Fourier transform f(x)=sinax/x, a>0

In summary, the question involves showing the transform of f(x)=(sinax)/x, where a>0, is 0 for |k|>a and (pi/2)^1/2 for |k|<a. The attempt at a solution involves using l'hopital's rule to evaluate the quotient at x=0 and then finding the integral (-inf, +inf) of exp[-ikx](sinax)/x. The suggestion of using contour integration is mentioned, but it is easier to show the answer by writing sin(ax)/x as (exp(iax)-exp(-iax))/2ix and comparing it to the Fourier transform of a step function.
  • #1

Homework Statement


I am trying to show given f(x)=(sinax)/x, a>0

that the transform is 0, |k|>a
(pi/2)^1/2, |k|<a

Homework Equations





The Attempt at a Solution



so far i have f transform =1/(2pi)^1/2.[integral from -inf to +inf]exp[-ikx](sinax)/x.dk, i am concerned about the singularity at x =0, does this compel me to use contour integration?
 
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  • #2
There's no singularity at 0 for sin(x)/x. (Use l'hopital's rule to give Lim x->0 =1)
 
  • #3
okay so I've used l'hopital to evaluate the quotient, so this tells me my rational function q->1 as x-> 0 right ? but i don't see how this helps me in the evaluation of the transform? I suppose I have to findintegral (-inf, +inf) of exp[-ikx](sinax)/x. First I use the fact that my integrand is an even function, I've ended up with lim R->inf of the integral(0,R) of (sinaxcoskx)/xdx. not really sure if this right and the form of the answer seems to suggest that contour integration was used??
 
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  • #4
You could do contour integration, but since you know the answer, it's easier to show it's right. Write sin(ax)/x as (exp(iax)-exp(-iax))/2ix. Now compare this expression with [tex]\int^a_{-a} e^{i k x} dk[/tex]. Do you see the relation between your function and the Fourier transform of a step function?
 

1. What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It is often used in signal processing and image analysis to analyze the frequency components of a given function.

2. What is the significance of "a" in the given function?

The "a" in the given function represents the frequency of the sine wave. A higher value of "a" will result in a higher frequency and a lower value of "a" will result in a lower frequency. The value of "a" is typically given in units of cycles per unit distance.

3. Why is "a" required to be greater than 0?

The function given, f(x)=sinax/x, is known as the sinc function and it has a singularity at x=0. Therefore, in order to avoid any mathematical complications, "a" must be greater than 0 so that the function is well-behaved and continuous.

4. What is the domain and range of the given function?

The domain of the given function is all real numbers except for x=0. The range of the function will depend on the value of "a" and will be between -1 and 1.

5. How is the Fourier transform of this function calculated?

The Fourier transform of f(x)=sinax/x can be calculated using the Fourier transform formula. However, since the function is not integrable, the Fourier transform cannot be directly calculated. Instead, it can be approximated using numerical methods such as the Fast Fourier Transform algorithm.

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