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After division by $x^2$, the equation is $y^{\prime \prime }+x^{-1}{y}^{\prime }-(1+{\alpha }^2{x}^{-2})y=0$. Every $x\ne 0$ is an ordinary point. The origin is singular, but it is a regular singular point because $xP(x)=1$ and $x^{2}Q(x)=-(x^2+{\alpha }^2)$ are analytic there. An ordinary point has analytic normalized coefficients $P,Q$; a singular point failing the displayed regularity test is irregular.

The Frobenius method ansatz $y=x^r{\sum }_{n\ge 0}{a}_n{x}^n$ gives the indicial equation $r^2-{\alpha }^2=0$, $a_1=0$, andFor nonintegral $\alpha$, two independent solutions are

For integral $\alpha$, put $m=|\alpha |$. In the $r=-m$ recurrence the denominator vanishes at $n=2m$, so that series fails or coincides in the exceptional $m=0$ case. Up to scale the single Frobenius series isFor $\alpha =1$, $y_1=x(1+x^2/8+\cdots )$, soIts integral contains both $s^{-2}$ and $\log s$; multiplying by $y_1$ leaves a pole and a logarithmic term. Thus the reduction-of-order solution is not a power series at zero.

Fora finite point $z_0$ is a regular singular point when $(z-z_0)P(z)$ and $(z-z_0{)}^{2}Q(z)$ extend holomorphically to $z_0$. A singularity at infinity is classified by applying this criterion after the change of variable $w=1/z$.

For the displayed Bessel differential equation,At $z=0$, the functions $zP(z)=1$ and $z^{2}Q(z)=z^2-1/4$ are holomorphic, so zero is regular singular. To inspect infinity, write $Y(w)=y(1/w)$. The equation becomesHere $w^{2}Q(w)=w^{-2}-1/4$ is not holomorphic at zero, so infinity is an irregular singular point. Thus

Set $y=z^{-1/2}f$. Direct differentiation makes every lower-order term cancel:Hence $f^{\prime \prime }+f=0$, and on any domain carrying a branch of $\sqrt{z}$ two linearly independent solutions areNear the regular singular point, these behave as $z^{-1/2}$ and $z^{1/2}$, the two Frobenius exponents. At infinity their combinations are $z^{-1/2}{e}^{\pm iz}$; after $w=1/z$ these contain $e^{\pm i/w}$, the exponential behavior characteristic of an irregular singular point.