Fourier sine and cosine transforms

[rsingh628]

Hi all, I am struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definition I have been used are provided below as well. I tried each part of the problem, but I'm only left in terms of limits as x -> infinity of sin or cos function, which are undefined. How do I approach this? Am I totally off track and missing some key properties of these transforms? Sorry for the poor formatting...any help appreciated.

Problem and definitions:


Attempt at a solution attached

 

[blamocur]

The limits you got don't exist, i.e., you cannot compute the transform by direct computation. My quick internet search showed that one trick is exploiting the fact that the derivative of the Heaviside function is Dirac delta function..

P.S. Looks like a slight of hand: in my view the transform simply does not exists, but neither does Dirac delta function

 

[rsingh628]

The limits you got don't exist, i.e., you cannot compute the transform by direct computation. My quick internet search showed that one trick is exploiting the fact that the derivative of the Heaviside function is Dirak delta function..

P.S. Looks like a slight of hand: in my view the transform simply does not exists, but neither does Dirac delta function

Thank you, and after doing some searching myself, I also stumbled on this link prior, but I'm not sure how the info there applies to finding the sine/cosine transforms in the problem statement? Seems like it's just for the Fourier transform.

 

[blamocur]

Thank you, and after doing some searching myself, I also stumbled on this link prior, but I'm not sure how the info there applies to finding the sine/cosine transforms in the problem statement? Seems like it's just for the Fourier transform.

[math]e^{i\omega t} = \cos \omega t + i \sin\omega t[/math]

 

[rsingh628]

[math]e^{i\omega t} = \cos \omega t + i \sin\omega t[/math]

Still not sure on how to approach, are you suggesting to rewrite cos(2*pi*s*x) and sin(2*pi*s*x) as complex exponentials using Euler's rule?

 

[blamocur]

Still not sure on how to approach, are you suggesting to rewrite cos(2*pi*s*x) and sin(2*pi*s*x) as complex exponentials using Euler's rule?

Yes I am, but I'd rewrite the complex exponential as a combination of [imath]\sin[/imath] and [imath]\cos[/imath], then look at the real and imaginary parts of the result. The referenced page relies on the property of the Fourier transform of derivatives. Alternately, you can ignore the paper (except for verifying the end result) and use integration by parts and the fact that [imath]H^\prime=\delta[/imath].