Fourier Transform Problem

Dec 2009
30
1
I have got the below fourier problem:

The Fourier transform of a function \(\displaystyle f(t)\) is defined as:

\(\displaystyle F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt\)

I have worked through it and got two different answers depending on whether I used a positive contour or negative contour.

For the positive contour:
\(\displaystyle j\sqrt{\frac{\pi}{2}} e^\omega\)

For the negative contour:
\(\displaystyle j\sqrt{\frac{\pi}{2}} e^{-\omega}\)

I'm not sure whether these need to be combined or which one to use for the final answer. Looked at Wolfram Alpha and this gave something completely different.


Can someone please point me in the right direction.
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
I have got the below fourier problem:

The Fourier transform of a function \(\displaystyle f(t)\) is defined as:

\(\displaystyle F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt\)

I have worked through it and got two different answers depending on whether I used a positive contour or negative contour.

For the positive contour:
\(\displaystyle j\sqrt{\frac{\pi}{2}} e^\omega\)

For the negative contour:
\(\displaystyle j\sqrt{\frac{\pi}{2}} e^{-\omega}\)

I'm not sure whether these need to be combined or which one to use for the final answer. Looked at Wolfram Alpha and this gave something completely different.


Can someone please point me in the right direction.
Just post the question as it is given please.

CB
 
Dec 2009
30
1
Just post the question as it is given please.

CB
The fourier transform of a function \(\displaystyle f(t)\) is defined as

\(\displaystyle

F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt
\)

use this formula and residue calculus to compute the Fourier transform of the function

\(\displaystyle f(t) = \frac{t}{1 + t^2}\)
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
The fourier transform of a function \(\displaystyle f(t)\) is defined as

\(\displaystyle

F(\omega) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^ \infty f(t) e^{-jwt}dt
\)

use this formula and residue calculus to compute the Fourier transform of the function

\(\displaystyle f(t) = \frac{t}{1 + t^2}\)
Think about which way you traverse the two contours and what that means for the part of the integral along the real axis, also which contour you use depends on the sign of \(\displaystyle \omega\) to make the integral over the semicircular arc go to zero, see the very similar problem on the Wikipedia page.

CB
 
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