let-f-t-1-a-2-t-2-witha-gt-0-give-the-fourier-transformfor-f-

Question Number 33704 by math khazana by abdo last updated on 22/Apr/18

l e t f (t) = \frac{1}{a^{2} + t^{2}} w i t h a > 0 g i v e t h e f o u r i e r

t r a n s f o r m f o r f .

Commented by prof Abdo imad last updated on 27/Apr/18

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

f o r f (x) = \frac{1}{a^{2} + x^{2}} w e g e t

F (f (x)) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} \frac{e^{- i x t}}{a^{2} + t^{2}} d t l e t c o n s i d e r t h e

c o m p l e x f u n c t i o n φ (z) = \frac{e^{- i x z}}{z^{2} + a^{2}} t h e p o l e s o f

φ a r e i a a n d - i a (s i m p l e s) s o

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i a)

φ (z) = \frac{e^{- i x z}}{(z - i a) (z + i a)} \Rightarrow R e s (φ, i a) = \frac{e^{- i x (i a)}}{2 i a}

= \frac{e^{a x}}{2 i a} \Rightarrow \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{e^{a x}}{2 i a} = \frac{π}{a} e^{a x} \Rightarrow

F (f (x)) = \frac{1}{\sqrt{2 π}} \frac{π}{a} e^{a x} = \sqrt{\frac{π^{2}}{2 π}} \frac{e^{a x}}{a}

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