Perturbative Methods for Modelling (M2)

Section 3.3,  p. 68, Reference Book

Describe the Frobenius method for $\fs2 L\, y = y'' +\frac{p(x)}{x} \, y' + \frac{q(x)}{x^2} \, y=0$  where $\fs2 p(x) = \sum_{n=0}^\infty p_n\, x^n$ and $\fs2 q(x) = \sum_{n=0}^\infty q_n\, x^n$ are analytic functions in a vicinity of the regular singular point $\fs2 x=0$.

Examples 3-7, p. 71-76, Reference Book

Apply the Frobenius methods for the modified Bessel equation :
$\fs2 y'' +\frac{1}{x} \, y' -\left( 1 +\frac{\nu^2}{x^2} \right) y =0$,
where $\fs2 \nu \in \mathbb R$. Consider the four cases:

 1) $\fs2 2\, \nu \notin\mathbb N$, 2) $\fs2 \nu = \frac{1}{2} + N$ with $\fs2N \in\mathbb N$, 3) $\fs2 \nu=0$ and 4) $\fs2\nu=1$.

We denote the modified Bessel function $\fs2 I _\nu(x)$ by:
$\fs2 I_\nu (x) = \sum_{n=0}^\infty \frac{\left(\frac{1}{2} \, x\right)^{2\, n + \nu}}{n!\, \Gamma(\nu+n+1)}$.