## Anas.P one year ago fourier transform question the first question please as soon as possible

1. anas.p

@mathmate

2. anas.p

@e.mccormick

3. anas.p

Find the fourier Transform of $e^{-a^{2}x^{2}} ,a>0$Given $\int\limits_{-\infty}^{\infty} e^{-t} dt = \sqrt{\pi}$

4. anas.p

@zzr0ck3r @Nnesha

5. Michele_Laino

hint: first step, we have to compute this integral: $\large g\left( \omega \right) = \int_{ - \infty }^{ + \infty } {{e^{ - {a^2}{x^2}}}{e^{i\omega x}}} dx = \int_{ - \infty }^{ + \infty } {{e^{ - \left( {{a^2}{x^2} - i\omega x} \right)}}} dx$

6. anas.p

i tried... this method substitution method. integration by parts but it keeps rotating back with nothing reducing. if you could show me a step by step process of solving this i would really be grateful

7. Michele_Laino

now, we can write this: $\Large \begin{gathered} {a^2}{x^2} - i\omega x = {a^2}{x^2} - i\omega x - \frac{{{\omega ^2}}}{{4{a^2}}} + \frac{{{\omega ^2}}}{{4{a^2}}} = \hfill \\ \hfill \\ = {\left( {ax - \frac{{i\omega }}{{2a}}} \right)^2} + \frac{{{\omega ^2}}}{{4{a^2}}} \hfill \\ \end{gathered}$ so we get: $\Large \begin{gathered} g\left( \omega \right) = \int_{ - \infty }^{ + \infty } {{e^{ - {a^2}{x^2}}}{e^{i\omega x}}} dx = \int_{ - \infty }^{ + \infty } {{e^{ - \left( {{a^2}{x^2} - i\omega x} \right)}}} dx = \hfill \\ \hfill \\ = {e^{ - \frac{{{\omega ^2}}}{{4{a^2}}}}}\int_{ - \infty }^{ + \infty } {{e^{ - {{\left( {ax - \frac{{i\omega }}{{2a}}} \right)}^2}}}} dx \hfill \\ \end{gathered}$

8. Michele_Laino

now we have to make this variable change: $\Large z = ax - \frac{{i\omega }}{{2a}}$

9. Michele_Laino

where z is the new variable

10. Michele_Laino

hint: we can use this identity: $\Large \int_{ - \infty }^{ + \infty } {{e^{ - {z^2}}}} dz = 2\int_0^{ + \infty } {{e^{ - {z^2}}}dz}$

11. anas.p

Thanks man... Really helpful... I had no idea that we had to use the complete the square method. Thanks again.