Regular singular points of 2nd order ODE

[Jerbearrrrrr]

Homework Statement

[PLAIN]http://img265.imageshack.us/img265/6778/complex.png

I did the coefficient of the w' term. What about the w term?

This seems like a fairly standard thing, but I can't seem to find it anywhere.
What ansatz should I use for q, if the eqn is written w''+pw'+qw?
C/(z-a)²+ D/(z-b)²?
Any conditions, except for the one generated by z->1/t substitution?
Or should I use C+cz on the top, etc?

 

[gabbagabbahey]

Homework Statement

[PLAIN]http://img265.imageshack.us/img265/6778/complex.png

I did the coefficient of the w' term. What about the w term?

This seems like a fairly standard thing, but I can't seem to find it anywhere.
What ansatz should I use for q, if the eqn is written w''+pw'+qw?
C/(z-a)²+ D/(z-b)²?
Any conditions, except for the one generated by z->1/t substitution?
Or should I use C+cz on the top, etc?

Well, if a 2nd order ODE has regular singular points at $z=a$ and $z=b$, then $q$ has poles up to 2nd order at those points, and the most general form of $q$ is then

$$q(x)=\frac{g(z)}{(z-a)^2(z-b)^2}$$

where $g(z)$ is analytic everywhere. You should have used similar reasoning to find

[tex]p(z)=\frac{f(z)}{(z-a)(z-b)}[/itex]

Then just apply the linearity condition to find $f$ and $g$.