Question Number 149853 by 0731619 last updated on 07/Aug/21

Answered by mindispower last updated on 08/Aug/21

(1/π)∫_0 ^π e^(izcos(θ)) dθ=J_0 (z) bassel function cos^2 (θ)=((1+cos(2θ))/2) ⇔Im((1/2)∫_0 ^π e^(i(((1+cos(2x))/2))) dx.(1/2) =Im(e^(i/2) /2)(∫_0 ^(2π) e^(i((cos(x))/2)) dx) ∫_0 ^(2π) e^((icos(x))/2) dx=2∫_0 ^π e^(i((cos(x))/2)) dx we get (1/2)Ime^(i/2) ∫_0 ^π e^(i(1/2)cos(x)) dx using that ∫_0 ^π e^(i((cos(x))/2)) dx=πJ_0 ((1/2))∈R we get (1/2)Ime^(i/2) ∫_0 ^π e^(i((cos(x))/2)) dx=(π/2)sin((1/2))J_0 ((1/2))≃0.707