Please check my solution for this Fourier transform (time diff. Prop.)

[YehiaMedhat]

The main question is to transform the following expression: [imath]\frac{it}{t^2 + 4}[/imath]
1. Using the symmetry property first: [imath]2\pi \frac{i}{4} e^{-2|-\omega|}[/imath]
2. Then applying the differentiation in time property: [imath]i\frac{d}{d\omega} [\frac{i\pi}{2} e^{-2|\omega|}][/imath]
3. Differentiating: [imath]\frac{\pi}{4} e^{-2|\omega|}[/imath]
Is that right, because the solution in the TA's sheet also shows that it's different answer.

 

[mario99]

The main question is to transform the following expression: [imath]\frac{it}{t^2 + 4}[/imath]
1. Using the symmetry property first: [imath]2\pi \frac{i}{4} e^{-2|-\omega|}[/imath]
2. Then applying the differentiation in time property: [imath]i\frac{d}{d\omega} [\frac{i\pi}{2} e^{-2|\omega|}][/imath]
3. Differentiating: [imath]\frac{\pi}{4} e^{-2|\omega|}[/imath]
Is that right, because the solution in the TA's sheet also shows that it's different answer.

What is the answer in the TA's sheet?

 

[YehiaMedhat]

What is the answer in the TA's sheet?

[math]2\pi e^{-2|\omega|}u(-\omega)[/math]

 

[mario99]

[math]2\pi e^{-2|\omega|}u(-\omega)[/math]

[imath]u(-\omega)[/imath] is there because [imath]f(t)[/imath] is an odd function and we want just negative frequencies.

The main question is to transform the following expression: [imath]\frac{it}{t^2 + 4}[/imath]
1. Using the symmetry property first: [imath]2\pi \frac{i}{4} e^{-2|-\omega|}[/imath]
2. Then applying the differentiation in time property: [imath]i\frac{d}{d\omega} [\frac{i\pi}{2} e^{-2|\omega|}][/imath]
3. Differentiating: [imath]\frac{\pi}{4} e^{-2|\omega|}[/imath]
Is that right, because the solution in the TA's sheet also shows that it's different answer.

How did you get [imath]\frac{\pi}{4}[/imath] in step three?

 

[mario99]

The main question is to transform the following expression: [imath]\frac{it}{t^2 + 4}[/imath]
1. Using the symmetry property first: [imath]2\pi \frac{i}{4} e^{-2|-\omega|}[/imath]
2. Then applying the differentiation in time property: [imath]i\frac{d}{d\omega} [\frac{i\pi}{2} e^{-2|\omega|}][/imath]
3. Differentiating: [imath]\frac{\pi}{4} e^{-2|\omega|}[/imath]
Is that right, because the solution in the TA's sheet also shows that it's different answer.

If I do the transformation by integration, I get [imath]\displaystyle F(\omega) = \frac{\pi}{2}e^{-2|\omega|} \text{sgn}(\omega)[/imath]

And because [imath]f(t)[/imath] is odd, we should include a negative sign to the final answer.

[imath]\displaystyle F(\omega) = -\frac{\pi}{2}e^{-2|\omega|} \text{sgn}(\omega)[/imath]

And if you will use symmetry, the answer [imath]2\pi\frac{i}{4}e^{-2|\omega|}[/imath] is for [imath]\frac{i}{t^2 + 4}[/imath]

For [imath]\frac{it}{t^2 + 4}[/imath], the answer should be [imath]2\pi\frac{i\omega}{4}e^{-2|\omega|}[/imath]