Fourier sine and cosine transforms of Heaviside function

  • #1
ashah99
60
2
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:
1664468467400.png

Attempt

1664468499323.png

1664468524407.png
 
Physics news on Phys.org
  • #2
ashah99 said:
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.
 
  • #3
ashah99 said:
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.
 
Back
Top