Principle Value integral

**strammer** · January 31st 2013, 11:04 AM

Prove that $PV \int_{-\infty}^\infty e^{-ix}dx = 0$ .

**hollywood** · February 1st 2013, 11:25 PM

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

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