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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
let ϕ(x)= (1/( (√(a^2 −x^2 ))))  if ∣x∣<a  and ϕ(x)=0 if ∣x∣≥a  find the fourier transform of ϕ .
letφ(x)=1a2x2ifx∣<aandφ(x)=0ifx∣⩾afindthefouriertransformofφ.
Commented by abdo mathsup 649 cc last updated on 21/May/18
F(f(x)) = (1/( (√(2π))))∫_(−∞) ^(+∞)   f(t)e^(−ixt)  dt .
F(f(x))=12π+f(t)eixtdt.
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...
wehaveF(φ(x))=12π+φ(t)eixtdtandφisevenF(φ(x))o=12π+φ(t)eixtdt=12πaaeixta2t2dt=2π0acos(xt)a2t2dtletfindw(x)=0acos(xt)a2t2dtwehsvew(x)=0atsin(xt)a2t2andbypartsw(x)=[a2t2sin(xt)]0a0aa2t2xcos(xt)dt=x0aa2t2cos(xt)dtchsngementt=asinαgivew(x)=0π2acos(α)cos(axsinα)acosαdα=a20π2cos2(α)cos(axsinα)dα=a20π2cos(α)(cos(α)cos(axsinα))dα=a2{1axsin(axsin(α))cos(α)]0π20π2sin(α)sin(axsinα)dα}=0π2sin(α)sin(axsinα)dα.becontinued

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