# Frobenius method; expansion at infinity, what happens when difference of roots of indicial equation is integer?

We can use the Frobenius method for an ODE like $$u'' + qu = 0$$ where the coefficients are functions if $q$ has a particular negative power series expansion. This is when we take infinity as a regular singular point. In this case, if the indicial equation gives us roots that integer difference, how do we write the solutions? In the case where we expand around 0, we have a term including $\ln(x)$ -- what happens in the infinity case?

-

First you make the substitution $x\to\frac{1}{z}$, transform the ODE to the one in the variable $z$, and get a solution $y(z)$ which contains some $\ln(z)$. Then you just substitute back $z\to\frac{1}{x}$ and get terms with $\ln\frac{1}{x}$.