Homework 8
114B
due Monday November 18, 2002:

Do Boas Ch 15 4.21, 4.24 (a) and (c), 10.14, 10.15 except do not compare 2.19 to 2.27.

1. What is

2. is 1 for x > 0, and 0 otherwise. Calculate the derivative of . Hint: use the previous problem.

In problems 3 through 9, calculate the Fourier transforms of the following functions

3. , where q is a real constant.

4. . Hint: use the previous problem.

5. where a is a real constant.

6. . Hint: use the previous problem.

7. .

8. . Hint: use the "shift theorem", problem 10.15 Ch 15 Boas.

9. 1 for |x| < 1/2, 0 otherwise. This is called the "top hat" function. Do a rough sketch of your answer.

10.

(a) Calculate the derivative of the top hat function (see the last problem).

(b) Calculate the Fourier transform of this derivative using problem 10.14 Ch 15 Boas.

(c) Compare this with problem 7 of this homework set.

11. Calculate the Fourier transform of a periodic function f(x) = f(x+1) as follows:

(a) It's Fourier series can be written

Solve for the 's (see formula (8.3) Ch. 7 of Boas).

(b) Every term in the Fourier series is now easy to Fourier transform, so now Fourier transform the whole series. Your answer should involve the sum of delta functions and the 's calculated in (a).

12. Define

Calculate the Fourier transform of , . Do this as follows:

(a) Show is a periodic function.

(b) Use the result of the previous problem. The formula for the 's should now be easy to integrate.

(c) Express your answer in terms of .

13. Consider convoluted with a Gaussian:

where ``*'' is the convolution operation.

(a) Draw a rough sketch of C(x) when .

(b) Draw a rough sketch of C(x) when .

(c) Using the convolution theorem and the last problem, calculate the Fourier transform of C(x).

(d) Draw a rough sketch of the Fourier transform when .

(e) Draw a rough sketch of the Fourier transform when .

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