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How to get the result?

  1. Jun 14, 2010 #1
    Example 1 in
    http://en.wikipedia.org/wiki/Chebyshev_polynomials

    to get the Chebyshev expansion coefficient of ln(1 + x) is actually to calculate
    \int_{-\pi}^{\pi}\ln\left(1+\cos\theta \right)\cos\left(n\theta \right) d\theta
    where -\pi is the singluar point of the integration.
    Using complex analysis technique, it can be changed into
    \int \left[2\ln\left(1+z \right)-\ln z -\ln2 \right]\frac{z^n+z^\left(-n\
    right)}{2z}dz
    where the integration path is the unit circle plus something else, since -1 is the branch point of ln(1 + z), just on the unit circle; 0 is the branch point of lnz and there should be a branch cut along the real axis and a infinitesmal small circle to get around the origin point. And I think the integration is divergent on this small circle.
    So I do not know how to get the result on that website.
    can anyone help?
     
  2. jcsd
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