Section 2.5 More on Frobenius methods (partial)
Subsection 2.5.1 Frobeius theory at regular singular points
More may end up here eventually. For now, it is sufficient to note that the method of reduction of order can be used to produce a second linearly independent solution from the first Frobenius series obtained. The form of the resulting answer will vary depending on the nature of the roots.
Theorem 2.5.1.
For \(x > 0\text{,}\) suppose that
and that \(p, q\) are analytic at 0 with a mutual radius of convergence \(R\text{.}\) Write \(p(x) = \sum p_n x^n, q(x) = \sum q_n x^n\text{.}\) Let \(r_1, r_2\) be roots of the indicial equation (2.4.1), and assume that \(r_1 \geq r_2\) if the roots are real.
Then (2.5.1) has two linearly independent solutions on at least the interval \((0,R)\text{.}\) Exactly one of the following cases holds.
- \(r_1 - r_2\) is not an integer.\begin{gather*} y_1(x) = x^{r_1} \sum_{n=0}^\infty a_n x^n, \,\,\, a_0 \neq 0\\ y_2(x) = x_{r_2} \sum_{n=0}^\infty b_n x^n, \,\,\, a_0 \neq 0. \end{gather*}
- \(r_1 = r_2 = r\text{.}\)\begin{gather*} y_1(x) = x^{r} \sum_{n=0}^\infty a_n x^n, \,\,\, a_0 \neq 0\\ y_2(x) = y_1(x) \ln x + x_{r} \sum_{n=0}^\infty b_n x^n, \,\,\, a_0 \neq 0. \end{gather*}
- \(r_1 - r_2\) a positive integer.\begin{gather*} y_1(x) = x^{r_1} \sum_{n=0}^\infty a_n x^n, \,\,\, a_0 \neq 0\\ y_2(x) = Ay_1(x) \ln x + x_{r_2} \sum_{n=0}^\infty b_n x^n, \,\,\, a_0 \neq 0. \end{gather*}
Subsection 2.5.2 Irregular points
Irregular points are important but outside the scope of this class. See this lecture for example as a starting point. The main idea is that solutions still involve Frobenius series, but now multiplied by exponential factors. That is, irregular points give rise to solutions that blow up or collapse!