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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 Problemsto, 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.

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