**4520.001, Fall 2001**

**Functions of One Complex Variable**

**Conley**

the University of North Texas. |

**INSTRUCTOR:** Charles Conley, GAB 475,
(940)
565-3326

**OFFICE HOURS:** Tuesday and Thursday, 1:30-3:30 and
by
appointment

**CLASS MEETS:** TR 3:30-4:50, GAB 201

**EXAMS, HOMEWORK, AND GRADING:** There will be two 100 point
midterms, on Tuesday, Oct. 9 and Tuesday, Nov. 13, and a comprehensive
180 point final,
on Thursday, Dec. 13, 1:30-3:30. There will also be twelve 10
point
homeworks, due Thursdays at the beginning of class. There will be
no
make-up exams except for emergencies, and late homework will be worth
half-credit.

**TEXT, PREREQUISITES, AND OTHER REFERENCES: **The text is *Complex
Analysis for Mathematics and Engineering* (fourth edition), by
Mathews and Howell. The prerequisites are Math 1710-20 and 2730,
the calculus sequence.

**TOPICS:** We will begin by going over
multiplication of
complex numbers in rectangular and polar coordinates. Then we
will
study differentiability of functions of one complex variable, followed
by
Taylor and Laurent series and contour integration. We will also
discuss
conformal maps and Julia and Mandelbrot sets if time permits. We
will
give some rigorous proofs, but no prior exposure to proofs will be
assumed.
Concrete examples and calculations will be emphasized.

*Chapters 1-2:* Complex numbers and functions.

*Chapters 3-5:* Analytic and harmonic functions, power
series functions, and the elementary functions such as e^z.

*Chapters 6-8:* Complex integration, contour integration,
and
Taylor and Laurent series. These chapters contain many surprising
and
beautiful theorems, and comprise the centerpiece of the course.

Since class was cancelled on Sept. 11, there will be only 11 problem sets.

FINAL EXAM: Thursday, Dec. 13,
1:30-3:30

HOMEWORK 11, Due Thursday, Dec. 6:

Section 7.3: 1 (only do this one way, not two), 3, 5, 8
(for
|z|<1, not |z|>1), 12

Section 7.4: 1abdg, 2bcj, 3bfgh, 4*

Section 8.2: 1acdefghi*j, 2, 3abce, 4, 5

HOMEWORK 10, Due Thursday, Nov. 29:

Section 6.6: 3, 5, 6 (part a should say n>1), 7, 10,
11

Section 7.2: 1c, 2acd, 3b, 4, 5, 7b, 8a (it should say
powers
of z, not z bar), 9*

HOMEWORK 9, Due Tuesday, Nov. 20 (this is a short homework due to Thanksgiving):

Section 6.5: 1-6, 8-15.

EXAM 2: Tuesday, Nov. 13

HOMEWORK 8, Due Thursday, Nov. 9:

Section 6.3: 1, 2, 4, 5, 7, 10, 11, 12, 13

Section 6.4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14

HOMEWORK 7, Due Thursday, Nov. 1:

Section 5.4: 1, 2, 5ab, 6abc, 7ab, 8ab, 9, 10b

Section 6.1: 1abde, 2, 3

Section 6.2: 1, 2, 6a, 7, 10, 15

HOMEWORK 6, Due Thursday, Oct. 25:

Section 5.1: 3, 4, 6ef, 7bc, 8, 9a, 13bc, 18

Section 5.2: 1abc, 3bc, 6, 10, 12

Section 5.3: 1, 4

HOMEWORK 5, Due Thursday, Oct. 18:

Section 4.1: 1, 6

Section 4.3: 1, 2, 4, 5

Section 4.4: 3 (skip parts c and g),
4, 5, 6

EXAM 1: Tuesday, Oct. 9. I will be away the week of Oct. 8-12, and so Professor
Douglass
(GAB 413) will proctor the exam on Tuesday and begin Chapter 4 on
Thursday,
Oct. 11. You may ask him questions any time between 10:00 and
2:00
on Monday (but please don't tax him too much!). Thursday, Oct 4
will be a review day. I will pass out a review sheet with a
practice exam.

HOMEWORK 4, Due Thursday, Oct. 4:

Section 3.1: 1bd, 3, 7bf, 8, 11, 14

Section 3.2: 1cd, 4, 7, 11b, 15b (10a
is
cancelled)

Section 3.3: 1b, 2, 5a, 6, 8, 10

HOMEWORK 3, Due Thursday, Sept. 27 (note that the due date has been extended, and that Section 2.5 is included):

Section 2.3: 1, 2, 6, 7, 8, 9, 11b, 16a, 18

Section 2.4: 1bc, 2, 6b, 7, 16

Section 2.5: 1, 5

HOMEWORK 2, Due Tuesday, Sept. 18 (note that the the due date has changed and the Section 2.3 problems have been dropped):

Section 1.5: 3 [there is a misprint in
the solution, sin(3 theta) is 3sin(theta)cos^2(theta) - sin^3(theta) ]

Section 1.6: 1abc, 2a, 3a, 5a, 9ab for parts ii-vi

Section 2.1: 1b, 2b, 3b, 4b, 5b, 6, 9, 10d

Section 2.2: 1a, 2, 3a, 4, 7, 8

HOMEWORK 1, Due Thursday, Sept. 6:

Section 1.2: 1abef, 2cfh, 3, 5

Section 1.3: 1, 2a, 3b, 5, 6ab, 9, 11, 12

Section 1.4: 1abdeh, 2b, 3abcd, 5fg

Section 1.5: 1, 4, 5