- **Calculus & Beyond Homework**
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- - **Chebyshev Density and Potential/Runge Phenomenon**
(*http://www.physicsforums.com/showthread.php?t=386777*)

Chebyshev Density and Potential/Runge Phenomenon1. The problem statement, all variables and given/known dataShow 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 informationThis topic comes from Chebyshev interpolation. p(x) is the Chebyshev density. 3. The attempt at a solutionThe 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. |

Re: Chebyshev Density and Potential/Runge PhenomenonHello,
Is this question from the exercises of chapter 5 of the text 'Spectral Methods' by Trefethen? If so, I believe equation (5.9) of that text is wrong. The potential should have the form phi(z) = log(|z+sqrt(z^2 - 1)/2|) so that the minus sign is actually a plus. I'm working on the first exercise. Your question seems to be pertaining to the fifth exercise. Not sure if this helps. |

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