Week 11 (March 29 - April 2)

This week we begin considering the Fourier transform. My goal for the week is to motivate the definition of the transform and to obtain enough properties to describe an application to the solutions of the one-dimensional heat equation on an infinite rod. For this reason, I will discuss the complex form of the Fourier series (a part of Section 14.6), define the transform and obtain a few of the properties discussed in Section 15.1, look at some properties in Section 15.2, and then jump ahead to Section 16.7. We will eventually discuss all the properties in Sections 15.1 and 15.2, but not all of them this week.

On Monday, we will discuss the complex form of the Fourier series and then use this to obtain the Fourier transform.

On Wednesday, we will introduce the application of the Fourier transform to the description of the solutions of the one-dimensional heat equation on an infinite rod. Then we will begin deriving the properties of the transform necessary to obtain this description.

On Friday, we will meet in the lab to examine the Heaviside and Dirac functions, convolution of functions, and solutions of the heat equation in an infinite rod in the module "Fourier Transform I."

Homework due on Friday, April 2 .

1. Section 14.6: (Note that we will have discussed some parts of this section, but not all of it.) Problems 1-3. Just calculate the complex Fourier coefficients and determine what the series converges to. You may want to use Maple to do the integral computations. If you do, compare the graph of the approximations to the graph of the function. Also, for Problems 1 and 3, show that the answers you obtain agree with the ones in the back of the text.

2. Section 15.1. Problem 1. Just calculate the Fourier transform, don't worry yet about the amplitude spectrum. Do this integral calculation by hand.

Homework due on Monday, April 5.

For Problems 1 and 2, use Theorem 15.2 and the known Fourier transforms.

1. Find the Fourier transform of 5 [H(x - 3) - H(x - 11)]. (See Problem 3 in Section 15.1.)

2. Find the Fourier transform of 5 exp(-3(x - 5)²).

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3. Find a function f(x) such that the Fourier transform of f is F, where F(w) is the product of 1/(1 + iw)) and 1/(2 + iw). Do this two ways. First, expand F(w) by partial fractions and then use the linearity of the Fourier transform. Second, use the result on the convolution of a product. Make sure your two approaches give the same answer.

4. Find the Fourier transform of d/dx f(x), where f(x) = H(x) exp(-3x). (See Problem 5 in Section 15.2. Watch out for the jump discontinuity in f.)

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   Lawrence C. Moore < lang@math.duke.edu>

   Last modified: March 29, 1999