What is the Fourier Transform of sin?

[asi123]

Homework Statement

Hey guys.
I need to find the Fourier transform of sin, is this right?

http://img156.imageshack.us/img156/5531/scan0004r.jpg

I searched the internet but all I could find is the answer with the dirac delta and I don't need that.

Thanks.

Homework Equations

The Attempt at a Solution

 

[dx]

The Fourier transform of sin(t) involves the Dirac delta function. What do you mean by "I don't need that"? And why did you change the limits from -∞ to ∞ to -π to π in your integral?

 

[asi123]

The Fourier transform of sin(t) involves the Dirac delta function. What do you mean by "I don't need that"? And why did you change the limits from -∞ to ∞ to -π to π in your integral?

Oh, sorry, I need to find it from -pi to pi.
Is there something wrong with what I did?

Thanks.

 

[dx]

I didn't read your whole solution, but there is a mistake in your first step. The Fourier transform integral goes from -∞ to ∞. Why did you change those limits?

 

[asi123]

I didn't read your whole solution, but there is a mistake in your first step. The Fourier transform integral goes from -∞ to ∞. Why did you change those limits?

Yeah, I need to find it from -pi to pi.
Is that way it doesn't involves Dirac function?

Thanks.

 

[dx]

No! It's not from -pi to pi. It's -∞ to ∞.

 

[asi123]

No! It's not from -pi to pi. It's -∞ to ∞.

But that is the question.
Find Fourier transform of sin in -pi<t<pi.

What do you mean?

Thanks.

 

[Cyosis]

Your question is to transform the function $$f(t) = \left\{ \begin{matrix} \sin t & \mathrm{if}\; -\pi < t < \pi \\ 0 & \mathrm{otherwise} \end{matrix} \right$$ ?

 

[asi123]

Your question is to transform the function $$f(t) = \left\{ \begin{matrix} \sin t & \mathrm{if} -\pi < t < \pi \\ 0 & \mathrm{otherwise} \end{matrix} \right$$ ?

Yeah, sorry for the misconfusion.

 

[Cyosis]

Then your approach is correct since the function is zero outside -pi<t<pi anyway so you may as well integrate from -pi to pi.

 

[dx]

Ah, now it makes sense! Thanks Cyosis!

 

[Cyosis]

You're welcome.