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Chapter 7: Q7-13-17P (page 388)

Show that the Fourier sine transform of x-12 is . Hint: Make the change of variablez=伪x . The integral 鈭�0鈭�z-1/2sinzdzcan be found by computer or in tables

Short Answer

Expert verified

Thus, the required Fourier series is g伪=2蟺伪12蟺2

Step by step solution

01

Determine the Fourier transformation for the given function. 

Fourier sine transform of f(x) is,

g伪=2蟺鈭�0鈭�f(x)sin伪xdx=2蟺鈭�0鈭�x12sin伪xdx

Let z=伪x,dz=伪dx

So, we get,

g伪=2蟺鈭�0鈭�x12sin伪xdx=2蟺鈭�0鈭�z伪12sinzdz伪=2蟺伪12鈭�0鈭�sinzzdzg伪=2蟺伪12蟺2

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Most popular questions from this chapter

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 6.

Let f(t)=ei蝇ton(鈭�蟺,蟺). Expandf(t)in a complex exponential Fourier series of period 2蟺. (Assume w鈮�integer.)

From the fact you know, find in your head the average value of

a) x3-3sinh2x+sin2蟺虫+cos3蟺虫on (-5,5)

b)2sin23x-4cosx+5xcosh2x-xcos2xon (-蟺,蟺)

If f(x)is complex, we usually want the average of the square of the absolute value of f(x). Recall that|f(x)|2=f(x)路f(x)炉wheref(x)炉means the complex conjugate of f(x). Show that if a complexf(x)=鈭�-鈭�鈭�cnein蟺x/l, then (11.5)holds

Sketch several periods of the corresponding periodic function of period2蟺. Expand the periodic function in a sine-cosine Fourier series.

f(x)={1,-蟺<x<0,...0,0<x<蟺,....
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