Chapter 7: Fourier Series and Transforms

Write out the details of the derivation of the formulas (8.3)

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 functionf_p of period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

21.

Find the fourier transform of$\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{-}{\mathbf{x}}^{\mathbf{2}}\mathbf{/}\mathbf{\left(}\mathbf{2}{\mathbf{\sigma }}^{\mathbf{2}}\mathbf{\right)}}$. Hint: Complete the square in the xterms in the exponent and make the change of variable$\mathit{y}\mathbf{=}\mathit{x}\mathbf{+}{\mathit{\sigma }}^{\mathbf{2}}\mathit{i}\mathit{\alpha }$ .Use tables or computer to evaluate the definite integral.

In Problems 17 to 22 you are given 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)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

The function${\mathit{j}}_{\mathbf{1}}\mathbf{\left(}\mathbf{\alpha }\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{\alpha }\mathbf{cos}\mathbf{\alpha }\mathbf{-}\mathbf{sin}\mathbf{\alpha }\mathbf{)}\mathbf{/}\mathit{\alpha }$is of interest in quantum mechanics. [It is called a spherical Bessel function; see Chapter 12, equation 17.4] Using problem 18, show that

$\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}{\mathit{j}}_{\mathbf{1}}\mathbf{\left(}\mathbf{\alpha }\mathbf{\right)}\mathbf{sin}\mathit{\alpha }\mathit{d}\mathit{\alpha }\mathbf{=}\{\begin{array}{l}\begin{array}{cc}\mathrm{\pi x}/2,& -1<x<1\end{array}\\ \begin{array}{cc}0,& \left|x\right|>1\end{array}\end{array}$

Write an equation for a sinusoidal sound wave of amplitude 1 and frequency 440 hertz ( 1hertz means 1 cycle per second). (Take the velocity of sound to be 350 m/sec).

Using problem 17,show that

$\begin{array}{l}\underset{0}{\overset{\infty }{\int }}\frac{1-\cos \pi \alpha }{\alpha }\sin \alpha d\alpha =\frac{\pi }{2}\\ \underset{0}{\overset{\infty }{\int }}\frac{1-\cos \pi \alpha }{\alpha }\sin \alpha d\alpha =\frac{\pi }{4}\end{array}$

If a violin string is plucked (pulled aside and let go), it is possible to find a formula f(x, t) for the displacement at time t of any point x of the vibrating string from its equilibrium position. It turns out that in solving this problem we need to expand the function f(x, 0), whose graph is the initial shape of the string, in a Fourier sine series. Find this series if a string of length l is pulled aside a small distance h at its center, as shown.

$\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{\int }}}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\alpha }\mathbf{x}}{{\mathbf{\alpha }}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathit{d}\mathit{\alpha }\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{\left|}\mathbf{x}\mathbf{\right|}}$

(b) Obtain the result in (a) by using the Fourier cosine transform equations (12.15).

(c) Find the Fourier cosine transform of $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{1}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}}$. Hint: Write your result in (b) with xand$\mathit{\alpha }$interchanged.

If, in Problem 23, the string is stopped at the center and half of it is plucked, then the function to be expanded in a sine series is shown here. Find the series. Caution: Note thatfor f(x,0) = 0 for l/2<x<l.