Please check my solution for this Fourier transform.

[YehiaMedhat]

[math]\mathcal{F}\{ e^{-4t^2 -8t} \}[/math][imath]e^{-t^2} = \frac{2}{1+\omega^2}[/imath] and [imath]e^{-4t^2 -8t} = e^{-4(t^2+2t+1)} e^4 = e^4 e^{-4(t+1)^2}[/imath]
[imath]\therefore\ \text{Applying the time shift property: } e^4\sqrt{\pi}\ e^{i\omega} e^{\frac{\omega^2}{4}}[/imath]
[imath]\therefore\ \text{Applying the scaling property: }\mathcal{F}\{ e^4 e^{-(2(t+1))^2} \} = \frac{e^4\sqrt{\pi}}{2}\ e^{i\frac{\omega}{2}} e^{\frac{(\frac{\omega}{2})^2}{4}}[/imath]
Final answer:
[math]\frac{e^4\sqrt{\pi}}{2}\ e^{i\frac{\omega}{2}} e^{\frac{\omega^2}{16}}[/math]
Is it right or not, because the solution in the sheets provided by my TA doesn't apply the scaling property to the [imath]e^{i\omega}[/imath], so, the final answer in the sheet is: [imath]\frac{e^4\sqrt{\pi}}{2}\ e^{i\omega} e^{\frac{\omega^2}{16}}[/imath]

 

[fresh_42]

I assume that you missed the factor [imath] 2 [/imath] in the exponent of the shift formula.

 

[mario99]

[math]\mathcal{F}\{ e^{-4t^2 -8t} \}[/math][imath]e^{-t^2} = \frac{2}{1+\omega^2}[/imath] and [imath]e^{-4t^2 -8t} = e^{-4(t^2+2t+1)} e^4 = e^4 e^{-4(t+1)^2}[/imath]
[imath]\therefore\ \text{Applying the time shift property: } e^4\sqrt{\pi}\ e^{i\omega} e^{\frac{\omega^2}{4}}[/imath]
[imath]\therefore\ \text{Applying the scaling property: }\mathcal{F}\{ e^4 e^{-(2(t+1))^2} \} = \frac{e^4\sqrt{\pi}}{2}\ e^{i\frac{\omega}{2}} e^{\frac{(\frac{\omega}{2})^2}{4}}[/imath]
Final answer:
[math]\frac{e^4\sqrt{\pi}}{2}\ e^{i\frac{\omega}{2}} e^{\frac{\omega^2}{16}}[/math]
Is it right or not, because the solution in the sheets provided by my TA doesn't apply the scaling property to the [imath]e^{i\omega}[/imath], so, the final answer in the sheet is: [imath]\frac{e^4\sqrt{\pi}}{2}\ e^{i\omega} e^{\frac{\omega^2}{16}}[/imath]

Original function:
[imath]g(t) = e^{-4t^2 - 8t} = e^{4}e^{-4(t + 1)^2}[/imath]

Let [imath]f(t) = e^{-t^2}[/imath]

Shifting
[imath]\mathcal{F}\{e^{4}e^{-(t + 1)^2}\} =e^{4}\sqrt{\pi} e^{-i\omega} F(\omega)[/imath], where [imath]F(\omega) = e^{-\frac{\omega^2}{4}}[/imath]

The scaling property, you only apply it to the transformed function [imath]F(\omega)[/imath].

Scaling
[imath]\mathcal{F}\{e^{4}e^{-(2[t + 1])^2}\} = \sqrt{\pi}e^{4} e^{-i\omega} \frac{F(\frac{\omega}{2})}{|2|} = \frac{\sqrt{\pi}}{2}e^{4} e^{-i\omega} e^{-\frac{\omega^2}{16}}[/imath]