quasar_4
Sep21-08, 10:04 PM
1. The problem statement, all variables and given/known data
Show that \int \frac{ln(x) dx}{x^{3/4} (1+x)} = - \sqrt{2} Pi^2
2. Relevant equations
Residue theorem - integral = 2*Pi*i * sum of residues
3. The attempt at a solution
I am so lost. I don't even know where to start. I don't understand how to construct my contour. I'm not sure what I'm supposed to avoid. I don't understand how they're finding the residue. I don't get anything about this. Except that I think (!) I can show that |z*f(z)| goes to 0 as z--> 0 and --> 0 as z --> infinity. They do this in the book, but I don't really understand why (something to do with contour radii disappearing in extreme limits). I really need a better text (we have a review book that has a PAGE on this, so it's not enough, but I don't know what books are good).
All I can think of is do some kind of substitution where z = exp(i*theta), but I don't know beyond that... the examples in my book are very confusing. I really need help. My professor is out of the country for a few weeks and gave us this to work on. I don't know what to do without being able to get help :cry:
Any help, even just pointing to a good lucid text, would be so appreciated. This is a physics class, not a math class, so it would have to be reading that was clear enough to someone without complex analysis background. Thanks!
Show that \int \frac{ln(x) dx}{x^{3/4} (1+x)} = - \sqrt{2} Pi^2
2. Relevant equations
Residue theorem - integral = 2*Pi*i * sum of residues
3. The attempt at a solution
I am so lost. I don't even know where to start. I don't understand how to construct my contour. I'm not sure what I'm supposed to avoid. I don't understand how they're finding the residue. I don't get anything about this. Except that I think (!) I can show that |z*f(z)| goes to 0 as z--> 0 and --> 0 as z --> infinity. They do this in the book, but I don't really understand why (something to do with contour radii disappearing in extreme limits). I really need a better text (we have a review book that has a PAGE on this, so it's not enough, but I don't know what books are good).
All I can think of is do some kind of substitution where z = exp(i*theta), but I don't know beyond that... the examples in my book are very confusing. I really need help. My professor is out of the country for a few weeks and gave us this to work on. I don't know what to do without being able to get help :cry:
Any help, even just pointing to a good lucid text, would be so appreciated. This is a physics class, not a math class, so it would have to be reading that was clear enough to someone without complex analysis background. Thanks!