let ϕ(x)= (1/(√(a^2 −x^2 ))) if ∣x∣<a and ϕ(x)=0 if ∣x∣≥a find the fourier transform of ϕ .


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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
$we have F(ϕ(x))= (1/(√(2π))) ∫_(−∞) ^(+∞) ϕ(t) e^(−ixt) dt and ϕ is even F(ϕ(x))o= (1/(√(2π))) ∫_(−∞) ^(+∞) ϕ(t) e^(−ixt) dt =(1/(√(2π))) ∫_(−a) ^a (e^(−ixt) /(√(a^2 −t^2 ))) dt =(√(2/π)) ∫_0 ^a ((cos(xt))/(√(a^2 −t^2 )))dt let find w(x) =∫_0 ^a ((cos(xt))/(√(a^2 −t^2 )))dt we hsve w^′ (x) = ∫_0 ^a ((−t sin(xt))/(√(a^2 −t^2 ))) and by parts w^′ (x) = [ (√(a^2 −t^2 )) sin(xt)]_0 ^a −∫_0 ^a (√(a^2 −t^2 )) x cos(xt)dt = −x ∫_0 ^a (√(a^2 −t^2 )) cos(xt)dt chsngement t =asinα give w^′ (x) = ∫_0 ^(π/2) a cos(α)cos(ax sinα) a cosα dα = a^2 ∫_0 ^(π/2) cos^2 (α) cos(ax sinα)dα =a^2 ∫_0 ^(π/2) cos(α) (cos(α) cos(ax sinα))dα =a^2 { (1/(ax)) sin(ax sin(α))cos(α)]_0 ^(π/2) − ∫_0 ^(π/2) −sin(α) sin(ax sinα) dα} = ∫_0 ^(π/2) sin(α) sin(ax sinα)dα ....be continued...$
$w e h a v e F (φ (x)) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} φ (t) e^{- i x t} d t a n d φ i s e v e n$ $F (φ (x)) o = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{+ \infty} φ (t) e^{- i x t} d t$ $= \frac{1}{\sqrt{2 π}} \int_{- a}^{a} \frac{e^{- i x t}}{\sqrt{a^{2} - t^{2}}} d t$ $= \sqrt{\frac{2}{π}} \int_{0}^{a} \frac{c o s (x t)}{\sqrt{a^{2} - t^{2}}} d t l e t f i n d$ $w (x) = \int_{0}^{a} \frac{c o s (x t)}{\sqrt{a^{2} - t^{2}}} d t w e h s v e$ $w^{^{'}} (x) = \int_{0}^{a} \frac{- t s i n (x t)}{\sqrt{a^{2} - t^{2}}} a n d b y p a r t s$ $w^{^{'}} (x) = {[\sqrt{a^{2} - t^{2}} s i n (x t)]}_{0}^{a} - \int_{0}^{a} \sqrt{a^{2} - t^{2}} x c o s (x t) d t$ $= - x \int_{0}^{a} \sqrt{a^{2} - t^{2}} c o s (x t) d t c h s n g e m e n t t = a s i n α$ $g i v e w^{^{'}} (x) = \int_{0}^{\frac{π}{2}} a c o s (α) c o s (a x s i n α) a c o s α d α$ $= a^{2} \int_{0}^{\frac{π}{2}} {c o s}^{2} (α) c o s (a x s i n α) d α$ $= a^{2} \int_{0}^{\frac{π}{2}} c o s (α) (c o s (α) c o s (a x s i n α)) d α$ $= a^{2} {{\frac{1}{a x} s i n (a x s i n (α)) c o s (α)]}_{0}^{\frac{π}{2}}$ $- \int_{0}^{\frac{π}{2}} - s i n (α) s i n (a x s i n α) d α}$ $= \int_{0}^{\frac{π}{2}} s i n (α) s i n (a x s i n α) d α . . . . b e c o n t i n u e d . . .$
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