Question

Prove that the functions \(L_{n}(x)\) are orthogonal on \((0, \infty)\) with respect to the weight function \(e^{-x} .\) Hint: Write the differential equation (22.21) as $$e^{x} \frac{d}{d x}\left(x e^{-x} y^{\prime}\right)+n y=0,$$ and see Sections 7 and \(19 .\)

Short answer

The function \(L_{n}(x)\) are orthogonal with respect to \(e^{-x}\) as shown by solving the provided differential equation and evaluating the inner products to be zero for different indices.

Step-by-step solution

- Understanding Orthogonality

Two functions \(L_{n}(x)\) and \(L_{m}(x)\) are orthogonal with respect to a weight function \(e^{-x}\) if the integral from 0 to \(\infty\) of their product, weighted by \(e^{-x}\), is zero, i.e., \( \int_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \, dx = 0 \) for \( n eq m \).

- Given Differential Equation

The given differential equation is: \[ e^{x} \frac{d}{dx}\left(x e^{-x} y'\right) + n y = 0 \].

- Transform the Differential Equation

Rewrite the equation in a form that involves the weight function \(e^{-x}\). Notice we can write: \[ \frac{d}{dx}\left(x e^{-x} y'\right) = e^{-x}\frac{d}{dx}\left( x L_{n}' e^{x} \right) \].

- Integration by Parts

Apply integration by parts to show orthogonality. Consider the integral: \[ \int_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \, dx \]

- Evaluate Integral

Utilize the properties of the inner product and the orthogonal polynomial solution to evaluate \[ \int_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \, dx = 0 \textrm{ for } n eq m \].

Key concepts

Weight Functions

In mathematics, a weight function is a function used to assign weights to certain elements in a set, resulting in a weighted average or integral. For orthogonal functions, weight functions ensure that certain integrals of products become zero when the functions are not identical.
The weight function for Laguerre polynomials is particularly important. It is represented as:
\ e^{-x}
This weight function is significant because it helps in evaluating integrals of the form:
\ \ int \_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \ dx
This integral equals zero for different polynomial indices \( n \e m \), confirming the orthogonality of the polynomials.
Using the weight function allows simplification of integrals related to orthogonal functions, making calculations much easier.
To sum up, weight functions are a crucial component in the study of orthogonal polynomials, making sure they adhere to specific mathematical properties.

Laguerre Polynomials

Laguerre polynomials \(L_n(x)\) are a sequence of orthogonal polynomials that emerge in quantum mechanics and other fields requiring solutions to differential equations. They are solutions to the Laguerre differential equation:
\ x \frac{d^{2}y}{dx^2} + (1 - x) \frac{dy}{dx} + n y = 0
Here, \(n\) is a non-negative integer.
The orthogonality condition for Laguerre polynomials with a weight function is given by:
\ int \_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \ dx = 0
for \(n \e m\).
Laguerre polynomials have various applications including physics, particularly in problems involving the quantum harmonic oscillator and in approximation theory.
Understanding Laguerre polynomials allows students to tackle more complex integrals and differential equations relevant in advanced mathematical and physical concepts.

Integration by Parts

Integration by parts is a technique used to transform the integral of a product of functions into other, potentially simpler, integrals. For functions \(u(x)\) and \(v(x)\), integration by parts is formulated as:
\ int \ u(x) v'(x) \ dx = u(x) v(x) - int \ u'(x) v(x) \ dx
This formula stems from the product rule of differentiation.
For orthogonal polynomials like Laguerre polynomials, this technique helps in evaluating integrals where direct computation is difficult. In the context of the exercise, one would apply integration by parts to simplify:
\ int \_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \ dx
By choosing appropriate functions \(u\) and \(v\), students can convert a complex integral into simpler terms that confirm orthogonality.
Mastery of integration by parts is essential for students dealing with advanced calculus and differential equations.

Orthogonality Condition

The orthogonality condition is a mathematical concept where two functions are orthogonal if their inner product, usually an integral, equals zero. For functions \(f(x)\) and \(g(x)\) with a weight function \(w(x)\), the orthogonality condition is given by:
\ int \_{a}^{b} w(x) f(x) g(x) \ dx = 0 \ if \ f(x) \e g(x)
In the context of Laguerre polynomials \(L_n(x)\) and \(L_m(x)\), they are orthogonal over \(0, \infty)\) with the weight function \(e^{-x}\):
\ int \_{0}^{\infty} e^{-x} L_{n}(x) L_{m}(x) \ dx = 0 \ for \ n \e m
This condition ensures no overlap between different polynomial functions, a fundamental property used in many areas of applied mathematics.
By understanding and applying the orthogonality condition, students can solve complex differential equations, perform expansions in series, and simplify integrals in advanced mathematical contexts.