Regular Singular Points and the Frobenius Method

When a singular point is "regular," the Frobenius method provides a systematic extension of power series techniques by allowing fractional and logarithmic terms.

Classification of Singular Points

Definition9.2Regular and irregular singular points

A singular point $x_0$ of $y'' + P(x)y' + Q(x)y = 0$ is regular if:

$(x - x_0)P(x) \quad \text{and} \quad (x - x_0)^2 Q(x)$

are both analytic at $x_0$ . Equivalently, $P(x)$ has at most a simple pole and $Q(x)$ has at most a double pole at $x_0$ .

A singular point that is not regular is called irregular.

ExampleClassifying singular points

Bessel equation $x^2 y'' + xy' + (x^2 - \nu^2)y = 0$ : $P = 1/x$ (simple pole), $Q = (x^2 - \nu^2)/x^2$ (double pole). So $x = 0$ is a regular singular point.
Euler equation $x^2 y'' + \alpha xy' + \beta y = 0$ : $P = \alpha/x$ , $Q = \beta/x^2$ . Regular singular point at $x = 0$ .
The equation $x^3 y'' + y = 0$ : $P = 0$ , $Q = 1/x^3$ (triple pole). So $x = 0$ is an irregular singular point. The Frobenius method does not apply here.

The Indicial Equation

Definition9.3Indicial equation

Near a regular singular point $x_0$ , write $(x - x_0)P(x) = \sum p_n(x-x_0)^n$ and $(x - x_0)^2 Q(x) = \sum q_n(x-x_0)^n$ . The indicial equation is:

$r(r-1) + p_0 r + q_0 = 0,$

whose roots $r_1, r_2$ (with $\text{Re}(r_1) \geq \text{Re}(r_2)$ ) are called the exponents (or indices) at $x_0$ .

Theorem9.2Frobenius theorem

Near a regular singular point $x_0$ , there exists at least one solution of the form:

$y_1(x) = (x - x_0)^{r_1} \sum_{n=0}^{\infty} a_n (x - x_0)^n, \quad a_0 \neq 0,$

converging for $0 < |x - x_0| < R$ . The nature of the second solution depends on the difference $r_1 - r_2$ :

$r_1 - r_2 \notin \mathbb{Z}$ : Second solution $y_2 = (x-x_0)^{r_2}\sum b_n(x-x_0)^n$ .
$r_1 - r_2 \in \mathbb{Z}_{>0}$ : Second solution may involve a logarithmic term: $y_2 = Cy_1 \ln|x-x_0| + (x-x_0)^{r_2}\sum b_n(x-x_0)^n$ .
$r_1 = r_2$ : Second solution necessarily has a logarithm: $y_2 = y_1 \ln|x-x_0| + (x-x_0)^{r_1}\sum b_n(x-x_0)^n$ .

Applications

ExampleBessel equation via Frobenius

For $x^2y'' + xy' + (x^2 - \nu^2)y = 0$ with $\nu \geq 0$ : the indicial equation at $x = 0$ is $r^2 - \nu^2 = 0$ , giving $r_1 = \nu$ , $r_2 = -\nu$ .

Substituting $y = x^\nu \sum a_n x^n$ yields the recurrence $a_{n+2} = -a_n / ((n+2)(n+2+2\nu))$ , giving the Bessel function:

$J_\nu(x) = \left(\frac{x}{2}\right)^\nu \sum_{k=0}^{\infty} \frac{(-1)^k}{k!\,\Gamma(\nu+k+1)} \left(\frac{x}{2}\right)^{2k}.$

When $2\nu \in \mathbb{Z}$ , the second solution involves logarithmic terms (Bessel functions of the second kind $Y_\nu$ ).

RemarkThe point at infinity

The point at infinity can be studied via the substitution $t = 1/x$ . The equation transforms, and $x = \infty$ corresponds to $t = 0$ . Classifying $t = 0$ for the transformed equation determines whether $x = \infty$ is ordinary, regular singular, or irregular singular. For example, the Bessel equation has an irregular singular point at $\infty$ , which accounts for the oscillatory behavior of Bessel functions for large $x$ .