## 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.

** Grading: ** 2/5 final + 3/5 homework; I have, however, the following (ingenious, in my view) grading scheme. The score of any particular homework which is below your score on the final will be dropped - together with its weight in the total grade. E.g. if 1/2 of your hw is below your final score, then only the other 1/2 counts, but it weighs 3/7 (rather than 3/5), while the final score
weighs 4/7 (rather than 2/5). An important intellectual component of this class is to understand and appreciate the consequences of this grading arrangement. The bottom line is that your successful hw can improve your grade, while your unsucessful hw cannot lower it.

** 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.)

Answers

** 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.