Fourier sine and cosine transforms of Heaviside function

[ashah99]

Homework Statement

Problem statement is given below.

Relevant Equations

Relevant equation used are given below.

Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. 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:

Attempt

 

[Philip Koeck]

Homework Statement:: Problem statement is given below.
Relevant Equations:: Relevant equation used are given below.

Hi, I am really struggling with the following problem on the Fourier sine and cosine transforms of the Heaviside unit step function. The definitions I have been using are provided below. 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:
View attachment 314809
Attempt

View attachment 314810
View attachment 314811

I would try using theorems rather than just brute-forcing from the definition.
The derivative of a step-function from -1/2 to +1/2 is a delta.
There's a theorem for the FT of a derivative.
Use the shift- theorem to move between 0 and x₀.
Maybe work with the full FT and then extract the Cos and Sine-transform from the result.

Just a few ideas. I haven't tried it.

 

[vela]

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?

To get the integrals to converge, you can introduce a convergence factor $e^{-\lambda x}$ and then take the limit as $\lambda \to 0^+$ after you integrate.