HW 8

1) The Fourier series of a Sin function should be trivial, right? Execute the following Mathematica code. Explain the result.

FourierTrigSeries[Sin[x], x, 4]

Remember to load the appropriate library before you try this.

2) Look at problem 5.2 in Chapter 7 of Boas. Verify the first nonzero Sin and Cosine terms by doing the integrals by hand. Then show the precise call to FourierTrigSeries that will reproduce the 8 terms shown in the answer. This will require to look carefully into the definitions used by Mathematica and the optional arguments of FourierTrigSeries.

3-4) Chapter 7, section 10 of Boas: Problems 4 and 9.

5) Prove Equation 11.5 on page 375.

6) Show that this function is normalized. That means that when you integrate it from -infinity to +infinity you get 1:

$\frac{1}{\sqrt{2 \pi} \sigma} e^{(-(t - m)^2/(2 \sigma^2))}$

7) Show that the Fourier transform of this function is: $\frac{e^{i m w-\frac{w^2 \sigma ^2}{2}}}{\sqrt{2 \pi }}$

Thus, the Fourier transform of a Gaussian is not a Gaussian itself unless the mean is zero.