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Question Number 86830 by Ar Brandon last updated on 31/Mar/20
Prove  that ∫_0 ^∞ ((sin x)/x)dx = (π/2)
Provethat0sinxxdx=π2
Answered by mind is power last updated on 31/Mar/20
let f(z)=(e^(iz) /z)  ∀ε,R>0   ε<R we have  ∫_C_ε  f(z)dz=2iπRes(f,0)  C_ε =]−R,−ε[∪[εe^(iθ) ,θ∈[−π,0]]]ε,+R[ ∪Re^(iθ) =2iπ  ∫_ε ^R (−f(−z)+f(z))dz=2iπ−∫_(−π) ^0 f(εe^(iθ) ).iεe^(iθ) dθ(∫_0 ^π ie^(iRcos(θ)−Rsin(θ))   ∫_ε ^R 2i((sin(x))/x)dx=2iπ−∫_0 ^π i((e^(iε(cos(θ)+isin(θ))) /(εe^(iθ) ))).εe^(iθ) dθ−∫_0 ^π ie^(iRcos(θ)−Rsin(θ)) dθ  2i∫_0 ^R ((sin(x))/x)dx=2iπ−lim_(ε→0) ∫_0 ^π ie^(iεe^(iθ) ) dθ−i∫_0 ^π e^(iRcos(θ)−Rsin(θ)) dθ=iπ  ⇒2i∫_0 ^∞ ((sin(z))/z)dz=iπ⇒∫_0 ^(+∞) ((sin(z))/z)=(π/2)
letf(z)=eizzϵ,R>0ϵ<RwehaveCϵf(z)dz=2iπRes(f,0)Cϵ=]R,ϵ[[ϵeiθ,θ[π,0]]]ϵ,+R[Reiθ=2iπϵR(f(z)+f(z))dz=2iππ0f(ϵeiθ).iϵeiθdθ(0πieiRcos(θ)Rsin(θ)ϵR2isin(x)xdx=2iπ0πi(eiϵ(cos(θ)+isin(θ))ϵeiθ).ϵeiθdθ0πieiRcos(θ)Rsin(θ)dθ2i0Rsin(x)xdx=2iπlimϵ00πieiϵeiθdθi0πeiRcos(θ)Rsin(θ)dθ=iπ2i0sin(z)zdz=iπ0+sin(z)z=π2
Commented by Ar Brandon last updated on 01/Apr/20
Thanks  Sir,  but  i  think  you  applied  a  theory  which  i  don′t  yet  know.  Can  you  please  emphasize  more  on  this?
ThanksSir,butithinkyouappliedatheorywhichidontyetknow.Canyoupleaseemphasizemoreonthis?
Commented by mathmax by abdo last updated on 01/Apr/20
sir ar brondon try to study complex anslysis step by step and  that the cocept bevome eazy for you...
sirarbrondontrytostudycomplexanslysisstepbystepandthatthecoceptbevomeeazyforyou
Commented by Ar Brandon last updated on 02/Apr/20
OK  thanks  so  much.
OKthankssomuch.
Answered by TANMAY PANACEA. last updated on 31/Mar/20
f(a)=∫_0 ^∞ ((sinx)/x)×e^(−ax) dx  ((df(a))/da)=∫_0 ^∞ ((sinx)/x)×e^(−ax) ×−x dx  (−1)(df/da)=∫_0 ^∞ e^(−ax) ×sinx dx    q=∫_0 ^∞ e^(−ax) ×sinx dx  p=∫_0 ^∞ e^(−ax) ×cosx dx  p+iq=∫_0 ^∞ e^(−ax) ×e^(ix) dx  =∫_0 ^∞ e^(−x(a−i)) dx  =∣(e^(−x(a−i)) /(a−i))∣_0 ^∞ =(1/(a−i))=((a+i)/(a^2 +1))  so q=∫_0 ^∞ e^(−ax) sinx dx=(1/(a^2 +1))  now ((−df(a))/da)=(1/(a^2 +1))  ∫df=∫((−da)/(a^2 +1))  f(a)=−tan^(−1) (a)+c  now look f(a)=∫_0 ^∞ ((sinx)/x)×e^(−ax)  dx  when a→∞  then e^(−ax) →0   so f(a)=0  hence..  0=−tan^(−1) (∞)+c  c=(π/2)  f(a)=−tan^(−1) (a)+(π/2)  f(a)=∫_0 ^∞ ((sinx)/x)×e^(−ax) dx=−tan^(−1) (a)+(π/2)  now look  f(0)=∫_0 ^∞ ((sinx)/x)×e^(−0)  dx=(π/2)  ∫_0 ^∞ ((sinx)/x)=(π/2)
f(a)=0sinxx×eaxdxdf(a)da=0sinxx×eax×xdx(1)dfda=0eax×sinxdxq=0eax×sinxdxp=0eax×cosxdxp+iq=0eax×eixdx=0ex(ai)dx=∣ex(ai)ai0=1ai=a+ia2+1soq=0eaxsinxdx=1a2+1nowdf(a)da=1a2+1df=daa2+1f(a)=tan1(a)+cnowlookf(a)=0sinxx×eaxdxwhenatheneax0sof(a)=0hence..0=tan1()+cc=π2f(a)=tan1(a)+π2f(a)=0sinxx×eaxdx=tan1(a)+π2nowlookf(0)=0sinxx×e0dx=π20sinxx=π2
Commented by Ar Brandon last updated on 01/Apr/20
got  it,  thanks
gotit,thanks

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