# Principle Value integral

• Jan 31st 2013, 12:04 PM
strammer
Principle Value integral
Prove that $PV \int_{-\infty}^\infty e^{-ix}dx = 0$.
• Feb 2nd 2013, 12:25 AM
hollywood
Re: Principle Value integral
If I remember correctly, $PV \int_{-\infty}^\infty e^{-ix}dx = \lim_{N\to\infty}\int_{-N}^N e^{-ix}dx$.

And to integrate, you treat -i as a constant $\int e^{-ix}=\frac{1}{-i}e^{-ix} = ie^{-ix}$

so $PV \int_{-\infty}^\infty e^{-ix}dx = \lim_{N\to\infty} ie^{-iN} -ie^{iN} = \lim_{N\to\infty} 2\sin{N}$

and it looks like it doesn't converge.

- Hollywood