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Math Help - Complex integral

  1. #1
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    Complex integral

    Hi,

    Problem with complex integral.Consider the integral:

    \int_{0}^{\infty}\frac{(x^a-1)}{x^3-1}\,dx

    with the branch -pi/0<phi<3pi/2 and the idented contour at z=0 and z=1

    (circular contour in the upper half plane)

    a)show that this integral can be written in terms of the integral:

    \pi\frac{(e^{2ia\pi/3}-1)}{(e^{2i\pi/3}-1)sin2\pi/3}

    +\int_{0}^{\infty}\frac{(x^ae^{ia\pi}-1)}{x^3+1}

    no problem with this.I have found it.

    b)Evaluate the second integral in part a)and fiond the value of the original

    integral.

    \frac{\pi sin(a\pi/3)}{3sin\pi/3)sin®[\pi/3(a+1)]}

    with this last integral I hav a problem.I used the contour z=x(0<x<R) the sectorcircle z=Re^i.phi(0<phi<2pi/3)and the line z=xe^2pÓ/3(0<x<R).

    I can't find the result.
    The choosen contour????????????
    Can somebody give me a suggestion?????????Thanks
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  2. #2
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    Quote Originally Posted by Gilbert View Post
    Hi,

    Problem with complex integral.Consider the integral:

    \int_{0}^{\infty}\frac{(x^a-1)}{x^3-1}\,dx
    I do not see how to do this, not even sure if it converges.
    It seems to me you are actually asking,
    \int_0^{\infty} \frac{x^a - 1}{x^3+1}dx.
    So that the denominator is defined.

    Also it might help if you say what a is? Is it 0<a<1?

    If it is like how I describe then write,
    \int_0^{\infty} \frac{x^a}{x^3+1} dx  - \int_0^{\infty} \frac{dx}{x^3+1}.
    Now use the fact,
    \int_0^{\infty} \frac{dx}{x^3+1} = \frac{2}{3}\pi.
    And we just have to compute,
    \int_0^{\infty} \frac{x^a}{x^3+1} dx.

    I think this is done by using a keyhole contour.
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  3. #3
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    Apr 2008
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    Complex integral

    Hi,

    yes it is 0<a<1.I don't think it can be done by the keyhole contout???????????
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