let-gt-0-find-the-fourier-transform-of-f-t-e-a-2-t-2-

Question Number 33705 by math khazana by abdo last updated on 22/Apr/18

l e t α > 0 f i n d t h e f o u r i e r t r a n s f o r m o f

f (t) = e^{- a^{2} t^{2}}

Commented by math khazana by abdo last updated on 22/Apr/18

F (f (x)) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} f (t) e^{- i x t} d t .

Answered by sma3l2996 last updated on 22/Apr/18

F (f (t)) (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} e^{- a^{2} t^{2} - i x t} d t = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} e^{- (a^{2} t^{2} + i x t)} d x

a^{2} t^{2} + i x t = a^{2} t^{2} + 2 \times a t \times \frac{i x}{2 a} + {(\frac{i x}{2 a})}^{2} - {(\frac{i x}{2 a})}^{2} = {(a t + \frac{i x}{2 a})}^{2} + \frac{x^{2}}{4 a^{2}}

l e t u = a t + \frac{i x}{2 a} \Rightarrow d t = \frac{d u}{a}

F (f (t)) (x) = \frac{1}{a \sqrt{2 π}} \int_{- \infty}^{\infty} e^{- u^{2} - \frac{x^{2}}{4 a^{2}}} d u = \frac{e^{- \frac{x^{2}}{4 a^{2}}}}{a \sqrt{2 π}} \int_{- \infty}^{+ \infty} e^{- u^{2}} d u

w e k n o w t h a t \int_{- \infty}^{+ \infty} e^{- x^{2}} d x = \sqrt{π}

s o

F (f (t)) (x) = \frac{\sqrt{2}}{2 a} e^{- \frac{x^{2}}{4 a^{2}}}