Complex Analysis (MATH 425/525, Fall 2022)

Teacher: Alexandre Eremenko

Syllabus

OFFICE: Math 600

OFFICE HOURS:  Tuesday and Thursday, 1:30-2:30, or by appointment

PHONE: (765) 494-1975

EMAIL: eremenko@math.purdue.edu

GENERAL INFORMATION

Brightspace: Fall 2022 MA 425/52500

Homework 1 (due August 30) p. 5: 7, 16, 20 bd, 22, 23; p. 12: 7 cef, 13; p. 22: 11, 13; p. 31: 4-10.

Additional material (NOT required reading, NOT a part of HW):
How complex numbers were discovered?
Is there anything else like the complex numbers?
Can you find the mistake?

HW2 (due September 6) p. 37: 5d, 7a, 10; p. 42: 2-8, 11, 15, 17; p. 56: 1, 3; p. 63: 7.

HW3 (due September 13) p. 56: 6(a-c), 13(a-c), p. 63: 11(a-c), p. 71: 7(a-e), 13(a-f).

HW4 (due September 20) p. 77: 1(a-b), 6, 13; p. 84: 3(a-f), 10; p. 108: 3(a,b), 11(a); p. 115: 5(a-c), 17; p. 123: 1(a-d), 5(a-c), 12.

Green formula and Cauchy theorem

Poisson formula
Uniqueness of solution of Dirichlet problem

HW5 (due September 27) p. 130: 3,4; p. 136: 1(abd),15(a); p. 160: 3,8; p. 170: 3(ab),8,13, p. 178: 1(a-e),4.

The book gives two alternative approaches to Cauchy's theorem: sections 4.4a and 4.4b. I mostly follow the second approach, based on the Green formula (section 4.4b). So 4.4a is an optional reading. See also the handout "Green formula and Cauchy theorem" above.

HW6 (due October 4) p. 200: 7,9a-f; p. 212 1,3(abdf), 7; p. 219: 4,5,6,13.

Midterm exam is scheduled on October 6, in class. Covers Chapters I-III, 4.1-4.4.

Sample old exam

Solutions

Additional material: Infinite series I
Infinite series II
Infinite series III. Abel's theorem
Infinite series IV. Bernoulli numbers
Infinite series V. Burmann-Lagrange formula

Midterm exam solved

HW 7 (due October 18) p. 225: 4,6,7,14(a-e); p. 239: 1(a,c,e), 11(a-d); p. 250: 1(a,b), 5(a-g).

Examples of Laurent expansions

Examples of Dirichlet problem

HW 8 (due October 25) p. 259: 3(a-f), 13a; p. 267: 3(a-f), 9; p. 276: 3(a-c), 4.

HW 9 (due November 1) p. 285: 1(a-g), 5(a-e); p. 290: 1(a-e); p. 313: 1(a-i), 3(a-f).
Since I did not cover p. 313 1(a-i), 3(a-f) in class, it will not be graded. So it will make no difference for the grade whether you sumbit it or not. But I strongly recommend to solve these problems for yourself.

HW 10 (due Noverber 8) p. 317: 1,3; p 325: 1, 11, 15ab, 19; p. 336: 5,9; p. 344: 2,10; p. 354: 4,8. Read sections 6.1-6.6.

HW 11 (due November 15) p. 364: 3, 7, 8, 10; p. 375: 2; p. 382: 3, 13(a-d).

Problem 3, p. 382 will not be graded. The statement of this problem in the book is not accurate.

Final exam, Fall 2021
More problems for preparation to the final
What to expect on the final exam

HW 12 (due November 22) p. 392: 1,2,8,9; p. 403: 3(a-c), 6,17;

HW 13 (due December 29) p. 416: 4,8; p. 430: 1,3,5, 7; p. 440: 3,5.

Why airplanes fly and ships sail and propellers pull, etc.