MCS 321 Problem Assignments

4. Integrals

Sections to read	Topics	Problems	Due Date
36-37	Calculus of complex functions of a real variable	(a) Prove the product rule for complex functions of a real variable. [Hint: see class notes.] (b) (d/dt)[(t + it)e^it] = ? p. 115 #3, 4	M 3/22
38-39	Contours and contour integrals	p. 120 #2, p. 129 #1(a)	M 3/22
39-40	Contour integrals	p. 129 #2, 7, 8	F 3/26
41	Upper bound for \|integral\|	p. 134 #4	F 3/26
42-43	Antiderivatives	p. 141 #1, 2	F 3/26
44-46	Cauchy-Goursat theorem!	p. 153 #1 Read Sec. 46.	W 3/31
47-48	Cauchy integral formula, higher derivatives	pp. 162-163 #1 Extra credit: For same C, integrate ((1+i)z+i)/(z²+1) and (3z+2+i)/(z²+(1-i)z-i). Handout (a), (b), and (c)	W 3/31
49-50	Liouville's theorem, Fundamental Theorem of Algebra, Maximum modulus principle

Homework rules

Students are encouraged to discuss course topics with one another. On assigned work, though, you are expected to do the work on your own. Your discussion with other students should be confined to matters related to understanding the problems and general approaches to doing problems, and you should acknowledge those who help you in this way. You should not collaborate on problem solutions unless directed otherwise.

Acknowledge your sources (people and texts).
In nontrivial problems, show how you get your answers.
Turn in neat, well-written solutions, not messy first drafts. Trim "fringes." Staple multiple-page submissions.
Do not copy collaborative solutions; write up solutions in your own words.
Turn in homework on time. Each class day late reduces the possible points by 33%.
Do extra credit problems entirely on your own.

Honor pledge: "On my honor, I pledge that I have not given, received, or tolerated others' use of unauthorized aid in completing this work."

Proofs

Clearly state the proposition to be proved.
A proof is a clear, logical argument that convinces the reader that the proposition is true.
Embed mathematical expressions in your sentences in a grammatically sensible way. Look at your math texts to see how this is done.
Every step in a proof is or can be justified by a reason. Valid reasons include assumptions, definitions, and previously established results. Consider the audience (or the professor's instructions) in deciding to what extent you make explicit or abbreviate your steps and reasons.