Series Solutions to an ODE, Ordinary Point, and Regular Singular Point — the Frobenius’s Method
This article is based on my unpublished book about Mathematics for Physics.
When you learned about the series solutions to the ordinary differential equation (ODE), you probably heard about the ordinary points and regular singular points. But in many cases, the explanation of the logic or motivation is not adequate. I hope that I can do a better job here for you.
So, what is a series solution to an ODE? Let’s consider a homogeneous linear differential equation:
, where the leading order coefficient is chosen to be unity. Roughly speaking, instead of finding the functional form of the solution y(x), we would like to assume our solution can be expended about a certain point x₀ as something like a Taylor series:
By doing so, we are turning our problem of finding y(x) into finding those coefficients aₙ in our series. However, if the actual solution y(x) is not well-behaved (analytic) at the point x₀, we may not be able to expand y(x) about this x₀ as a Taylor series from the first place. In that case, if you blindly expand it, you will find that when you try to solve aₙ, you cannot find reasonable aₙ. So, the series solution is not going to work in this case.
Now, you may ask: “how do we know whether the actual solution y(x) is well-behaved at the point x₀ or not without actually knowing the solution y(x)?” (Of course, the reason why we use series solution is exactly that we don’t know the actual solution y(x)) Thanks to a great mathematician, Lazarus Fuchs, he told us how to know whether the actual solution y(x) is well-behaved or not at a certain x₀ by looking at those coefficients A_{n −1}(x), A_{n −2}(x), . . . , A_{0}(x) in our homogeneous linear differential equation, i.e., the first equation in this article, without actually knowing the solution y(x).
Series solution about an Ordinary Point
If A_{n −1}(x), A_{n −2}(x), . . . , A_{0}(x) are all analytic in a neighborhood of x₀ in the complex plane, we call x₀ is an ordinary point of this homogeneous linear differential equation. According to Fuchs, all n linearly independent solutions to this differential equation are analytic in a neighborhood of x₀ and can be expanded as Taylor series:
Therefore, we can differentiate this formal solution and obtain the expression of its derivatives:
By substituting them back to our original problem, and requiring the equation is satisfied for each order of (x-x₀)ⁿ we can solve those aₙ, putting them into the Tayor series, thereby obtaining our solutions y(x).
Series solution about a Regular Singular Point
If x₀ is not an ordinary point, however, if all of
are analytic in the neighborhood of x₀ in the complex plane, we call x₀ is a regular singular point of this homogeneous linear differential equation. Although a solution y(x) to our ODE might still be analytic at a regular singular point x₀, it’s not necessary analytic. If it’s not, nevertheless, it doesn’t go that crazy at point x₀, and that’s the reason why we call it a “regular” singular point. Its singularity must be either a pole or an algebraic or logarithmic branch point [1]. As you learned before, e.g., if y(x) has an isolated pole at x₀, we cannot expand y(x) about x₀ as a Taylor series. Instead, we expand y(x) as a Laurent series about x₀ to catch up or mimic the behavior of the pole. Here, Fuchs tells us that there is at least one solution y(x) can be written as:
, where Y(x) is analytic at x₀, therefore can be further expanded about x₀ using Taylor series:
The wisdom here is to use (x-x₀)ˢ to catch up the singular or non-analytic behavior of y(x) at the regular singular point x₀ and the number s is called the indicial exponent and is to be determined when we solve the differential equation. In general, the indicial exponent s need not be an integer. If s is not an integer, the solution y(x) has an algebraic branch point at x₀. If s is a negative integer, y(x) has a pole at x₀. If s is a positive integer or zero, y(x) is analytic at x₀ and this series is reduced to a Taylor series. The last equation, which is a series expansion with power n+s, is called the Frobenius series, and the method using the Frobenius series to solve a homogeneous linear differential equation is called the Frobenius’s Method.
Reference
[1] Carl Bender and Steven Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory
