principal value and contour integration

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February 6th 2011, 06:06 AM
Random Variable

principal value and contour integration

I've never really understood when can you say that $\displaystyle P.V. \int^{\infty}_{-\infty} f(x) \ dx = \int^{\infty}_{-\infty} f(x) \ dx$ . And what if you can't say that? Does that mean that the integral only exists on the complex plane?
February 6th 2011, 06:23 AM
FernandoRevilla

We verify:

$\int_{-\infty}^{+\infty}f(x)dx\;\textrm{convergent}\;\Rig htarrow VP\left(\int_{-\infty}^{+\infty}f(x)dx\right)=\int_{-\infty}^{+\infty}f(x)dx$

When we integrate on the complex plane using the residues method we usually compute:

$\lim_{R \to{+}\infty}{\int_{-R}^{R}f(x)dx=VP\left(\int_{-\infty}^{+\infty}f(x)dx\right)$

So, we need to prove previously that the integral is convergent.

Fernando Revilla

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