Jan 15: Conway p3: 4, p4: 2 (explain what this means geometrically), 3, p5:2,3,6, p10: 4. Also
Find the modulus, argument, real and im parts of ...
Show that z and z' corresponds to diametrically opposite points on the sphere iff z times the conjugate of z' is -1.
Jan 24: p33: 6,7, p44: 6,7,9,12,18,20. Also
Show that the image of sin(z) is the entire complex plane. What is the image under sin(z) of the line given by Re(z)=pi/4? What is the image under sin(z) of the line given by Im(z)=pi/4?
Feb 4 (Friday): p33: 19; p54: 11, 14, 16, 26, 30; also...
a. Explain why the diagram on p 51 is a geometric way to determine the point symmetric to z with respect to the circle.
b. Reflect the imaginary axis, the line Re(z)=Im(z), and the circle |z|=1 with respect to the circle |z-2|=1.
c. Let G_1 and G_2 be a pair of nonintersecting circles or lines in (the complex plane union infinity). When can they be mapped by a Mobius transformation to concentric circles? Explain.
d. Pick a pair on nonintersecting, nonconcentric circles in the complex plane. Write out explicitly a Mobius transformation that maps them to concentric circles. To what extent is this map unique?
Feb 14 (Monday): p 67: 7, 9 12, 13, 16, 17, 21, 22
Mar 14 (Monday): p74: 5,9; p80: 6, 8; p 87: 5, 6, 8, 9; also . . .
Write a proof of the Cauchy integral formula (Conway's version 1 of the Cauchy integral formula), including proofs of the two lemmas we did not prove in class. One is written in Conway, the other is a generalization of the Leibniz rule.
Mar 21 (Monday): p 83: 3; p 96: 5, 6, 8, 10, 11; p 110: 4, 5, 6, 13.
Apr 4 (Monday) p 110: 1bdehj, 15; p 121: 2a, 2b, 2c, 2d, 2f, 4, 5, 12
To consider handing in (let me know if you plan to do so): p 126: 2, 8, 9, 10; p 130: 2, 3, 6, 7; p 132: 2,3.
Math 676, Spring 2004
1:00-1:50pm in 6620 Everett
Instructor:
Dr. Melinda Koelling
office: 5525 Everett Tower
email: melinda.koelling at wmich.edu
phone: 387-4509
office hours: MThF 10am-11am and W 2-3.
Course Description: Cauchy Theory, series expansion, power series, types of singularities, calculus of residues. Other topics on the basis of student interest, if time permits.
Prerequisites: MATH 571 or equivalent.
Text: Functions of One Complex Variable I, Second Edition, by John Conway.
Homework: You will be expected to read relevant sections of the book, review your notes, and complete homework before every class. You are encouraged to talk with other people to learn the material, but you should always write a final draft of your homework on your own.
Grades: Homework 35% , Exams (at least two) 40% , and Final exam 25% . I reserve the right to lower your grade for poor attendance.
Makeups: Makeup exams will be made only in the case of a genuine medical or personal emergency. It is your responsibility to prove that your absence is due to an emergency.
Communications: Academic correspondence must be through wmich.edu accounts. I will respond to email at least once per weekday.
Academic Dishonesty: You are responsible for making yourself aware of and understanding the policies and procedures in the Graduate Catalog that pertain to Academic Integrity. These policies include cheating, fabrication, falsification and forgery, multiple submission, plagiarism, complicity and computer misuse. If there is reason to believe you have been involved in academic dishonesty, you will be referred to the Office of Student Conduct. You will be given the opportunity to review the charge(s). If you believe you are not responsible, you will have the opportunity for a hearing. More details about this procedure can be found at website of the office of student conduct, and an outline of the procedure can be found at http://www.osc.wmich.edu/faq/academic.html . You should consult with me if you are uncertain about an issue of academic honesty prior to the submission of an assignment or test.
Disabilities: Any students with a documented disability (e.g. physical, learning, psychiatric, vision, hearing, etc.) who needs to arrange reasonable accommodations must contact Ms. Beth Denhartigh at telephone 387-2116 or by email at the beginning of the semester. A disability determination must be made by that office before any accommodations are provided by the instructor.