Chapter 7: Fourier Series and Transforms

For each of the periodic functions in Problems5.1to 5.11, use Dirichlet's theorem to find the value to which the Fourier series converges at $\mathit{x}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{\pm }\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{\pm }\mathit{\pi }\mathbf{,}\mathbf{}\mathbf{\pm }\mathbf{2}\mathit{\pi }$.

Let $\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathit{e}}^{\mathbf{i}\mathbf{\omega }\mathbf{t}}$on$\mathbf{(}\mathbf{-}\mathbf{\pi }\mathbf{,}\mathbf{\pi }\mathbf{)}$. Expand$\mathit{f}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$in a complex exponential Fourier series of period $\mathbf{2}\mathit{\pi }$. (Assume $\mathit{w}\mathbf{\ne }$integer.)

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance from the origin is the given function.

$\mathit{s}\mathbf{=}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{2}\mathit{t}\mathbf{+}\mathit{\pi }\mathbf{/}\mathbf{8}\mathbf{)}\mathbf{+}\mathbf{3}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{2}\mathit{t}\mathbf{-}\mathit{\pi }\mathbf{/}\mathbf{8}\mathbf{)}$

In Problems 4 to 10, the sketches show several practical examples of electrical signals (voltages or currents). In each case we want to know the harmonic content of the signal, that is, what frequencies it contains and in what proportions. To find this, expand each function in an appropriate Fourier series. Assume in each case that the part of the graph shown is repeated sixty times per second.

6. Triangular wave; the graph consists of two straight lines whose equations you must write! The maximum voltage of occurs at the middle of the cycle.

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 $\left(-\pi ,\pi \right)$and verify in each case that the answer is equivalent to the one found in Section 5.

Find the exponential Fourier transform of the given f(x) and write f(x) as a Fourier integral [that is, find $\mathbf{\text{g}}\mathbf{\left(}\mathbf{\text{α}}\mathbf{\right)}$in equation (12.2) and substitute your result into the first integral in equation (12.2)]

role="math" localid="1664338250973" $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\begin{array}{ll}\mathbf{x}\mathbf{,}& \mathbf{\left|}\mathbf{x}\mathbf{\right|}\mathbf{<}\mathbf{1}\end{array}\\ \begin{array}{ll}\mathbf{0}\mathbf{,}& \mathbf{\left|}\mathbf{x}\mathbf{\right|}\mathbf{>}\mathbf{1}\end{array}\end{array}$

In each of the following problems you are given a function on the interval $\mathbf{-}\mathbf{\pi }\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{\pi }$. Sketch several periods of the corresponding periodic function of period$\mathbf{2}\mathit{\tau }\mathit{\tau }$ . Expand the periodic function in a sine-cosine Fourier series,

role="math" localid="1659239194875" $\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{\pi }\mathbf{<}\mathbf{x}\mathbf{<}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{and}\mathbf{}\mathbf{0}\mathbf{<}\mathbf{x}\mathbf{<}\frac{\mathbf{\pi }}{\mathbf{2}}\\ \mathbf{0}\mathbf{,}\mathbf{}\mathbf{-}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{\pi }\end{array}$

Use a trigonometry formula to write the two terms as a single harmonic. Find the period and amplitude. Compare computer plots of your result and the given problem.

$\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{x}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\left(x+\pi /3\right)$

In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the x axis are the same.

$\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{x}\mathbf{}\mathit{o}\mathit{n}\mathbf{}\left(0,\pi \right)$

Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series $\mathbf{\sum }_{\mathbf{n}\mathbf{=}\mathbf{1}}^{\mathbf{\infty }}\frac{\mathbf{1}}{{\mathbf{n}}^{\mathbf{4}}}$ ,using problem 9.9.