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Chapter 7: Q10P (page 384)

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

Short Answer

Expert verified

Answer

The exponential Fourier transform of the given function is g伪2i蟺伪2伪a-sin伪aand fxas a Fourier integral isfx=鈭�-鈭�鈭�伪a-sin伪ai蟺伪2ei伪xd伪.

Step by step solution

01

Given information

The graph of the given function is as shown below.

In mathematical form, the function can be written as shown below.

fx=-2x+a,x鈭�-a,0-2x-a,x鈭�0,a.

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=22蟺鈭�-鈭�鈭�fxe-i伪xdxto find the Fourier transform.

Solve further to obtain.

Now, writeas a Fourier integral using the formula.

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Most popular questions from this chapter

For each of the periodic functions in Problems 5.1to 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.

Use Parseval鈥檚 Theorem and the results of the indicated problems to find the sum of the series in Problems 5to 9

The series 132+1152+1352+..., using Problem 5.11

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

f(x)={sinx,|x|<蟺/20,|x|>蟺/2

Verify Parseval鈥檚 theorem (12.24) for the special cases in Problems 31 to 33.

32. f(x)and g(伪)as in problem 21.

Show that if (12.2) is written with the factor 12蟺multiplying each integral, then the corresponding form of Parseval鈥檚 (12.24) theorem is 鈭�-鈭�鈭�|f(x)|2dx=鈭�-鈭�鈭�|g(伪)|2d伪.
See all solutions

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