Charles Kingsley

Answered

2022-01-02

How can I find the principal value of the following?

$PV{\int}_{-\mathrm{\infty}}^{\mathrm{\infty}}\frac{{e}^{ix}}{x({x}^{2}+x+1)}dx$

Answer & Explanation

Vasquez

Expert

2022-01-09Added 457 answers

With

nick1337

Expert

2022-01-09Added 573 answers

You can use the residue theorem. Consider the complex contour integral

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

As

Only

Thus,

karton

Expert

2022-01-09Added 439 answers

and both integrals giving the real and imaginary part can be computed by using the Laplace transform. We have:

and:

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