Question

The Legendre Equation. Problems 22 through 29 deal with the Legendre equation $$ \left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\alpha(\alpha+1) y=0 $$ As indicated in Example \(3,\) the point \(x=0\) is an ordinaty point of this equation, and the distance from the origin to the nearest zero of \(P(x)=1-x^{2}\) is 1 . Hence the radius of convergence of series solutions about \(x=0\) is at least 1 . Also notice that it is necessary to consider only \(\alpha>-1\) because if \(\alpha \leq-1\), then the substitution \(\alpha=-(1+\gamma)\) where \(\gamma \geq 0\) leads to the Legendre equation \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\gamma(\gamma+1) y=0\) Show that for \(n=0,1,2,3\) the corresponding Legendre polynomial is given by $$ P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{d x^{n}}\left(x^{2}-1\right)^{n} $$ This formula, known as Rodrigues' \((1794-1851)\) formula, is true for all positive integers \(n .\)

Short answer

Based on the step-by-step solution provided above, we now have verification that for n=0,1,2,3, Rodrigues' formula accurately yields the correct Legendre polynomials. These polynomials are: - \(P_0(x) = 1\) - \(P_1(x) = x\) - \(P_2(x) = \frac{3}{2}x^2 - \frac{1}{2}\) - \(P_3(x) = 5x^3 - 3x\) These polynomials demonstrate that the Rodrigues' formula holds true for the given values of \(n\).

Step-by-step solution

Compute the Legendre polynomial using Rodrigues' formula for n=0

Let \(n=0\). Then according to Rodrigues' formula, the Legendre polynomial \(P_0(x)\) is given by: $$ P_0(x) = \frac{1}{2^0 \cdot 0!} \frac{d^0}{dx^0} (x^2 - 1)^0 = 1 $$

Compute the Legendre polynomial using Rodrigues' formula for n=1

Let \(n=1\). Then according to Rodrigues' formula, the Legendre polynomial \(P_1(x)\) is given by: $$ P_1(x) = \frac{1}{2^1 \cdot 1!} \frac{d^1}{dx^1} (x^2 - 1)^1 = \frac{1}{2} \cdot (2x) = x $$

Compute the Legendre polynomial using Rodrigues' formula for n=2

Let \(n=2\). Then according to Rodrigues' formula, the Legendre polynomial \(P_2(x)\) is given by: $$ P_2(x) = \frac{1}{2^2 \cdot 2!} \frac{d^2}{dx^2} (x^2 - 1)^2 $$ First, we need to find the second derivative: $$ \frac{d^2}{dx^2} (x^2 - 1)^2 = \frac{d^1}{dx^1} \left[ 2(x^2 - 1)\cdot(2x) \right] = 12x^2 - 4 $$ Now, substituting it into the formula for \(P_2(x)\): $$ P_2(x) = \frac{1}{2^2 \cdot 2!} (12x^2 - 4) = \frac{3}{2}x^2 - \frac{1}{2} $$

Compute the Legendre polynomial using Rodrigues' formula for n=3

Let \(n=3\). Then according to Rodrigues' formula, the Legendre polynomial \(P_3(x)\) is given by: $$ P_3(x) = \frac{1}{2^3 \cdot 3!} \frac{d^3}{dx^3} (x^2 - 1)^3 $$ To find the third derivative of \((x^2 - 1)^3\), we will first find the first and second derivatives $$ \frac{d^1}{dx^1} (x^2 - 1)^3 = 6(x^2 - 1)^2 \cdot (2x) $$ $$ \frac{d^2}{dx^2} (x^2 - 1)^3 = \frac{d^1}{dx^1} \left[ 6(x^2 - 1)^2 \cdot (2x) \right] = 24x(x^2 - 1)(2x^2 - 1) $$ Now the third derivative: $$ \frac{d^3}{dx^3} (x^2 - 1)^3 = \frac{d^1}{dx^1} \left[ 24x(x^2 - 1)(2x^2 - 1) \right] = 120x^3 - 60x $$ Now, substituting it into the formula for \(P_3(x)\): $$ P_3(x) = \frac{1}{2^3 \cdot 3!} (120x^3 - 60x) = 5x^3 - 3x $$ Now, we have shown that the Rodrigues' formula for n=0,1,2,3 gives the Legendre polynomials as: - \(P_0(x) = 1\) - \(P_1(x) = x\) - \(P_2(x) = \frac{3}{2}x^2 - \frac{1}{2}\) - \(P_3(x) = 5x^3 - 3x\) Thus, we have showed that for \(n=0,1,2,3\), the corresponding Legendre polynomial is given by Rodrigues' formula.

Key concepts

Rodrigues' formula

The Rodrigues' formula is a powerful tool used for finding explicit formulas of orthogonal polynomials. It was named after the 19th-century French mathematician Olinde Rodrigues. For Legendre polynomials, Rodrigues' formula is as shown in the textbook solution:

\[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2 - 1)^n \]
This formula is particularly beneficial because it provides a methodical approach to generate Legendre polynomials for any non-negative integer value of \( n \). The process involves taking the \( n \)-th derivative of the binomial \( (x^2 - 1)^n \), which leverages the fundamental calculus operation of differentiation to construct polynomials. The significance of Rodrigues' formula lies in its clarity and efficiency in producing polynomials with desirable orthogonality properties, making it indispensable in fields such as quantum mechanics, where these polynomials often appear.

Differential equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are a cornerstone of modern sciences and engineering, modeling the behavior of physical, biological, and economic systems. The textbook problem involves the Legendre differential equation, a particular second-order ordinary differential equation (ODE) with significant applications in physics, especially in problems that exhibit spherical symmetry like the gravitational potential and the electrostatic potential.

The Legendre equation is given as: \[ (1 - x^2)y'' - 2xy' + \alpha(\alpha + 1)y = 0 \]
As the solution steps indicate, finding solutions to such ODEs often involves verifying that a function, in this case, a Legendre polynomial, fulfills the equation for specified initial or boundary conditions, which is an essential skill in both theoretical investigations and practical applications.

Series solutions

When dealing with differential equations, series solutions are a method to express the solution as an infinite sum of terms, typically powers of variables. This method is handy when a closed-form solution cannot be readily obtained. The radius of convergence plays a pivotal role in determining where the series solution is valid, which is the distance within which the series converges to a finite value.

In the context of Legendre polynomials and the corresponding differential equation provided in the exercise, series solutions are relevant because they can illustrate the behavior of the Legendre polynomials around the ordinary point \( x = 0 \). The fact that the radius of convergence is at least 1 indicates that the series solution will be reliable and meaningful within this interval for any Legendre polynomial, providing precise information about the solution's behavior and its various applications.

Orthogonal polynomials

Orthogonal polynomials are a class of polynomials that have the property of being orthogonal with respect to a given inner product on a certain interval. The concept of orthogonality is similar to that in geometry where two vectors are orthogonal if their dot product is zero. In the context of Legendre polynomials, they are orthogonal with respect to the inner product defined on the interval \( [-1, 1] \), which is critical for their numerous properties and applications.

Legendre polynomials, as generated by Rodrigues' formula, make up a sequence of orthogonal polynomials. This orthogonality means that when two different Legendre polynomials are multiplied together and integrated over the interval from -1 to 1, the result is zero. This property is symbolically expressed as: \[ \int_{-1}^{1} P_m(x) P_n(x) dx = 0 \quad \text{if} \quad m eq n \]
The orthogonality property of Legendre polynomials makes them invaluable in solving problems in physics and mathematics that involve expansions over function spaces, such as Fourier series, and in procedures like least squares fitting and numerical quadrature.