Irregular singularity

In mathematics, in the theory of ordinary differential equations in the complex plane ${\displaystyle \mathbb {C} }$, the points of ${\displaystyle \mathbb {C} }$ are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.

More precisely, consider an ordinary linear differential equation of n-th order

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