How can I find the principal value of the following? PV∫−∞∞eixx(x2+x+1)dx

Charles Kingsley

Charles Kingsley

Answered

2022-01-02

How can I find the principal value of the following?
PVeixx(x2+x+1)dx

Answer & Explanation

Vasquez

Vasquez

Expert

2022-01-09Added 457 answers

With

p=1+3i2:P.V.eixx(x2+x+1)dx=eixx2+x+1[1x+i0++iπδ(x)]dx=eix(x+i0+)(xp)(xp)dx+iπ=iπ+2πieipp(pp)=iπ+2πiei/23/2[(1+3i)/2](i3)=iπ+2π33ei/23/213i2=(i3+3i3ei/23/2)π1.3030+2.3477i

nick1337

nick1337

Expert

2022-01-09Added 573 answers

You can use the residue theorem. Consider the complex contour integral
Cdzeizz(1+z+z2)
where C is a semicircle of radius R in the upper half plane with a small semicircular detour about the origin into the upper half plane of radius ϵ. Then the contour integral is equal to
PVRRdxeixx(1+x+x2)+iϵπ0dϕeiϕeiϵeoϕϵeiθ(1+ϵeiϕ+ϵ2ei2ϕ
As R, the third integral vanishes as 1/R3. As ϵ0, the second integral becomes iπ. By the residue theorem, the contour integral is equal to i2π times the residue of any poles inside C. The poles of the integrand are at
z2+z+1=0z=1±i32
Only z+ is inside C, with residue
ei/2e3/2ei2π/3(i3)
Thus,
PVdxeixx(1+x+x2)=2π3e3/2ei(2π/3+1/2)+iπ

karton

karton

Expert

2022-01-09Added 439 answers

PV+eixx(1+x+x2)dx=PV0+(2i(1+x2)1+x2+x4sinxx21+x2+x4cos(x))
and both integrals giving the real and imaginary part can be computed by using the Laplace transform. We have:
0+2cosx1+x2+x4dx=π3e32(3sin12+3cos12)
and:
0+1+x21+x2+x4sinxxdx=π6(3+23e32sin12)

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