Chapter 7: Fourier Series and Transforms

The functions in Problems 1 to 3 are neither even nor odd. Write each of them as the sum of an even function and an odd function.

(a) ${\mathit{e}}^{\mathbf{i}\mathbf{n}\mathbf{x}}$

(b)$\mathit{x}{\mathit{e}}^{\mathbf{x}}$

Show that if f(x) has period p , the average value of f is the same over any interval of length p . Hint: Write${\mathbf{\int }}_{\mathbf{a}}^{\mathbf{a}\mathbf{+}\mathbf{p}}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathit{d}\mathit{x}$ as the sum of two integrals ( a top, andpto a+p ) and make the change of variable x = t + p in the second integral.

In problem 1to 3, the graphs sketched represent one period of the excess pressure p(t)in a sound wave. Find the important harmonics and their relative intensities. Use a computer to play individual terms or a sum of several terms of the series.

Following a method similar to that used in obtaining equations(12.11) to (12.14), show that if f(x)is even, then$g\left(\alpha \right)$is even too. Show that in this case f(x)and$\mathit{g}\left(\alpha \right)$can be written as Fourier cosine transforms and obtain (12.15).

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials${\mathit{e}}^{\mathbf{i}\mathbf{n}\mathbf{x}}$on the interval $\mathbf{(}\mathbf{-}\mathbf{\pi }\mathbf{,}\mathbf{\pi }\mathbf{)}$and verify in each case that the answer is equivalent to the one found in Section 5.

Prove (11.4)for a function of period 2Lexpanded in a sine-cosine series.

Sketch several periods of the corresponding periodic function of period$\mathbf{2}\mathit{\pi }$. Expand the periodic function in a sine-cosine Fourier series.

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.

In Problems 17 to 22 you are given f(x) on an interval, say 0 < x < b. Sketch several periods of the even function f_cof period 2b, the odd function f_sof period 2b, and the function f_pof period b, each of which equals f(x)on0 < x < b. Expand each of the three functions in an appropriate Fourier series.

20. f(x) = x², 0 < x < 1.

In Problems 17to 20, find the Fourier sine transform of the function in the indicated problem, and write f(x)as a Fourier integral [use equation (12.14)]. Verify that the sine integral for f(x)is the same as the exponential integral found previously.

Problem 12