Chapter 7: Fourier Series and Transforms

For each of the periodic functions in Problems 5.1 to 5.11 , use Dirichlet's theorem to find the value to which the Fourier series converges at$\mathbf{\text{x=0,±π/2,±π,±2π}}$ .

(a) Sketch at least three periods of the graph of the function represented by cosine series for f(x)in Problem 9.

(b) Sketch at least three periods of the graph of the exponential Fourier series of period2 for f(x)in Problem 9.

(c) To what value does the cosine series in (a) coverage at x=0? At x=1? At x=2? At x=-2?

(d) To what value does the exponential series in (b) converge at x=0? At x=1? At$\mathit{x}\mathbf{=}\frac{\mathbf{3}}{\mathbf{2}}$? At x=-2.

Find the three Fourier series in problem9 and 10.

What would be the apparent frequency of a sound wave represented by

_{$\mathit{p}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{60}\mathbf{n}\mathbf{\pi }\mathbf{t}}{\mathbf{100}{\left(n-3\right)}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{?}$}

(a) Given $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{}\left(\pi -x\right)\mathbf{/}\mathbf{2}$on$\left(0,\pi \right)$ , find the sine seriesof period$\mathbf{2}\mathbf{\pi }$ for $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$.

(b) Use your result in (a) to evaluate$\underset{}{\overset{}{\mathbf{\sum }\mathbf{1}\mathbf{/}{\mathbf{n}}^{\mathbf{2}}}}$ .

Show that the Fourier sine transform of ${\mathit{x}}^{\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}}}$ is . Hint: Make the change of variable$\mathit{z}\mathbf{=}\mathit{\alpha }\mathit{x}$ . The integral ${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathit{z}}^{\mathbf{-}\mathbf{1}\mathbf{/}\mathbf{2}}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{z}\mathbf{}\mathit{d}\mathit{z}$can be found by computer or in tables

Let f(x)and$\mathit{g}\left(\alpha \right)$be a pair of Fourier transforms. Show that$\frac{\mathbf{\text{df}}}{\mathbf{\text{dx}}}$and$\mathit{i}\mathit{\alpha }\mathit{g}\left(\alpha \right)$are a pair of Fourier transforms. Hint: Differentiate the first integral in (12.2)under the integral sign with respect to x. Use to show that

${\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\mathit{\alpha }{\left|g\left(\alpha \right)\right|}^{\mathbf{2}}\mathit{d}\mathit{\alpha }\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{\pi }\mathbf{i}}{\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\stackrel{\mathbf{-}}{\mathbf{f}}\left(x\right)\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\mathit{f}\left(x\right)\mathit{d}\mathit{x}$.

Find the form of Parseval’s theorem(12.24)for sine transforms (12.14)and for cosine transforms(12.15).

Find the exponential Fourier transform of

$\mathbf{\text{f}}\left(\text{x}\right)\mathbf{\text{=}}\{\begin{array}{l}2a-\left|x\right|,\left|x\right|<2a\\ 0,\left|x\right|>2a\end{array}$

And use your result with Parseval’s theorem to Evaluate${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{4}}\left(\alpha a\right)}{{\mathbf{\alpha }}^{\mathbf{4}}}\mathit{d}\mathit{\alpha }$

Use Poisson’s formula (Problem 21b) and Problem 20 to show that$\mathbf{\sum }_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{n}\mathbf{\theta }}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{=}\mathit{\pi }\mathit{\theta }\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{0}\mathbf{<}\mathit{\theta }\mathbf{<}\mathit{\pi }$.(This sum is needed in the theory of scattering of light in a liquid.) Hint: Consider$\mathit{f}\left(x\right)$and$\mathit{g}\left(\alpha \right)$as in Problem 20. Note that$\mathit{f}\left(2k\pi \right)\mathbf{=}\mathbf{0}$except for$\mathit{k}\mathbf{=}\mathbf{0}$if$\mathit{a}\mathbf{<}\mathit{\pi }$. Put$\mathit{\alpha }\mathbf{=}\mathit{n}$,$\mathit{a}\mathbf{=}\mathit{\theta }$.