Complex Variables

MATH 4283/5273

Time and Place: TR 12:00 - 1:15 in MSCS 509
Professor: Igor E. Pritsker
Office: MSCS 519C
Office Hours: TR 10:30-11:30
Office Phone: 744-8220
E-mail: igor@math.okstate.edu
Web: https://math.okstate.edu/people/igor/
Textbook: Fundamentals of Complex Analysis with Applications to Engineering and Science, by E. B. Saff and A. D. Snider, 3rd ed.

Complex analysis is a classical and beautiful part of mathematics with numerous applications in mechanics, thermodynamics, electrostatics, fluid mechanics and many other sciences. The course and our textbook are tailored for mathematics, science and engineering students who completed the calculus sequence.

Grading: There will be two semester tests and the Final Exam. The break up of your course grade is as follows:

Tests 1-2 50% (25% each)

Homework 20%

Final Exam 30%

Your grade will be determined according to the scale

A 90-100

B 80-89

C 70-79

D 60-69

F 59 and lower

Note that the above numbers are percentages of the highest possible score in the course.

Attendance is mandatory in this class.

Homework will be assigned daily (see the detailed schedule below), and will be collected periodically. It is required that you complete all homework.

Make-up Exams are given only in cases of serious illness or extreme emergency that prevents you from taking a test at the specified time. You have to contact me before the test and communicate all circumstances.

University Syllabus Attachment: Contains drop deadlines and procedures, as well as many other important dates and university policies.

Brief Schedule
Chapter 1 Chapter 2 Chapter 3 Test 1 Chapter 4 Chapter 5 Test 2 Chapter 6 Chapter 7 Final Exam

**Brief Schedule**
Chapter 1	Chapter 2	Chapter 3	Test 1	Chapter 4	Chapter 5	Test 2	Chapter 6	Chapter 7	Final Exam

Note: The homework problems below are assumed to be odd numbered, unless it is indicated otherwise.

Detailed Schedule
Date Sec Page Topic Homework

T, Aug 22
R, Aug 24 1.1-2 1, 7 Algebra and Representation of Complex Numbers 3-15, 19-23 (p. 5); 1-13 (p. 12)

1.3 14 Vectors and Polar Forms 3-11, 17

1.4 26 The Complex Exponential 1-13

T, Aug 29
R, Aug 31 1.5 33 Powers and Roots 1-13

1.6 39 Planar Sets 1-15

2.1-2 53, 58 Functions of a Complex Variable, Limits and Continuity 1-7 (p. 56); 1-5, 9-11 (p. 63)

T, Sep 5
R, Sep 7 2.3 65 Analyticity 3-11

2.4 73 The Cauchy-Riemann Equations 1-11

2.5 79 Harmonic Functions 1-11

T, Sep 12
R, Sep 14 2.6 87 Steady State Temperature as a Harmonic Function 1, 3

3.1 99 Polynomials and Rational Functions 1-7, 11, 13

3.2 110 The Exponential, Trigonometric, and Hyperbolic Functions 1-9, 15, 17

T, Sep 19
R, Sep 21 Test 1 - Sep 19 (Sections 1.1-1.6, 2.1-2.5, 3.1-3.2)

3.3 118 The Logarithmic Function 1-5, 9-15

3.4 125 Washers, Wedges, and Walls 1, 2, 3, 4, 5

T, Sep 26
R, Sep 28 3.5 131 Complex Powers and Inverse Trigonometric Functions 1-11

4.1 149 Contours 1-11

4.2 161 Contour Integrals 1-11

T, Oct 3
R, Oct 5 4.3 173 Independence of Path 1-7

4.4 180 Cauchy's Integral Theorem 1-5, 9-17

T, Oct 10
R, Oct 12 4.5 204 Cauchy's Integral Formula 3-13

4.6 214 Bounds for Analytic Functions 1-7, 11

4.7 221 Applications to Harmonic Functions 1-5, 11

T, Oct 17
R, Oct 19 5.1 235 Sequences and Series 1-11

5.2 242 Taylor Series 1-7, 13

5.3 252 Power Series 1-7, 11

T, Oct 24
R, Oct 26 5.5 269 Laurent Series 1-9

Test 2 - Oct 26 (Sections 3.3-3.5, 4.1-4.7, 5.1-5.3, 5.5)

T, Oct 31
R, Nov 2 5.6 277 Zeros and Singularities 1-11

6.1 307 The Residue Theorem 1-7

T, Nov 7
R, Nov 9 6.2 314 Trigonometric Integrals 1-9

6.3 318 Improper Integrals 1-7

T, Nov 14
R, Nov 16 6.7 355 The Argument Principle and Rouche's Theorem 1-13

7.1 369 Invariance of Laplace's Equation 1, 3, 7

7.2 377 Geometric Considerations 1-7, 11

T, Nov 21
R, Nov 23 Fall Break

T, Nov 28
R, Nov 30 7.3 383 Moebius Transformations 1-7, 11

7.4 395 Moebius Transformations, Continued 1-9, 15, 17

T, Dec 5
R, Dec 7 7.5 407 The Schwarz-Christoffel Transformation 1-5

7.6 419 Applications to Electrostatics, Heat Flow, and Fluid Mechanics 1-5

T, Dec 12 Final Exam (MSCS 509, 10-11:50 a.m.)

Date	Sec	Page	Topic	Homework
T, Aug 22 R, Aug 24	1.1-2	1, 7	Algebra and Representation of Complex Numbers	3-15, 19-23 (p. 5); 1-13 (p. 12)
	1.3	14	Vectors and Polar Forms	3-11, 17
	1.4	26	The Complex Exponential	1-13
T, Aug 29 R, Aug 31	1.5	33	Powers and Roots	1-13
	1.6	39	Planar Sets	1-15
	2.1-2	53, 58	Functions of a Complex Variable, Limits and Continuity	1-7 (p. 56); 1-5, 9-11 (p. 63)
T, Sep 5 R, Sep 7	2.3	65	Analyticity	3-11
	2.4	73	The Cauchy-Riemann Equations	1-11
	2.5	79	Harmonic Functions	1-11
T, Sep 12 R, Sep 14	2.6	87	Steady State Temperature as a Harmonic Function	1, 3
	3.1	99	Polynomials and Rational Functions	1-7, 11, 13
	3.2	110	The Exponential, Trigonometric, and Hyperbolic Functions	1-9, 15, 17
T, Sep 19 R, Sep 21	Test 1 - Sep 19 (Sections 1.1-1.6, 2.1-2.5, 3.1-3.2)
	3.3	118	The Logarithmic Function	1-5, 9-15
	3.4	125	Washers, Wedges, and Walls	1, 2, 3, 4, 5
T, Sep 26 R, Sep 28	3.5	131	Complex Powers and Inverse Trigonometric Functions	1-11
	4.1	149	Contours	1-11
	4.2	161	Contour Integrals	1-11
T, Oct 3 R, Oct 5	4.3	173	Independence of Path	1-7
T, Oct 3 R, Oct 5	4.4	180	Cauchy's Integral Theorem	1-5, 9-17
T, Oct 10 R, Oct 12	4.5	204	Cauchy's Integral Formula	3-13
	4.6	214	Bounds for Analytic Functions	1-7, 11
	4.7	221	Applications to Harmonic Functions	1-5, 11
T, Oct 17 R, Oct 19	5.1	235	Sequences and Series	1-11
	5.2	242	Taylor Series	1-7, 13
	5.3	252	Power Series	1-7, 11
T, Oct 24 R, Oct 26	5.5	269	Laurent Series	1-9
T, Oct 24 R, Oct 26	Test 2 - Oct 26 (Sections 3.3-3.5, 4.1-4.7, 5.1-5.3, 5.5)
T, Oct 31 R, Nov 2	5.6	277	Zeros and Singularities	1-11
T, Oct 31 R, Nov 2	6.1	307	The Residue Theorem	1-7
T, Nov 7 R, Nov 9	6.2	314	Trigonometric Integrals	1-9
T, Nov 7 R, Nov 9	6.3	318	Improper Integrals	1-7
T, Nov 14 R, Nov 16	6.7	355	The Argument Principle and Rouche's Theorem	1-13
	7.1	369	Invariance of Laplace's Equation	1, 3, 7
	7.2	377	Geometric Considerations	1-7, 11
T, Nov 21 R, Nov 23	Fall Break
T, Nov 28 R, Nov 30	7.3	383	Moebius Transformations	1-7, 11
T, Nov 28 R, Nov 30	7.4	395	Moebius Transformations, Continued	1-9, 15, 17
T, Dec 5 R, Dec 7	7.5	407	The Schwarz-Christoffel Transformation	1-5
T, Dec 5 R, Dec 7	7.6	419	Applications to Electrostatics, Heat Flow, and Fluid Mechanics	1-5
T, Dec 12	Final Exam (MSCS 509, 10-11:50 a.m.)