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Complex Analysis - Cauchy Integral? Which technique do I use?

  1. Dec 19, 2012 #1
    1. The problem statement, all variables and given/known data

    [tex]\int_0^\infty\frac{x^{p-1}}{1+ x}dx[/tex]
    ** I could not get p-1 to show as the exponent; the problem is x raised to the power of
    p-1.

    [tex]\int_0^\infty\frac{ln(x) dx}{(x^2+1)^2}[/tex]


    3. The attempt at a solution

    There is no attempt, but I would like to make one! I'm asking for guidance. I've been given these integrals from my engineering professor and was told to solve them over the winter break. That by researching these integrals, I can begin to prepare myself for complex analysis next semester. Unfortunately, I'm not sure how to approach them. What complex analysis technique is used to solve these? I'd really like to read up on it as well as everything I need to know to understand the technique.. Thanks !!
     
    Last edited: Dec 20, 2012
  2. jcsd
  3. Dec 20, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Put the p-1 (or anything you want kept together) in { }, not ( ). Also, it is better to put an entire formula in Latex, not just part. The Latex for the integral sign is "\int_0^\infty".

    \int_0^\infty\frac{x^{p-1}}{1+ x} dx gives
    [tex]\int_0^\infty\frac{x^{p-1}}{1+ x} dx[/tex]

    Assuming that [itex]p\ge 1[/itex], that has a pole only at x=-1. One way to do this is to write it as part of contour integral, with the complex z in place of x, going along the upper quarter-circle [itex]|z|= \epsilon[/itex] from [itex]\epsilon i[/itex] to [itex]\epsilon[/itex], 0), the real axis from [itex]\epsilon[/itex] to R, the quarter circle of |z|= R, from R to Ri, then the imaginary axis from Ri to [itex]\epsilon i[/itex]. End by taking the limits as R goes to infinity and [itex]\epsilon[/itex] goes to 0.

     
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