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Chapter 7: Q18P (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 fcof period 2b, the odd function fsof period 2b, and the function fpof period b, each of which equals f(x)on 0 < x < b . Expand each of the three functions in an appropriate Fourier series.

Short Answer

Expert verified

The even function fc of period 2b is the odd function fs of period 2b is and the function f(x) on 0 , x < b is

The graph of the 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 = 1/3.

As a result, the series is

04

Find the odd function fs.

The value of a0=0.

Thus, the series of the function is

05

Find the complete function.

The value of a0/2 = 1/3 and the other coefficients are:

Find the value of bn.

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 interval 0 < x < b as:

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Most popular questions from this chapter

In each of the following problems you are given a function on the interval-π<x<π.

Sketch several periods of the corresponding periodic function of period 2π . Expand the periodic function in a sine-cosine Fourier series.

f(x)={0,-π<x<0,sinx,0<x<π..

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

In each case, show that a particle whose coordinate is (a) x=Rez, (b) y=lmz is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion.

z=2eiπt

(a) Find the exponential Fourier transform off(x)=e−|x|and write the inverse transform. You should find

∫0∞cosαxα2+1dα=π2e−|x|

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

(c) Find the Fourier cosine transform of f(x)=11+x2. Hint: Write your result in (b) with xandαinterchanged.

The diagram shows a “relaxation” oscillator. The chargeqon the capacitor builds up until the neon tube fires and discharges the capacitor (we assume instantaneously). Then the cycle repeats itself over and over.

(a) The charge q on the capacitor satisfies the differential equation

, here R is the Resistance, C is the capacitance and Vis the

Constant d-c voltage, as shown in the diagram. Show that if q=0 when

t=0 then at any later time t (during one cycle, that is, before the neon

Tube fires),

(b) Suppose the neon tube fires at. Sketch q as a function of t for

several cycles.

(b) Expand the periodic q in part (b) in an appropriate Fourier series.
See all solutions

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