let-x-1-a-2-x-2-if-x-lt-a-and-x-0-if-x-a-find-the-fourier-transform-of-

Question Number 35632 by abdo mathsup 649 cc last updated on 21/May/18

l e t φ (x) = \frac{1}{\sqrt{a^{2} - x^{2}}} i f ∣ x ∣< a a n d φ (x) = 0 i f ∣ x ∣⩾ a

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 φ .

Commented by abdo mathsup 649 cc last updated on 21/May/18

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

Commented by abdo mathsup 649 cc last updated on 24/May/18

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

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

= \frac{1}{\sqrt{2 π}} \int_{- a}^{a} \frac{e^{- i x t}}{\sqrt{a^{2} - t^{2}}} d t

= \sqrt{\frac{2}{π}} \int_{0}^{a} \frac{c o s (x t)}{\sqrt{a^{2} - t^{2}}} d t l e t f i n d

w (x) = \int_{0}^{a} \frac{c o s (x t)}{\sqrt{a^{2} - t^{2}}} d t w e h s v e

w^{^{'}} (x) = \int_{0}^{a} \frac{- t s i n (x t)}{\sqrt{a^{2} - t^{2}}} a n d b y p a r t s

w^{^{'}} (x) = {[\sqrt{a^{2} - t^{2}} s i n (x t)]}_{0}^{a} - \int_{0}^{a} \sqrt{a^{2} - t^{2}} x c o s (x t) d t

= - x \int_{0}^{a} \sqrt{a^{2} - t^{2}} c o s (x t) d t c h s n g e m e n t t = a s i n α

g i v e w^{^{'}} (x) = \int_{0}^{\frac{π}{2}} a c o s (α) c o s (a x s i n α) a c o s α d α

= a^{2} \int_{0}^{\frac{π}{2}} {c o s}^{2} (α) c o s (a x s i n α) d α

= a^{2} \int_{0}^{\frac{π}{2}} c o s (α) (c o s (α) c o s (a x s i n α)) d α

= a^{2} {{\frac{1}{a x} s i n (a x s i n (α)) c o s (α)]}_{0}^{\frac{π}{2}}

- \int_{0}^{\frac{π}{2}} - s i n (α) s i n (a x s i n α) d α}

= \int_{0}^{\frac{π}{2}} s i n (α) s i n (a x s i n α) d α \dots . b e c o n t i n u e d \dots