In physics and engineering, problems involving spherical shapes—like calculating the electric field around a sphere or modeling the vibrations of a planet—often lead to the same mathematical challenge. That challenge is the Legendre differential equation.

This is a second-order linear ODE. The variable $x$ often represents a spatial coordinate (like the cosine of an angle in spherical coordinates), while $n$ is a constant that is determined by the physical constraints of the problem. Our goal is to find the function $y(x)$ that satisfies this equation.

A quick look at the coefficient of $y''$, which is $(1-x^2)$, tells us we have singular points at $x = \text{±}1$. At these points, the equation behaves strangely. However, the point $x=0$ is an ordinary point, making it a good place to build a solution using a power series.

To substitute this into the Legendre equation, we need its first and second derivatives:

Plugging these into the equation and re-indexing the sums so that each term contains $x^k$ leads to a single, unified expression. After some algebra, we find that for the entire series to equal zero, the coefficient of each power of $x$ must be zero. This requirement gives us a powerful tool: a recurrence relation.

The algebraic work yields the following recurrence relation connecting the coefficients:

Notice this relation links coefficients that are two indices apart. $a_2$ depends on $a_0$, $a_4$ depends on $a_2$, and so on. Similarly, $a_3$ depends on $a_1$, $a_5$ depends on $a_3$, etc. This splits our solution into two independent parts:

1. An even solution ($y_{even}$) built from $a_0$ with only even powers of $x$ ($x^0, x^2, x^4, \text{...}$).
2. An odd solution ($y_{odd}$) built from $a_1$ with only odd powers of $x$ ($x^1, x^3, x^5, \text{...}$).

The general solution is the sum of these two: $y(x) = y_{even}(x) + y_{odd}(x)$. The initial coefficients, $a_0$ and $a_1$, are arbitrary constants we would determine from initial conditions.

For most values of $n$, both the even and odd series continue infinitely. When analyzed, these infinite series solutions diverge (go to infinity) at the singular points $x = \text{±}1$. In physical contexts where $x$ often relates to $\cos(\theta)$, these points correspond to the poles of a sphere. A physical quantity like temperature or potential cannot be infinite, so these solutions are not physically meaningful.

There's a special case, however. Look at the numerator of the recurrence relation: $k(k+1) - n(n+1)$. If the constant $n$ is an integer, say $n=N$, then when $k$ reaches $N$, the numerator becomes zero.

When $a_{N+2}$ becomes zero, the recurrence relation ensures that all subsequent coefficients in that series ($a_{N+4}, a_{N+6}, \text{...}$) will also be zero. The infinite series truncates, becoming a finite polynomial of degree $N$.

• If $N$ is an even integer, the even series terminates, becoming a polynomial. The odd series remains an infinite, divergent series.
• If $N$ is an odd integer, the odd series terminates. The even series remains infinite.

In physical applications, we select the value of $n$ to be an integer $N$ and discard the divergent series, keeping only the finite polynomial solution. These terminating solutions are known as the Legendre polynomials, denoted $P_n(x)$. They are the well-behaved, physically realistic solutions to the Legendre equation on the interval $[-1, 1]$, first studied by Adrien-Marie Legendre.

By requiring our solutions to be physically sensible, we discover that only integer values of $n$ produce valid results. This process of finding special values (eigenvalues) that lead to well-behaved solutions (eigenfunctions) is a recurring theme in mathematical physics.

Ready to test your understanding?

This foundational understanding of the Legendre equation and its polynomial solutions is a stepping stone to solving complex problems in fields ranging from quantum mechanics to geophysics.