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Chapter 7: Q8P (page 384)

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

Short Answer

Expert verified

Answer

The exponential Fourier transform of the given function isg伪=1e+i伪1+i伪-1伪2and fxas a Fourier integral isfx=12蟺鈭�-鈭�鈭�e-i伪1+i伪-1伪2ei伪xd伪.

Step by step solution

01

Given information

The given function isfx=x,0<x<10,otherwise . The exponential Fourier transform of the function is to be found and the function is to be written as a Fourier integral.

02

The significance  of Fourier transforms

The following are the formulas for the Fourier series transforms.

Here,is called the Fourier transform of.

03

Find the Fourier transform

Use the formula g伪=12蟺鈭�-鈭�鈭�fxe-i伪xdxto find the Fourier trg伪=12蟺鈭�-鈭�鈭�fxe-i伪xdx=12蟺xi伪e-i伪x+1伪2e-i伪x01=12蟺e-i伪i伪+1伪2e-i伪-1=12蟺e-i伪-1+i伪e-i伪伪2

Solve further to obtain.

Now, writeas a Fourier integral using the formula.

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Most popular questions from this chapter

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.

1-e-xon(0,1)

For each of the periodic functions in Problems 5.1 to 5.11 , use Dirichlet's theorem to find the value to which the Fourier series converges atx=0,卤蟺/2,卤蟺,卤2蟺 .

Normalize f(x)in Problem 21; that is find the factor Nso that 鈭�鈭�鈭�鈭�|Nf(x)|2=1.Let 蠄(x)=Nf(x), and find 蠒(p)as given in Problem 35. Verify Parseval鈥檚 theorem, that is, show that鈭�鈭�鈭�鈭�|蠒(p)|2dp=1.

Given f(x)={x,0<x<1-2,1<x<2

a) Sketch at least three periods of the graph of the function represented by the sine series for f(x). Without finding any series, answer thefollowing question:

b) To what value does the sine series in (a) converge at x=1? At x=2? At x=0? At x=-1?

c)If the given function is continued with the period 2and then is represented by a complex exponential series 鈭�n=-鈭�鈭�Cne颈苍蟺虫, what is the value of 鈭�n=鈭�鈭�鈭�|cn|2?

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.

sinx+sin2xon(0,2蟺)
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