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 \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\gamma(\gamma+1) y=0\) The Legendre polynomials play an important role in mathematical physics. For example, in solving Laplace's equation (the potential equation) in spherical coordinates we encounter the equation $$ \frac{d^{2} F(\varphi)}{d \varphi^{2}}+\cot \varphi \frac{d F(\varphi)}{d \varphi}+n(n+1) F(\varphi)=0, \quad 0<\varphi<\pi $$ where \(n\) is a positive integer. Show that the change of variable \(x=\cos \varphi\) leads to the Legendre equation with \(\alpha=n\) for \(y=f(x)=F(\arccos x) .\)

Short answer

In conclusion, by relating the derivatives of \(F(\varphi)\) and \(f(x)\), then substituting and simplifying, we have shown that the given equation associated with Laplace's equation can be transformed into a Legendre equation with \(\alpha = n\) when the variable \(x=\cos \varphi\).

Step-by-step solution

Relate derivatives of \(F(\varphi)\) and \(f(x)\)

Since \(x = \cos{\varphi}\), we have \(\varphi = \arccos{x}\). To find the relationship between \(F(\varphi)\) and \(f(x)\), we need to relate their derivatives. We know that: $$ f(x) = F(\arccos{x}) $$ Differentiating \(f(x)\) with respect to \(x\): $$ \frac{df(x)}{dx} = \frac{dF(\arccos{x})}{d(\arccos{x})} \frac{d(\arccos{x})}{dx} $$ In order to simplify, notice that \(\frac{d(\arccos{x})}{dx} = -\frac{1}{\sqrt{1-x^2}}\) and \(\frac{dF(\arccos{x})}{d(\arccos{x})} = \frac{dF(\varphi)}{d\varphi}\). Therefore, $$ \frac{df(x)}{dx} = -\frac{1}{\sqrt{1-x^2}} \frac{dF(\varphi)}{d\varphi}. $$ Now, differentiating \(\frac{df(x)}{dx}\) with respect to \(x\): $$ \frac{d^2f(x)}{dx^2} = -\frac{1}{\sqrt{1-x^2}} \frac{d^2F(\varphi)}{d\varphi^2} - 2x \frac{dF(\varphi)}{d\varphi} \frac{1}{(1-x^2)^{3/2}}. $$

Substitute relationships into the given equation

Given the equation, $$ \frac{d^{2} F(\varphi)}{d \varphi^{2}}+\cot \varphi \frac{d F(\varphi)}{d \varphi}+n(n+1) F(\varphi)=0, $$ we now substitute known values involving \(f(x)\) and rearrange the terms: $$ -\sqrt{1-x^2} \frac{d^2f(x)}{dx^2} + 2x \frac{df(x)}{dx} + \frac{x}{\sqrt{1-x^2}} \frac{df(x)}{dx} + n(n+1) f(x) = 0. $$

Simplify to obtain Legendre equation with \(\alpha = n\)

In this step, we will simplify the equation to obtain the Legendre equation. $$ -\sqrt{1-x^2} \frac{d^2f(x)}{dx^2} + \left( 2x + \frac{x}{\sqrt{1-x^2}} \right) \frac{df(x)}{dx} + n(n+1) f(x) = 0. $$ Multiplying this entire equation by \(-\sqrt{1-x^2}\) yields: $$ \left(1-x^2\right) \frac{d^2f(x)}{dx^2}- 2x\frac{df(x)}{dx} - n(n+1) (1-x^2) f(x)=0, $$ which is the Legendre equation for \(y= f(x)\) with \(\alpha=n\).

Key concepts

Ordinary Points

An ordinary point in the context of differential equations is where the behavior of the solution is typical and well-behaved. For the Legendre equation \( (1-x^2) y'' - 2xy' + \alpha(\alpha +1)y = 0 \), an ordinary point is where the coefficients of the equation are analytic, meaning they can be expressed as a power series. In this equation, the point \(x = 0\) is an ordinary point. This is because \(P(x) = 1 - x^2\) has no zero at \(x=0\), making it possible to find a power series solution at this point. Essentially, at ordinary points, you can expect solutions to behave in a nice, predictable manner, typically expressible as a power series.

Radius of Convergence

The radius of convergence is a critical concept in series solutions of differential equations. It refers to the distance from the center of convergence (usually an ordinary point) within which the power series solution is valid. For the Legendre equation, \( P(x) = 1 - x^2 \) determines the radius of convergence because it defines where the singularities (or zeros) occur. Since the closest singularity to \( x = 0 \) is at \( x = 1 \), the radius of convergence is at least 1. This distance ensures that within this radius, the series solution will reliably approximate the true behavior of the function, and does not diverge, providing accurate mathematical and physical interpretations.

Spherical Coordinates

Spherical coordinates are a system of three-dimensional coordinates, used typically for things like finding the position of a point in space. The coordinates are given by \((r, \theta, \varphi)\), where \(r\) is the radius, \(\theta\) is the azimuthal angle, and \(\varphi\) is the polar angle. They are useful in solving problems with spherical symmetry, such as spherical shells or regions. When solving Laplace's equation or other potential equations in spherical coordinates, the substitution \( x = \cos \varphi \) simplifies the problem, potentially transforming it into a form like the Legendre equation. This is useful because it allows known solutions like Legendre polynomials to be applied, linking together spherical coordinates with polynomial solutions.

Laplace's Equation

Laplace's equation \( abla^2 V = 0 \) is a second-order partial differential equation that appears frequently in physics, especially in problems concerning gravity and electromagnetism. It states that the divergence of the gradient of \(V\) is zero, indicating that \(V\) is a harmonic function. In spherical coordinates, solving Laplace's equation means encountering the equation \( \frac{d^2 F(\varphi)}{d \varphi^2} + \cot \varphi \frac{d F(\varphi)}{d \varphi} + n(n+1) F(\varphi) = 0 \). Through the substitution \( x = \cos \varphi \), this form becomes the Legendre equation. Thus, the ability to use Legendre polynomials serves as a powerful method to solve these physical problems, showcasing the broad applicability of harmonic functions in various coordinate systems.