Physics Forums  Physics Forums ( -   Calculus & Beyond Homework
( -   -   Chebyshev Density and Potential/Runge Phenomenon (
brru25 Mar15-10 09:44 AM  Chebyshev Density and Potential/Runge
Phenomenon  Show that the integral from -1 to 1 of p(x)*log|z-x|
dx equals log|z - sqrt(z^2 -1)| / 2, where p(x) = 1 /
(pi*sqrt(1-x^2))  2. Other information  This topic comes from
Chebyshev interpolation. p(x) is the Chebyshev density.  3. The
attempt at a solution  The best idea I could come up with was to
use z = x + iy and substitute that into log|z-x| to get
log|x+iy-x| = log|iy| = log(y) since |iy| = sqrt(0^2 + y^2) =
sqrt(y^2) = y. That left me with just one term with an x to
integrate. Then I used trig substitution of x = sin(theta) to
have the integral become:  integral from -pi/2 to pi/2 of
log(y)/pi d(theta). That left me with an answer of just log(y)
which is clearly not right, or at least not in the form that
should be. I think my approach to the integration is completely
off.  bemer42 Oct16-11 08:15 PM  Re: C