# How to get the result?

1. Jun 14, 2010

### tghg

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?