Question

Derive the orthogonality of Legendre polynomials directly from the differential equation they satisfy.

Short answer

The orthogonality of Legendre polynomials is proven by the integral of the product of two different Legendre polynomials \(P_m(x)\) and \(P_n(x)\) in the interval \[-1, 1\], which turns out to be zero, given \(m\) not equal to \(n\). This is achieved by conducting integration by parts twice, with the boundary terms disappearing due to the defined boundary conditions.

Step-by-step solution

Definition of Legendre Differential Equation

The Legendre differential equation is given by: \[ (1 - x^2) y'' - 2x y' + n(n+1) y = 0 \] where \(y''\) is the second derivative with respect to \(x\), \(y'\) is the first derivative, and \(n\) is the order of the polynomial. The constants \(n(n+1)\) stand for the eigenvalues. Legendre polynomials, represented as \(P_n(x)\), are the solutions to this differential equation.

Orthogonality Condition

Two functions \(f(x)\) and \(g(x)\) are orthogonal in an interval \([-a, a]\) if their integral over that interval is zero, which means \[\int_{-a}^{a} f(x)g(x) dx = 0\]. In the context of Legendre polynomials, they are orthogonal over the interval \([-1, 1]\). The task is to prove that \[\int_{-1}^{1} P_m(x) P_n(x) dx = 0\] for \(m \neq n\).

Integrate by Parts to Prove Orthogonality

We can integrate by parts two times. The formula for integration by parts is \[\int u dv = uv - \int v du\]. Define \(u\) to be \(P_m(x)\), and \(dv\) to be the derivative of \(P_n(x)\) multiplied by \(d x\). Integrate two times and apply the boundary terms to see that they disappear (due to the boundary conditions of the Legendre polynomials). Since \(m\) and \(n\) are not equal, the integral is zero, thus proving the orthogonality of Legendre polynomials.

Key concepts

Legendre Differential Equation

The Legendre differential equation plays a crucial role in mathematical physics and engineering. It is a type of second-order differential equation given by \[ (1 - x^2) y'' - 2x y' + n(n+1) y = 0 \] where the terms represent successive derivatives of a function \(y\) with respect to \(x\), and \(n\) denotes the degree of the Legendre polynomial, which is also associated with the eigenvalue in this context.

The solutions to this equation are the well-known Legendre polynomials \(P_n(x)\). These polynomials are significant because they form a set of orthogonal basis functions over the interval \([-1, 1]\), making them especially useful for solving problems involving spherical geometries or when expanding functions into series for approximation purposes.

Eigenvalues in Differential Equations

Understanding eigenvalues in the context of differential equations is essential. Eigenvalues are special numbers that are part of the solution to matrix equations, but they also appear in differential equations as fixed values which, for a given linear operator, characterize the scale factor by which a corresponding eigenfunction is stretched.

In the Legendre differential equation, the term \(n(n+1)\) represents the eigenvalues. For each non-negative integer value of \(n\), there is an associated Legendre polynomial, which is the eigenfunction. The concept of eigenvalues and eigenfunctions is fundamental in many areas of physics and engineering, explaining phenomena such as resonance and wave behavior.

Orthogonality Condition in Function Spaces

The notion of orthogonality in function spaces extends the familiar concept of perpendicular vectors in three-dimensional space to an infinite-dimensional context. Two functions \(f(x)\) and \(g(x)\) are said to be orthogonal on an interval if the integral of their product over that interval is zero:

\[\int_{-a}^{a} f(x)g(x) dx = 0\]

For Legendre polynomials, they satisfy the orthogonality condition within the interval \([-1, 1]\). This property is vital for solving problems in Fourier series, quantum mechanics, and other areas where orthogonal functions are used to represent complex functions or signals. The orthogonality implies that the functions do not interfere with each other, which is particularly useful in modal analysis and separation of variables technique.

Integration by Parts

Integration by parts is a technique often used to transform the integral of a product of functions into an integral that is easier to evaluate. The formula is derived from the product rule of differentiation and is expressed as:

\[\int u dv = uv - \int v du\]

This technique is leveraged in the proof of the orthogonality of Legendre polynomials. By carefully choosing \(u\) and \(dv\) and applying the formula correctly, we can shift from an intimidating integral involving Legendre polynomials to a more manageable form. In our context, after integrating twice by parts and evaluating at the boundaries, we can demonstrate that the integral equals zero, thus confirming the orthogonality between different Legendre polynomials.