Using this, you can easily get various chain rules.
In class, I mentioned that the complex numbers form a field. A field is particular type of algebraic structure. We are not going to use this fact in any essential way. If you are curious about algebraic structures such as groups, rings, and fields, you can get definitions and references at MathWorld. MathWorld is generally reliable as a source of definitions and theorems. Everything is linked and each entry has lots of useful references so this can be a good starting point. MathWorld is operated by Wolfram, the company that produces Mathematica.
Several people have comed to talk with me about proof by induction. In linear algebra, I distribute this handout on induction that might be useful. Come talk with me if you have questions about induction.
Several of the assigned problems call for you to prove an inequality. Two main strategies for this are similar to how you might approach proving an identity. One strategy is to start with an inequality you know to be true and then do legal manipulations to get the inequality you are trying to prove. In doing this, be careful with the effect of multiplying by negative quantities. Another strategy is to start with the expression on one side of the desired inequality and then develop a string of equalities and inequalities (all "less thans" or all "greater thans") to arrive at the expression on the other side of the desired inequality.
Here's a web site I just learned about: simpsonsmath.com. Check it out.
On homework problems that you submit, I want you to put some focus on writing style. You might find it useful to read this handout with some notes on writing mathematics.
We will have our first exam on Thursday, February 9 from 8:00-9:20 am. It will cover material in the text through Section 21 (except Sections 13 and 16).
We are going to skip over the material in Section 16 for now. We'll come back to this after the first exam.
In class, Andrew asked about the definition of set. A rigorous approach to set theory is based on an axiomatic system consisting of undefined terms and axioms. In the standard axiomatic systems for set theory, set is an undefined term. There are different axiomatic systems for set theory in use throughout mathematics. The most common ones are variants of the Zermelo-Fraenkel axioms. See the MathWorld entry on Set Theory for more information, links, and references.
In class, we proved Property (8) in Theorem 2 on page 47 (using different notation). The text describes an alternate proof for Theorem 2 that is based on Theorem 1 and results about limits for functions of two real variables.
In class today, we looked at a broad view of differentiation for complex-valued functions. One feature of the story (which has no direct analog in the world of real-valued functions) is the relation between the derivative of a complex-valued function and the partial derivatives of the real and imaginary parts of the function. We developed the following basic result
The following converse of the first statement is false:
I've fixed the Old exams link.
In class today, I used the notation U(r,θ)=u(x(r,θ),y(r,θ)). This notation distinguished between the functions U and u. Many people, including the authors of our text, use u on both sides of this relation. Distinguishing between u and U can be useful but also introduces some notational complexity. Here's a specific example to make this less abstract. Suppose u(x,y)=x2+y2. What is U(r,θ)? What are two possible interpretations of u(r,θ)?
I'm introducing the idea of conformal mappings in order to give a more detailed geometric interpretation of the complex derivative.
The ideas we talked about today are in Sections 94 and 95 of the text. I've assigned a few computational problems from these sections.
In Problem 7 on p. 79, the suggestion involves thinking about y as a function of x along a specific level curve. Then one can think about u(x,y(x)) as a function of x and then can use a chain rule to express the derivative with respect to x of this function.
I've moved the due date for Section 25 back to Wednesday.
We are making heavy use of chain rules from multivariate calculus. You
should become comfortable with how to think about the relation
Using this, you can easily get various chain rules.
I've moved some homework dues dates back since we just got to complex log in detail today.
You have at least three strategies available when trying to establish that a given function is analytic:
Section 89 looks at sin z as a mapping. We'll talk about this briefly on Monday.
Section 34 deals with complex hyperbolic functions and Section 35 deals with inverses of complex trig and complex hyperbolic functions. In class, we won't talk about the material from these sections other than addressing questions you bring up about the reading or assigned homework problems.
In class, we'll introduce contour integrals from a different viewpoint than that used in the text. We'll end up in the same place. The approach we use in class has the advantage of being a more direct analog of the definition of definite integral for real-valued functions but has the disadvantage of involving limits. The approach used in the text is less intuitive but cleaner since contour integrals get defined in terms of definite integrals for real-valued functions. This approach is developed in Sections 36-40.
You should read of Sections 36-40 and begin to make connections between what we have done in class and what is done in the text. You should also begin working on problems assigned from these sections.
Our second exam will be Thursday, March 9 from 8:00-9:20. It will cover material through Section 43 except the following sections that we have not covered: 16,26,27,41. It will also cover Sections 89, 94, and 95 on functions as mappings and a geometric interpretation of complex derivative.
You will need to be careful with the details on Problem 6 from the handout we worked on in class today. You can break the square into four line segments and then deal with each segment individually. I've included a few comments about this on the version of the handout you can download here. (It's too much work to typeset mathematics here on the web page itself.)
I bungled the limits of integration at the end of class today. At one point, I did have it right. We have φ : [a,b] → [c,d] with φ(c)=a and φ(d)=b. Our substitution is t=φ(τ) or τ=φ-1(t). The lower limit of integration in the variable τ is φ-1(a)=c and the upper limit of integration in τ is φ-1(b)=d.
On Monday, we will discuss one more big idea that will be covered on Thursday's exam. This is the theorem in Section 42 about antiderivatives and path independence.
Section 41 contains a technical result that is first used to prove part of the "existence of antiderivative equivalent to path independence" theorem of Section 42. We'll come back to this after the exam.
There are many web resources for learning about this material. Two nice applets to explore Julia sets for and the Mandelbrot set are
There are also many excellent texts on this material at a variety of levels. One nice choice for undergraduates is A First Course in Chaotic Dynamical Systems by Robert Devaney. The book Chaos: Making a New Science by James Gleick is a very well-written popularization of this area.
The text refers to the theorem in Section 44 as the Cauchy-Goursat theorem. Others refer to it as Cauchy's theorem or the Cauchy Integral Theorem (not to be confused with the Cauchy Integral Formula in Section 47). Some use Cauchy's theorem for the version that includes continuity of f ' as a hypothesis and use Cauchy-Goursat theorem for the stronger version that does not include this hypothesis.
The text has two proofs, one in Section 44 of the weak version (requiring continuity of f ') and the other in Section 45 of the strong version. The proof of the weak version uses Green's theorem from multivariate calculus. We'll look at some consequences of the theorem and then next week in class, we'll sketch a proof similar to the one in Section 45.
Exam 3 will be on Thursday, April 6. It will cover material through the end of Chapter 4.
On Monday, we'll finish the last topic for the exam, namely the maximum modulus theorem. To prepare for this, you should recall the Extreme Value Theorem from calculus (both single-variable and multivariate versions).
An important consequence of the maximum modulus theorem is that it implies a nonconstant harmonic function u(x,y) cannot have a maximum value or a minimum value on the interior of a closed, bounded region of the plane. A harmonic function defined on a closed bounded region must therefore have its maximum and minimum values on the boundary of the region. In other words, the graph of a harmonic function has no peaks or pits.
In class, I stumbled in answering Jane's question about the size of the neighborhoods in the proof of the global maximum modulus theorem. Here's how to address the issue: Our proof of the local maximum modulus theorem shows that f is constant throughout the neighborhood of z0 for which |f(z0)|≤|f(z)|. From this point of view, the theorem is a bit more than local. When using this result in the proof of the global theorem, we can conclude that |f(z)| is constant throughout the entire neighborhood N0 in the first step since |f(z0)|≤|f(z)| for all z in D by hypothesis. The same argument works for each of the neighborhoods Ni, all of radius d.
Our text makes a subtle shift in dealing with sequences and series. The authors begin looking at sequences and series of constants and then, without explicitly acknowledging it, shift to looking at sequences and series of functions. When dealing with a sequence or series of constants, the fundamental question is Does the sequence or series converge? When dealing with a sequence or series of functions that depend on a variable z, the relevant question becomes For what values of z does the sequence or series converge?
We will use the geometric series result
often in the rest of the course. You will need to become comfortable with
manipulating expressions into the form on the right side so you can then
use the series expression on the left side.
An "extended" power series (based at z0) has both positive and negative powers of z-z0. A Laurent series is an "extended" power series with coefficients defined in terms of a given function. Laurent series are to "extended" power series as Taylor series are to power series.
Give a Laurent series, we ask two questions:
The material in Sections 57-61 gives more results on power series. We will not cover this material in detail. Here are the highlights. We know that the derivative of a (finite) sum of functions is equal to the sum of the derivatives of the functions. We also know that the integral of a (finite) sum of functions is equal to the sum of the integrals of the functions. Do these results extend to convergent series (i.e., infinite sums) of functions. In general, the answer is no. That is, in many cases one cannot interchange the order of operations in the derivative of a convergent series or the integral of a convergent series. To formulate a general theorem about this issue, a new version of convergence is useful. This version is called uniform convergence. Uniform convergence becomes the essential hypothesis in theorems about derivatives and integrals of series.
We will also skip the material in Section 64.
Exam #4 will be take-home. I'll give it out next Wednesday and it will be due on the last day of classes, Wednesday, May 3.
You have probably noticed that the nature of problems has changed to a more computational orientation. Because of this and because you will have a take-home exam next week, I will not assign any more problems to be submitted.
We will skip the material in Section 70.
The final exam will be in-class with one essay question that you can write in advance of the exam. I'll give details on this early next week.
What we did in class with Riemann surfaces was for your cultural benefit.
Our final exam is scheduled for Monday, May 8 from 8 to 10 am. I will let you work on the exam until 10:30. The final exam will be comprehensive. You can bring one sheet (standard notebook size) of notes to use on the final. You can use both sides of the sheet. The essay portion of the final exam is due at the time of the final exam. If you want an extension, ask for it by Sunday, May 7.
| Sections | Pages | Problems to do | Submit | Target or due date | Comments |
|---|---|---|---|---|---|
| 1-2 | 4-5 | 1-4,6,7,9,10 | None | Thursday, January 19 | |
| 3 | 7-8 | 1,2,5,7 | None | Friday, January 20 | |
| 4 | 11 | 1,4,5 | 3 | Wednesday, January 25 | |
| 5 | 13 | 1,2,5,9,11,15,16 | 8,12 | Wednesday, January 25 | |
| 6-7 | 21-22 | 1,2,4,5,6 | 9 | Friday, January 27 | |
| 8-9 | 28-29 | 2,3,6,8,9 | 7 | Monday, January 30 | |
| 11 | 35-36 | 1-4 | None | Monday, January 30 | |
| 12 | 42-43 | 1,3,8 | None | Monday, January 30 | |
| 10 | 31-32 | 1,2,3,4 | None | Wednesday, February 1 | |
| 14-15 | 53-54 | 1,2,3,5,6(a),7,8 | 9 | Friday, February 3 | |
| 18-19 | 59-60 | 1,3,4,5,6 | 9 | Wednesday, February 8 | |
| 20-21 | 68-69 | 1,2,3,5,6 | None | Wednesday, February 8 | |
| 22 | 69 | 4,7,8,9 | None | Monday, February 13 | |
| 23-24 | 73-74 | 1,2,4,5,6 | 7 | Thursday, February 16 | |
| 94-95 | 350 | 1,2,3 | None | Friday, February 17 | |
| 25 | 78-80 | 1,4,5,6,7,8 | 9 | Wednesday, February 22 | |
| 28 | 89-90 | 1,2,4,6,7,12,14 | None | Wednesday, February 22 | |
| 13 | 42 | 4,5,6 | None | Wednesday, February 22 | |
| 29-30 | 94 | 1-8 | 9 | Friday, February 24 | |
| 31 | 96 | 1,3 | None | Friday, February 24 | |
| 32 | 99-100 | 1-6,8 | None | Monday, February 27 | |
| 33 | 103-105 | 1,2,3,6,8 | None | Monday, February 27 | |
| 89 | 322-324 | 1,2 | None | Wednesday, March 1 | |
| 34 | 107-108 | 1,2,5,10,14 | None | Wednesday, March 1 | |
| 35 | 110 | 1 | 5,6 | Thursday, March 2 | |
| 36-37 | 115-116 | 2(a),2(b),4,5 | None | Thursday, March 2 | |
| 38 | 120-121 | 1,2,3 | None | Friday, March 3 | |
| 39-40 | 129-130 | 1-6,10 | 11 | Wednesday, March 8 | |
| 42-43 | 141-142 | 1-5 | None | Wednesday, March 8 | |
| 41 | 133-134 | 1,2,4 | 5 | Thursday, March 23 | |
| 44-46 | 153-156 | 1,2,6,7 | None | Monday, March 27 | |
| 47-48 | 162-164 | 1,2,5,7 | 9 | Friday, March 31 | |
| 49-50 | 171-173 | 1,2,4,5,6,7 | None | Wednesday, April 5 | |
| 51-52 | 181-182 | 1,2,3,4,6 | None | Wednesday, April 12 | |
| 53-54 | 188-190 | 1,2,3,5,7,12,13 | 10 | Wednesday, April 19 | |
| 55-56 | 198-200 | 1-8 | None | Wednesday, April 19 | |
| 62-63 | 230 | 1,2 | None | Friday, April 21 | |
| 65 | 233-234 | 1,2,3 | None | Friday, April 21 | |
| 66-67 | 238-239 | 1,2,3,4,5 | None | Monday, April 24 | |
| 68-69 | 245-246 | 1,3,4,5,6 | None | Wednesday, April 26 | |
| 71-72 | 1,4,6,7 | None | Friday, April 28 | ||
| 73 | 5 | None | Monday, May 1 |
The Mathematical Atlas describes the many fields and subfields of mathematics. The site has numerous links to other interesting and useful sites about mathematics.
If you are interested in the history of mathematics, a good place to start is the History of Mathematics page at the University of St. Andrews, Scotland.
Check out the Astronomy Picture of the Day.
You can look at exams from last time I taught Math 352. You can use these to get some idea of how I write exams. Don't assume I am going to write our exams by just making small changes to the old exams. You should also note that we were using a different textbook so some of the notation is different. There are also differences in the material covered on each exam. You can use this old exam to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.