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Chapter 7: Q20P (page 364)

Each of the following functions is given over one period. Sketch several periods of the corresponding periodic function and expand it in an appropriate Fourier series.

Short Answer

Expert verified

The sketch of the function is shown below.

Step by step solution

01

Given information

The function,

02

Concept of Fourier series

In terms of an infinite sum of sines and cosines, a Fourier series is an expansion of a periodic function.

The orthogonality relationships of the sine and cosine functions are used in Fourier series.

03

Evaluate the given function with the help of Fourier series and complex co-efficient

The complex coefficients are:

04

Draw the graph

The sketch of the function is shown below.

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Most popular questions from this chapter

When a current Iflows through a resistance, the heat energydissipated per secondis the average value ofRI2. Let a periodic (not sinusoidal) current I(t) be expanded in a Fourier seriesI(t)=∑-∞∞cne120inπt.Give a physical meaning to Parseval’s theorem for this problem.

In Problem 26 and 27, find the indicated Fourier series. Then differentiate your result repeatedly (both the function and the series) until you get a discontinuous function. Use a computer to plot f(x)and the derivative functions. For each graph, plot on the same axes one or more terms of the corresponding Fourier series. Note the number of terms needed for a good fit (see comment at the end of the section).

26.f(x)={3x2+2x3,-1<x<03x2-2x3,0<x<1

Question:

  1. Let f(x) on (0,2I) satisfy f (2I -x) = f(x), that is, is symmetric about x = I. If you expand f(x) on in a sine series , ∑bnsinnπx2Ishow that for even n,bn=0 . Hint: Note that the period of the sines is 4I . Sketch an f(x) which is symmetric about x = I, and on the same axes sketch a few sines to see that the even ones are antisymmetric about X = I. Alternatively, write the integral for bn as an integral from 0 to I plus an integral from I to 2I, and replace x by 2I -x in the second integral.
  2. Similarly, show that if we define f(2l-x)=-f(x), the cosine series has an=0for even n .

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 10.

Find the exponential Fourier transform of the given f(x)and write f(x)as a Fourier integral.

f(x)={x,0<x<10,otherwise
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