Ordinary Point and Singular Point
For , is an ordinary point when and are both analytic at , and otherwise a singular point. A singular point is regular when and are analytic at , and irregular otherwise.
The point is an ordinary point of the ODE
if and are both analytic at .
[For the purpose of this course, a function is analytic at if it have a convergent Taylor series about .]
Otherwise, is a singular point.
If is a singular point, but
are analytic, then is a regular singular point; otherwise, it is an irregular singular point.
Regularity relative to equidimensional equations
An equidimensional equation has a regular singular point at . A regular singular point is therefore no more singular than in an equidimensional equation.
For , both
diverge as , so and are singular points. Yet
are analytic at with values and , so are regular singular points.
By contrast, for the ratio tends to as , so may appear ordinary; but has no second derivative at and has no Taylor series about . Hence is an irregular singular point.
Related
Stated in
- Definition 8.1 (Ordinary Point and Singular Point)§8.1 Classification of Singular Points