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Complex Analysis: Math 413A Fall Semester 2008, Assignments of Advanced Calculus

Christian Brothers University (CBU)Advanced Calculus

Information about a university course named complex analysis (math 413a) offered in the fall semester 2008 at cbu. The course covers topics such as complex sequences, derivatives, cauchy-riemann equations, integration in the complex plane, series representations, and residue theory. Students are expected to have a solid background in calculus iii (math 232). The course goals include gaining a thorough understanding of complex analysis concepts and their applications to engineering and science problems.

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Pre 2010

Uploaded on 08/18/2009

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Math 413A – Complex Analysis
Fall Semester 2008
August 25 – December 12
Instructor. Dr. Leigh C. Becker Office. Science Building CW 312
Phone. 321–3443 Hours. Monday 2:00 – 4:00
Tuesday 9:00 – 12:00 & 1:00 – 4:00
Wednesday 2:00 – 4:00
(Other days & times by appointment)
E-mail. lbecke[email protected]du Home Page. www.cbu.edu/~lbecker
Catalog Description.
This course concerns itself with the rudiments and techniques of complex analysis. Topics that are covered include: complex
sequences, the derivative of a complex function, the Cauchy-Riemann equations, integration in the complex plane and the
Cauchy-Goursat theorem, Cauchy’s integral formula, Morera’s theorem, Taylor and Laurent series, residue theory, and the
evaluation of definite integrals.
Prerequisite. Math 232 (Calculus III)
Course Goals.
This course should give the student a thorough understanding of basic concepts in complex analysis such as: the
properties of the complex number system, continuity, analyticity, elementary complex-valued functions, series
representations for analytic functions, and residue theory. The student should develop an ability to read and understand
these concepts and to apply the methods of complex analysis to solve engineering and science problems.
Textbook.
Fundamentals of Complex Analysis with Applications to Engineering and Science by E.B. Saff and A.D. Snider, 3rd edition,
2003, Prentice Hall.
Class Schedule. M W F 1:00–1:50 in CW 212.
Syllabus.
Chapter 1. Complex Numbers
algebra of complex numbers, vectors & polar forms, the complex
exponential, planar sets
1. 1-6
Chapter 2. Analytic Functions
limits & continuity, analyticity, the Cauchy-Riemann equations, harmonic
functions
2. 1-5
Chapter 3. Elementary Functions
polynomials & rational functions; exponential, trigonometric, &
hyperbolic functions; logarithmic function; complex powers & inverse
trig functions; washers, wedges, and walls
3. 1-5
Chapter 4. Complex Integration
contours, contour integrals, Cauchy’s Integral Theorem, harmonic
functions
4. 1-6
Chapter 5. Series Representation for Analytic Functions
sequences & series, Taylor series, power series, Laurent series, zeros &
singularities
5. 1-6
Chapter 6. Residue Theory
The Residue Theorem, trig integrals over
[ ]
0,2p
, improper integrals,
additional sections in this chapter if time permits
6. 1-3
Classroom Policy and Attendance.
Please put your “turn-in” homework on the designated lectern or table as soon as you enter the classroom. Bring your
notebook for this course, textbook, calculator, ruler, and pencil to each and every class. Be seated before class begins. Do
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Math 413A – Complex Analysis

Fall Semester 2008

August 25 – December 12

Instructor. Dr. Leigh C. Becker Office. Science Building CW 312 Phone. 321–3443 Hours. Monday 2:00 – 4: Tuesday 9:00 – 12:00 & 1:00 – 4: Wednesday 2:00 – 4: (Other days & times by appointment) E-mail. [email protected] Home Page. www.cbu.edu/~lbecker Catalog Description. This course concerns itself with the rudiments and techniques of complex analysis. Topics that are covered include: complex sequences, the derivative of a complex function, the Cauchy-Riemann equations, integration in the complex plane and the Cauchy-Goursat theorem, Cauchy’s integral formula, Morera’s theorem, Taylor and Laurent series, residue theory, and the evaluation of definite integrals. Prerequisite. Math 232 (Calculus III) Course Goals. This course should give the student a thorough understanding of basic concepts in complex analysis such as: the properties of the complex number system, continuity, analyticity, elementary complex-valued functions, series representations for analytic functions, and residue theory. The student should develop an ability to read and understand these concepts and to apply the methods of complex analysis to solve engineering and science problems. Textbook. Fundamentals of Complex Analysis with Applications to Engineering and Science by E.B. Saff and A.D. Snider, 3rd^ edition, 2003, Prentice Hall. Class Schedule. M W F 1:00–1:50 in CW 212. Syllabus. Chapter 1. Complex Numbers algebra of complex numbers, vectors & polar forms, the complex exponential, planar sets

  1. 1- Chapter 2. Analytic Functions limits & continuity, analyticity, the Cauchy-Riemann equations, harmonic functions
  2. 1- Chapter 3. Elementary Functions polynomials & rational functions; exponential, trigonometric, & hyperbolic functions; logarithmic function; complex powers & inverse trig functions; washers, wedges, and walls
  3. 1- Chapter 4. Complex Integration contours, contour integrals, Cauchy’s Integral Theorem, harmonic functions
  4. 1- Chapter 5. Series Representation for Analytic Functions sequences & series, Taylor series, power series, Laurent series, zeros & singularities
  5. 1- Chapter 6. Residue Theory The Residue Theorem, trig integrals over [^0 ,2 p^ ], improper integrals, additional sections in this chapter if time permits
  6. 1- Classroom Policy and Attendance. Please put your “turn-in” homework on the designated lectern or table as soon as you enter the classroom. Bring your notebook for this course, textbook, calculator, ruler, and pencil to each and every class. Be seated before class begins. Do

not sharpen pencils during class. If you bring a cell phone to class, turn it off or put it on a silent setting before class begins. You may not listen to an iPod or similar device during class or exams. Food and drinks are not allowed in the classroom. I expect you to take notes. I encourage you to be an active listener and to ask me questions. However, if your questions take up too much class time, it would be best to come to my office or to go to the Math Center for help in view of time constraints and out of consideration for other students. Talking is not allowed unless you raise your hand and are given permission. It should be clear that talking to someone else while class is going on is impolite and disruptive to everyone. Finally, class is not over until I dismiss you. Woody Allen once said that 90% of life is just showing up. Take this to heart. If you are going to succeed in a mathematics course or, for that matter, in any science or engineering course, you must come to class— prepared and ready to listen, think, and participate. In my classes, attendance is mandatory! So, don’t forget to sign (your complete signature) the attendance sheet; otherwise, you will be marked absent. If you are tardy, remind me after class to change “absent” to “0.5 absent.” See the class attendance policy in the 2008–2009 CBU Catalog for more details. If you absolutely have to miss a class, obtain the notes from a classmate and study them. If the notes are not clear, see me for further assistance or clarification. Absence from a class does not excuse you from turning in your assignment on time. It is your responsibility to find out the assignment—either from a classmate or by going to my Web site www.cbu.edu/~lbecker; if that is not possible, call me at 321-3443. For example, if you miss a Wednesday class, the “turn-in” homework assignment will still be due on Friday. Homework. You are expected to do all assigned homework—whether it is collected or not. This is a crucial part of the course and the best way to learn mathematics and to prepare for exams. If you fail to follow the rules listed in the Homework Guidelines for a particular homework assignment , there will be a penalty such as a deduction of points or possibly a grade of 0 for habitual or fragrant violations. Late homework will not be accepted! (An assignment that is turned in after class has begun is considered late.) Quizzes. If you are absent or tardy for a quiz, your grade for that particular one will be 0. There will be no make-ups for quizzes unless there are extenuating circumstances. Quizzes are counted as homework grades. Hour Exams. There will be three 50-minute exams (possibly four if needed). If you miss an exam, you will not be allowed to make it up unless you have a valid, verifiable excuse and notify me as soon as possible. If you know in advance that you will be absent on a certain day (e.g., for a scheduled surgery), notify me ahead of time. If there is an emergency, call me the day of the absence if possible, certainly by the next day. If I am not in my office, leave a short telephone message (321-3443) that includes your name and a local telephone number where you can be reached. Or inform me by e-mail ([email protected]). If you take all of the hour exams, miss no more than three hours of class, and your final exam score exceeds your lowest hour exam score, then it will replace your lowest hour exam score. Expectations.  I expect you to take notes in class and then to annotate and study them before the next lecture.  I expect you to read the textbook.  I expect you to study the material associated with each lecture for at least two hours (this excludes the time it takes to do the homework). Final Exam. The final exam is comprehensive. It will take place sometime between December 15 and December 19. Under no circumstances will it be given before the scheduled date. Final Grade. The semester average ( Sem Avg ) will be computed using the following formula: Sem Avg  .25 [Homework & Quiz Avg]  .45 [Exam Avg]  .3 0 [Final Exam] (^). The final letter grade will be assigned to x^  Sem Avg^ as follows: A : x 90; B :80  x  90; C : 70  x  80; D : 60  x  70; F : x  (^60). Math Center. Web site: http://www2.cbu.edu/cbu/resources/MathCenter.