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Chapter 7: Q21P (page 371)

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.

Short Answer

Expert verified

The even function is , the odd function is

and the proper function isThe graph of these functions is shown below:

Step by step solution

01

Given Information

The given function is

02

Meaning of Fourier series.

A periodic function f(x) is expanded using the Fourier series formula in terms of an infinite sum of sines and cosines. Any periodic function or periodic signal is decomposed into the sum of a set of simple oscillating functions, mainly sines and cosines. The formula for a function's Fourier series is as follows:

03

Find the even function fc .

The average value of the function is a₀/2 = 1/2, and the other values are:

As a result, the series is

04

Find the odd function fs.

The value of a₀ = 0 and then the value of b_n:

Thus. the series of the function is

05

Find the complete function.

Again, the value of a₀/2 = 1/2 and the other coefficients are:

Find the value of b_n.

Thus, the series of the function is

06

Plotting the functions fc, fs and fp on the graph.

Now sketch the even function the odd function and the function on the graph as:

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Most popular questions from this chapter

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

Starting with the symmetrized integrals as in Problem 34, make the substitutions $α = \frac{2 π p}{h}$ (where pis the new variable, his a constant), $f (x) = ψ (x)$ , localid="1664270725133" $g (α) = \sqrt{\frac{h}{2 π}} ϕ (p)$ ; show that then

$\begin{matrix} ψ (x) = \frac{1}{\sqrt{h}} \int_{- \infty}^{\infty} ϕ (p) e^{\frac{2 π i p x}{h}} d p \\ ϕ (p) = \frac{1}{\sqrt{h}} \int_{- \infty}^{\infty} ψ (x) e^{- \frac{2 π i p x}{h}} d x \\ \int_{- \infty}^{\infty} {| ψ (x) |}^{2} d x = \int_{- \infty}^{\infty} {| ϕ (p) |}^{2} d p \end{matrix}$

This notation is often used in quantum mechanics.

In Problems13 to 16, find the Fourier cosine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.15)]. Verify that the cosine integral for f(x)is the same as the exponential integral found previously.

13. Problem 4

(a) Represent as an exponential Fourier transform the function

$f (x) = {\begin{cases} \begin{array}{l} \sin x, & 0 < x < π \end{array} \\ \begin{array}{l} 0, & otherwise \end{array} \end{cases}$

Hint: write $\sin x$ in complex exponential form.

(b) Show that your result can be written as

$f (x) = \frac{1}{π} \int_{0}^{\infty} \frac{\cos α x + \cos α (x - π)}{1 - α^{2}} d α$ .

From the fact you know, find in your head the average value of

a) $x^{3} - 3sinh2x + {sin}^{2} πx + cos3πx$ on $(- 5,5)$

b) ${2sin}^{2} 3x - 4cosx + 5xcosh2x - {xcos}^{2} x$ on $(- π,π)$

See all solutions

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