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Math Help - principal value and contour integration

  1. #1
    Super Member Random Variable's Avatar
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    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?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    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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