## Singular Point in Differential Equations

A differential equation of the form y′′ + p(x)y′ + q(x)y = 0 has a singular point at x_{0} if either of the following limits do not exist [1]:

What this means for second order differential equations is that an initial value problem will not have a unique solution. Alternatively, it may not have *any* solution, or its solution or derivatives might be discontinuous. For linear homogeneous differential equations, a singular point happens when at least one coefficient is either undefined (i.e. discontinuous) or multivalued at that point [2].

## Regular and Irregular Singular Point

Singular points can be regular or irregular. **Regular singular points** are well-behaved, defined in terms of ratios of the differential equation’s polynomial coefficients Q(x)/P(x) and R(x)/P(x) [3]. Irregular singular points exhibit bizarre behavior and cannot easily be pinned down or defined, other than to say that if a point isn’t regular, then it is irregular. To put this more concretely, a regular singular point can be defined as follows: It is where a singularity of P(x) is no worse than 1 / (x – x_{0}) and the singularity of Q(x) is no worse than 1 / (x – x_{0})^{2}. In other words, it’s where both of the following limits exist:

Otherwise, the point is an **irregular singular point.**

## Examples of Regular and Irregular Singular Points

- Example of a regular single point x
_{0}[4]:

Here, p_{2}(x) is singular but xp_{0}(x) = -1 is analytic* is x_{0}= 0 (and for all x). - Example of an irregular single point x
_{0}:

Here, x_{0}= -1 is an irregular singular point because (x + 1)p_{1}(x) is singular at x = -1.

**Example question:** Are the singular points of (x^{3} − 3x^{2})y′′ + y′ + 2y = 0 regular or irregular?

Step 1: Find the singular points. As every coefficient is a polynomial, the singular points (0 and 3) are the roots of the leading coefficient, x^{3} – 3x^{2}.

Step 2: Find the limits of each point.

x = 0 is an irregular singular point because the limit is undefined:

x = 0 is a regular singular point because both limits exist:

*Most functions you come across are “analytic.” All polynomial functions, rational functions, exponential functions, logarithmic functions, and trigonometric functions are analytic away from their singularities.

## Singular Point in Complex Analysis

A **singular point**, also called a *singularity*, is a point where a complex function isn’t analytic. In other words, it’s an obstacle to analytic continuation where the function can’t be expressed as an infinite series of powers of *z*. Singular points can be classified as regular points or irregular points (also called essential singularities).

A singular point may be an isolated point, or a point on the curve (e.g. a cusp). If there aren’t any other singular points in the neighborhood of z, the point is called an isolated singularity. In some cases, you might be able to assign a value to the discontinuity to fill in the “gap”. If that’s the case, the point is called a removable singularity.

## References

[1] Binegar, B. Lecture 19: Regular Singular Points and Generalized Power Series. Retrieved August 11, 2021 from: https://math.okstate.edu/people/binegar/4233/4233-l19.pdf

[2] Dobrushkin, V. MATHEMATICA TUTORIAL for the First Course. Part V: Singular and ordinary points. Retrieved August 11, 2021 from: https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch5/singular.html

[3] 9.2. Classifying Singular Points as Regular or Irregular. Retrieved August 11, 2021 from: https://www.oreilly.com/library/view/differential-equations-workbook/9780470472019/9780470472019_classifying_singular_points_as_regular_o.html

[4] Bertherton, C. Regular and Irregular Singular Points of ODEs. Retrieved August 11, 2021 from: https://atmos.washington.edu/~breth/classes/AM568/lect/lect15.pdf