Cornell University

Spring 2000

The aim of this course is to introduce certain ideas from mathematical analysis, which have been proven to be useful tools in applications in the areas of natural sciences as well as engineering. Those ideas are as deep as they are useful, and it is hoped that the beauty and the elegance of the subjects be conveyed in the course of learning the material.

The first subject to be presented and mastered is complex analysis,
which corresponds to the Chapter 1 to 7 of the textbook *Complex
Variables and Applications* by Brown/Churchill. The central topic is so
called Cauchy's theorem.

The second topic is the subject of Distribution and Fourier Transform.
It is treated in Chapter 1 to 4 of Strichartz's book *A Guide to
Distribution Theory and Fourier Transforms*.

The third topic, as time permits, will be the subject of Fundamental Solutions to Laplace/heat/wave equations. They often are called Green's functions. We will see how the theory of Distribution and Fourier Transform can be effectively used to solve the equations. You can find the relavant material in Chapter 5 of Strichartz's book.

Those three subjects are loosely related to each other, and I will try to explain the connections as much as I can.

The textbooks for the course are the two books mentioned above. I will be following the relavant parts of the books more or less, yet presentations of certain subjects may differ substantially from the way they are explained in the books.

The grading will be based on weekly assignments (50%) one in-class midterm (20%) and one take-home final (30%). The date for the midterm exam will be announced shortly.

In this course, the emphasis will *not* be on proving statements
formally/rigorously. Instead mastery of the technical aspects (i.e. ability
to compute) will be emphasized and you will be expected to train yourself
in various computational skills through the homework assignments as we
proceed through the semester.

You can also download the course information sheet (Postscript, PDF).

**Instructor:** Sumio Yamada (yamada@math.cornell.edu).

590 Malott Hall, Cornell University, Ithaca, New York.

Office Hours: Wednesdays from 1:30 to 3:30pm, in 590 Malott Hall.

**Teaching Assistant:** Lek-Heng Lim (lekheng@math.cornell.edu).

101 Malott Hall, Cornell University, Ithaca, New York.

- Homework 1 (Friday, February 4)
- page 5: 1(a), 1(c), 3, 4, 12, 13
- page 11: 1(a), 1(c), 2, 3, 5, 6, 7, 10
- page 17: 1, 2, 3, 5, 6, 7
- Homework 2 (Friday, February 11)
- page 42: 1(g), 4, 9, 11
- page 47: 1, 4
- page 54: 1, 6, 7, 8, 10
- page 63: 12, 13
- Homework 3 (Friday, February 18)
- page 68: 3, 4, 5, 7
- page 79: 1, 2, 3, 4, 7, 9, 10, 14
- page 84: 2, 3
- Homework 4 (Friday, February 25)
- page 92: 3, 6
- page 102: 1, 4, 6, 14
- page 119: 1, 2, 3, 4, 5, 7
- Homework 5 (Friday, March 3)
- page 128: 1(a), 1(b), 1(c), 2, 4, 9
- page 63: 7
- page 136: 1, 2, 4, 5, 10
- page 149: 3, 5, 10
- In-class Quiz (Monday, March 13)
- Evaluate 10 path integrals
- Homework 6
- Homework 7
- Homework 8
- page 226: 1, 2, 4, 6, 8, 9

Both Postscript and PDF are specialized languages for describing printable documents. Postscript is the older one, and it has traditionally been used to distribute electronic versions of mathematical texts.

Readers for both formats are available for several operating systems and platforms. You can learn more about Postscript here. PDF stands for "Portable Document Format" and it has been developed by Adobe. You can get free readers from their website.

Note that if you download and install Ghostscript you get both a Postscript and PDF reader.