Differential Equations

Series Solutions and Special Functions

Frobenius at a regular singular point. Legendre and Bessel functions turn up here, and not by coincidence.

Updated July 2026

Most differential equations that come out of physics cannot be solved in elementary functions, and the honest response is not to give up but to lower the bar. Instead of a closed form, look for a power series and solve for its coefficients one at a time. Where the series works cleanly, and where it needs a repair, is decided entirely by the equation's singular points.

Ordinary points are easy

At an ordinary point of a second-order linear equation, one where the coefficients are well-behaved, you may simply assume

y (x) = n = 0 \sum \infty a_{n} x^{n},

substitute, and match powers of $x$ . That gives a recurrence relating $a_{n + 2}$ to earlier coefficients, and two free constants, $a_{0}$ and $a_{1}$ , that carry the two initial conditions. The series converges out to the nearest singular point and no further.

Singular points need Frobenius

At a regular singular point a plain power series usually fails, because the solution behaves like a fractional or logarithmic power there. Frobenius's fix is to allow the series to start at a shifted power:

y (x) = x^{s} n = 0 \sum \infty a_{n} x^{n}, a_{0} \neq = 0.

Substituting and reading off the lowest power gives the indicial equation, a quadratic for the exponent $s$ . Its two roots are the two allowed leading behaviours.

Definition

The indicial equation

For an equation written as $x^{2} y^{''} + x p (x) y^{'} + q (x) y = 0$ with $p, q$ analytic, the indicial equation is $s (s - 1) + p (0) s + q (0) = 0$ . When the two roots differ by a non-integer, each gives an independent Frobenius series. When they coincide or differ by an integer, the second solution can pick up a logarithm, which is the method warning you honestly about a case it cannot cover with a bare series.

Where the named functions come from

The point of all this is that the standard special functions are not handed down from nowhere. Each is the series solution of an equation that drops out of separating a partial differential equation in a particular geometry.

Consequence

Two you meet constantly

Separating Laplace's equation in spherical coordinates produces the Legendre equation, whose polynomial solutions $P_{ℓ} (cos θ)$ are the angular building blocks of every multipole field. Separating the wave or heat equation in cylindrical coordinates produces Bessel's equation, whose solutions $J_{ν} (x)$ describe the modes of a drumhead and the diffraction pattern of a circular aperture.

The Legendre polynomials terminate, becoming actual polynomials, precisely for the integer values of $ℓ$ that keep the solution finite at the poles. That termination is not a convenience; it is the boundary condition selecting a discrete set of allowed values, which is where the integer $ℓ$ in the angular momentum of quantum mechanics ultimately comes from.

The series method rarely gives the prettiest form of an answer. What it gives is a guarantee: a way to construct a solution, term by term, for equations where no shortcut exists, and a clear account of exactly where and why that construction has to be handled with care.