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This is the webpage for my Spring 2020 Complex Variables course. The text for the course is "Fundamentals of Complex Analysis" by Saff and Snider, 3rd edition.
Fun links:
Visualization of Arg(z) versus arg(z)
The GeoGebra graphing website.
Amazing art by CS professor Jeff Ely.
Using color to visualize complex valued functions of a complex variable.
Engineering article using complex conjugates. EIT on Wikipedia.
Riemann surface of complex square root function video.
Wikipedia page of Maryam Mirzakhani. She was the first woman to earn the Fields Medal (2014), in part for her work with Riemann surfaces.
Application of harmonic functions to epidemic modeling.
Geometric meaning of a line integral visualization
Videos for class:
Theorem 4-7 Motivation
Here are the Homework assignments for the course. Here is the Google folder of solution keys.
Homework 11 -- Due Friday 3/20
Read: Sections 4.1, 4.2, 4.3, 4.4
Do:
Section 4.1 #3, 8, 11
Section 4.2 #3ac, 5, 8, 10
Section 4.3 #1abgh, 2, 4, 5
Section 4.4 #1
Homework 10 -- Due Friday 3/6
Read: Sections 2.6, 3.4, 3.5, 4.1
Do:
Section 2.6 #1acd (these take speculation about how heat behaves, they will be graded gently)
Section 3.4 #1, 2, 4
Section 3.5 #1bcde, 4
Section 4.1 #1
Homework 9 -- Due Monday 3/2
Read: Sections 2.5, 3.2, 3.3, 3.5
Do:
Section 2.5 #4, #21 (see below)
For #4 suppose you have some other harmonic function w(x,y) so that w is a harmonic conjugate of u. Argue that w(x,y)-v(x,y) is constant.
For #21 use #4.
Section 3.2 #10, 11, 15, 19 (For 19, suppose z,w are in such a disk and assume e^z=e^w, given all of this show z=w.)
Section 3.3 #1, 2 (just for equation 6), 5ac
Section 3.4 #1bcde, 4
Homework 8 -- Due Monday 2/24
Read: Sections 2.5, 3.1, 3.2.
Do:
Section 2.4 #6 (ignore hint in book, and follow hint from class)
Section 2.5 #3f, #11, #14 (see below)
3f hint: No integration nor differentiation needed, rather use the fact that e^(z^2) is analytic
11 addition: In Desmos please graph some level curves at levels from -2 to 2 with increment 0.5 (or something like that)
14 hint: Again, an indirect approach like in 3f. Consider the question, "Can we find an analytic function g(z) that has real part ln|f(z)|?" Note that z=x+iy in f(z) leads to using the Calc III Chain Rule.
Section 3.1 #3a (quadratic formula is OK for complex z), #11c
Section 3.2 #4, #5cd (please do not use sinh/cosh in your answers), #7
Homework 7 -- Due Friday 2/14 (if you want graded feedback before the weekend, please turn it in Thursday by 5:00 p.m. at my office)
Read: Sections 2.3, 2.4 and 2.5.
Do: Section 2.3 #8, 11aeg; Section 2.4 # 1c, 4, 5, 10; Section 2.5 #1b, 3ac.
Homework 6 -- Due Monday 2/10
Read: Section 2.2 and Section 2.3.
Do: Section 2.2 #13 (see note below), 19, 21ad; Section 2.3 #4a, 7ab, 9ab, 13, 16.
For #13 in Section 2.2 please justify your conclusions by citing theorems about limits and continuity, or by providing computations. One way to confirm that the limit of a function f(z) at a given point is not finite is to show that the magnitude |f(z)| does not have a finite limit at the given point. Also note that #14 in Section 2.3, which is not assigned, confirms that L'Hopital's Ruled does work for functions of a complex variable. Problem #16 just below is assigned and shows that the Mean Value Theorem fails in this setting!
Homework 5 -- Due Wednesday 2/5
Read: Section 2.1 and Section 2.2.
Do: Section 2.1 #13 (Note 13 says "into" and not "onto." So it's sufficient to show that a point in the specified domain set gets mapped into the specified codomain set. No need to show that all points in the codomain set get 'hit' by the function.)
Section 2.2 #11, 12 (for 12 a convincing argument is enough), 17 (for 17 its fine to use the method from class where we ran the limit in t in order to approach along the real and imaginary axes).
Homework 4
Read: Section 1.6 and Section 2.1. Section 1.7 is a great topic, but it is optional at the moment.
Do: Section 1.6 #15, 17; Section 2.1 #1ace, 2ace, 3abc, 4ab, 7, 8, 9, 12.
NOTE: At the end of class today we did an example that in fact is Section 1.6 #4c. One of you kindly pointed out that we need both of the arguments that we sketched in class. The first argument showed that points on the circle go to the line and nowhere else. The second argument showed that all points on the line were 'mapped to' by a point on the circle. Together these arguments complete the required proof. Please keep this in mind as you compute parts (a) and (b) of this problem.
Homework 3
Read: Section 1.6
Do: Section 1.6 #2-8, 11 (informal approach OK for these, formal proofs not needed)
Homework 2
Read: Sections 1.1 through 1.5
Do: Section 1.2 #13; Section 1.4 #20a, 23b; Section 1.5 #4a, 5, 11, 14, 15
Homework 1
Read: Sections 1.1 through 1.4
Do: Section 1.1 #7, 15, 16ac, 22; Section 1.2 #7acegi, 14; Section 1.3 #1, 6, 7adg; Section 1.4 #2a, 5, 11, 18.
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