write them the old-fashioned way and photograph with your smartphone.
Typically your graded homework will be returned to you in a week from the due date, with each problem graded by one of the scores 0 (unslolved), 1 (unsolved, but there is substantil progress toward a complete solution), 2 ("half-full or half-empty"?), 3 (essentially solved, but there are some relevant flaws/omissions), 4 (solved).
To gauge your mid-term performance, think of 4,3,2,1,0 scores as roughly within the range of A,B,C,D,F respectively.
• Academic honesty policy: 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.
• Final Exam: I am planning to run it "take home"

#### HOMEWORK

It is recommended that you skip no exercises from the book. However, the numbers listed below are those for which you are asked to submit written solutions.

HW1, due Mon., Jan. 25: Read handout on complex numbers, and Section I.1 (pages 9-16) from the textbook (use Amazon's "look inside" feature, if you don't have the book yet). Solve:
1. z^2-2z+\sqrt{3}i=0.
2. Sketch the set of complex numbers z satisfying simultaneously the following three conditions: |z|<1, |z+1|<1, Re(iz)<0.
3. Given Taylor expansions of sin X and cos X, compute Taylor expansions of tan X and arctan X up to order X^5 inclusively.
4. Express coefficients of the formal series (1+X+X^2+...)^p in terms of binomial coefficients. Hint: Start with the derivative of 1/(1-X).

HW2, due Mon., Feb. 1: Read Ch. I to the end. Solve: 4; 8bc; 11; 16(i,ii) and use 16(ii) to compute the sum 1-1/2+1/3-1/4+... (do not forget to thoroughly justify your answer).

HW3, due Mon., Feb. 8: We are moving to Ch. II. Section II.1 is 17 page long, and combines the standard material about line integrals and the Green theorem from the 2D multivariable calculus, with some improvements which sound rather technical (and unmotivated until they are put to use in Section II.2). You are welcome to read this meterial "as is", but in my lectures, I plan to rearrange it and will try to cut some corners. So, I suggest to jump right away to Section II.2 (whose subsections 1-3 are independent of Section II.1), and then return to some material on curvilinear integrals, but only as it becomes necessary for the rest of Section II.2.
Solve:
1. Show that tanh(z):=(e^z-e^{-z})/(e^z+e^{-z}) is meromorphic on the entire complex plane, find its periods (if any), zeroes, poles, and compute its Taylor coefficients (tanh(z))^(k)/k! at z=-i\pi (k-th derivative divided by k! at negative i times "pi") for k=0,1,2,3,4,5.
2. Construct a function, meromorhic in some open subset of the complex plane, which has poles at the points 1/n, n=1,2,3,...
3. II.6.
4. II.8.
HW4, due Mon., Feb. 15: Read the entire Chapter II (in any order) and work on exercises to it. Start reading Chapter III (perhaps Sections III.1 and III.2). Solve: II.3 (you may ignore the last question -- about non-differentiable paths), II.7, II.10, II.11.

HW5, due Mon., Feb. 22: Read Chapter III, sections 1-4. Solve:
1. Let F be a continuous map of the square [0,1]x[0,1] to a metric space D. Suppose that some "property P" is satisfied locally in D, i.e. for every point x in D there is a neighborhood U_x where P is satisfied. ( Example of P: a given differential 1-form on open D in \R^2 is locally exact.) Show that for sufficiently large N, the partition of the square into NxN little squares (of size 1/N x 1/N), property P is satisfied on the image under F of each little square (i.e. F(that little square) lies in some U_x).
2. Find all fractional-linear transformations w=(az+b)/(cz+d) which map the unit disk |z|<1 bijectively onto itself.
3. Prove that any (not necessarily invertible) holomorphic map f from the unit disk to itself which has two fixed points must be the identity map: f(z)=z.
4. Let f be a non-constant entire function. A level set |f| = const is a curve in the complex plane, which divides its complement into open connected components. Show that in every bounded connected component there lies at least one zero of f. ( Hint: This is a variant of exercise III.5 from the textbook.)

HW6, due Mon., Mar. 1: Read sections 4,5 of Chapter III. Solve:
1. Show that in every neighborgood of z=0 the range of the function e^{1/z} consists of all non-zero complex numbers.
2. Compute the residue at z=0 of 1/(sin z)^5
3. Expand 1/(z-1)(z-2) into its Laurent series in the annulus 2<|z+1|<3.
4. Prove that meromorphic functions on the entire Riemann sphere are necessarily rational.

HW7, due Mon., Mar. 8: Read section 6 of Chapter III. Solve Exercise 20 (i,ii,iii,iv) from Chapter III.

HW8, due Mon., Mar. 15: Read sections 3 and 4 of Chapter IV. Solve:
1. III.21
2. III.22
3. Prove that a positive function g(x,y) harmonic on the entire plane is constant.
4. Prove that the function g(x,y) from Example on page 127 is harmonic.

HW9, due Mon., Mar. 29: Read Sections 1 and 2 of Chapter V. Solve:
1. V.1
2. V.2
3. (a) Prove that every compact subset K in an open subset D of the complex plane is covered by finitely many closed disks contained in D which have rational radii and whose centers have rational coordinates. (b) Give an example of a series of holomorphic functions which converges on compact subsets uniformly, but not normally.
4. V.4

HW10, due Mon., Apr. 5: Read Subsection 2.5 and Section 3 of Chapter V. Solve:
1. Does there exist a doubly-periodic meromorphic function with one 1st order pole per each parallelogram of the period lattice (and no other poles)?
2. Find the solution to the Newton equation d^2 x/dt^2=x+3x^2/2 satisfying the initial conditions x(0)=-1, dx/dt (0)=0.
3. Show that replacing the period lattice \Omega with k\Omega, where k is a non-zero complex number, results in an elliptic curve E in \C^2 (here the embedding given by the Weierstrass P-function and its derivative is assummed) which is isomorphic to the original one (namely, related to it by a simple linear transformation of the ambient plane).
4. Show that the elliptic curves y^2=x^3-x and y^2=x^3-1 correspond to repectively a square period lattice and a "hexagonal" one (i.e. formed by rhombic parallelograms with 60/120 degree angles). Hint: Show that these elliptic curves admit cyclic groups of symmetries of the orders 4 and 6 respectively.

HW11, due Mon., Apr. 12: Read Section 4 of Chapter V. Solve:
1. V.6
2. Prove that a non-equicontinuous family of continuous functions on a compact space contains a sequence which has no uniformly convergent subsequence.
3. Prove that the bounded sequence sin nx contains no uniformly convergent subsequence.
4. Prove that the series 1+\sum_{n>0} P(n) q^n, where P(n) is the number of different partitions of n into positive integer summands has convergence radius 1. Namely, show that the series factors into the infinite product 1/(1-q)(1-q^2)(1-q^3)...(1-q^k)... which converges in the unit disk |q|<1 uniformly on compact subsets.

HW12, due Mon., Apr. 19: Read Sections 1 and 2 of Chapter VI. Solve:
1. Permutations of a typical configuration of 4 distinct points in CP^1 give 6 pairwise-non-equivalent configurations (which manifests in 6 different values of the corresponding cross-ratios). Describe (up to automorphisms of CP^1) all the configurations of 4 distinct points which are more symmetric than typical (i.e. whose cross-ratios take on less than 6 different values under the permutations of the points.)
2. Find explicitly a conformal isomorphism between the semi-disk |z|<1, Re z>0 and the upper half-plane Im w>0.
3. For any positive integer n>1, find a finite group of fractional-linear transformations which has order n and fixes two points z=1 and z=-1.
4. Prove that there is no conformal isomorphism between the punctured disk 0<|z|<1 and the annulus 1<|w|<2.

HW13, due Mon., Apr. 26: Read Sections 3,4 of Chapter VI. Solve:
1. Find explicitly a conformal isomorphism of the left half-plane Re z<0 onto the interior of the disk |w-1|<2.
2. VI.2
3. Find explicitly a conformal isomorphism of the infinite strip 0< Re z<1 onto the right half-plane Re w>0.
4. Check that on the complex curve y^2=4x^3-20a_2x-28a_4, the differential 1-form (dx)/y extends without singularity to the point at infinity and does not vanish at that point either.