Question Number 86830 by Ar Brandon last updated on 31/Mar/20

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)](https://www.tinkutara.com/question/Q86834.png)
Commented by Ar Brandon last updated on 01/Apr/20

Commented by mathmax by abdo last updated on 01/Apr/20

Commented by Ar Brandon last updated on 02/Apr/20

Answered by TANMAY PANACEA. last updated on 31/Mar/20

Commented by Ar Brandon last updated on 01/Apr/20
