Files and Resources for Math Methods (TECP TTVN)

Spring 2007 Here are the notes:

2007-03-26 Most of the following are scanned examples and notes. They are PNG files, which should open in any recent browser. They are scanned at 300 dpi, scaling has been a bit of a problem but if you use a print preview and scale to about 25 percent, they should fit on a page:
• A typeset PDF file of a couple of residue examples: Rational function residue examples.
• How to find residues (3 pages): Page 1, Page 2, Page 3,
• A trig function example, similar to 17.3 problem f,g,h: Simple trig function example.
• Integrating a rational function of sine and cosine using the unit circle. This is kind of an algebra horror, because it does a general case (2 pages): Page 1, Page 2. Not all of the algebra is worked out explicitly in this one, the spots where I wrote check! are missing steps. Snieder does not do an example like this, but it is another major type of integral that can be done with residues.
• Wikipedia has descriptions of two lemmas used to check that the semicircular arcs in contour integrals go to zero: the estimation lemma, also known as the ML-inequality, and Jordan's lemma.

The estimation lemma is used between problems 17.3 a and b, and Jordan's lemma is basically problem 17.3 f and 17.4 d, although Snieder does not use these names (nor do all texts.)

• Wikibooks has a short summary of Residue theory. It looks good but use with caution.
• Even better: The math department at Cal State Fullerton has a set of Complex variables modules for undergrads that look excellent, particularly chapters 6, 7, and 8 (scroll down there) for what we are doing.
On Wed. 2007-03-28 I plan to overview the way more general complex functions are defined, a topic not touched on in the book. Here is a draft of an article about that: Complex functions and multifunctions. This has been updated Wed. morning with more figures.

Homework for residues: 070328_HW.pdf I originally had this due Friday, Apr. 6, but I forgot that Easter was this close, so it is due Monday, Apr. 9, since there probably won't be anyone here Friday either. The problems are fairly short and just like the examples, so you can probably get it done early.

Here is a single-sheet: Basic terminology and formulas for complex variables (PDF).

'07-04-02 (Monday)We started on Ch. 15 -- Fourier analysis, covering section 15.1. We also talked about analogies with finite vector spaces.

The fourier series applet is here.

'07-04-04 (Wednesday)We will briefly continue talking about vector and function spaces and the complex form of Fourier series.

We will also go over some concepts and properties of the Dirac delta function. Chapter 14 covers this in detail but we mainly just want results, in particular equations 14.11,17,20,24, and 31.

There is an uncorrected error in Ch. 14: on page 206, bottom paragraph under problem a, the sentence should be: 'This expression states that the "surface area" under the delta frequence is one.'

Here is Test 2,

due Tuesday, April 17.

'07-04-13: Here is some info on Fourier transform conventions for the most common systems, with some pointers to resources on the web. This is a work in progress, covering differences between the various locations of pi, etc.

The Final Exam is here. It is due Wednesday, May 2.

The Errata page for the textbook.

Formularies

These are free formula compendia off of the web. The original address is in the introduction to each file. I will probably replace them with links to the original site soon.