Question

Show that $$\int_{-\infty}^{\infty} x^{2} e^{-x^{2}} H_{n}^{2}(x) d x=\sqrt{\pi} 2^{n}\left(n+\frac{1}{2}\right) n !$$

Short answer

The solution of the integral is \(\sqrt{\pi}2^{n}\left(n+\frac{1}{2}\right) n !\) as given, utilizing the properties of the Hermite polynomial, integration by parts and the concept of factorials.

Step-by-step solution

Understand Hermite polynomial properties

The Hermite polynomials \(H_{n}(x)\) satisfy the following recursions: \(H_{n+1} = 2xH_{n}(x) - 2nH_{n-1}(x)\) and \(H_{n}^{'}(x) = 2nH_{n-1}(x)\). This can be integrated by parts using \(u = H_{n}(x)H_{n}(x)\) and \(v = x^{2} e^{-x^{2}}\).

Use integration by parts

For integration by parts, the formula used is \(\int udv = uv - \int vdu\). In our case, calculating \(du\), we find a sum of terms from the product rule and simplify using the properties of Hermite polynomials. For \(dv\), we simply calculate the integral of \(x^{2} e^{-x^{2}}\) to get \(v\).

Simplify the integral

Evaluate the integral at the limit of negative and positive infinity, using the property that \(e^{-x^{2}}\) approaches zero faster than any power of \(x\). This means that all the terms that remain with \(x\) will vanish. The remaining integral can be simplified using the properties of Hermite polynomial once more.

Evaluate the factorial

The term \(n!\) can be simplified according to its definition. \(n!\) is the product of all positive integers up to \(n\).

Key concepts

Integration by Parts

Integration by parts is a fundamental technique in calculus used to compute integrals. It's especially useful for integrals involving a product of functions where a straightforward integration isn't possible. The formula for integration by parts is given by: \[ \int u \, dv = uv - \int v \, du \] where:

• \( u \) and \( v \) are differentiable functions of a variable (e.g., \( x \)).
• \( du \) is the derivative of \( u \), and \( dv \) is the derivative of \( v \) times the differential of the variable.

The choice of \( u \) and \( dv \) is crucial for simplifying the problem effectively. In cases like the current problem, where we integrate \( x^{2} e^{-x^{2}} H_{n}^{2}(x) \), \( u \) is chosen as a polynomial, allowing its derivative to reduce the degree of the polynomial. Meanwhile, \( dv \) incorporates the rest of the integrand, which is usually cumbersome to differentiate directly. This technique simplifies the integration process by reducing the complexity of the integral step by step. Once \( uv \) is calculated, the challenge often lies in evaluating the remaining integral \( \int vdu \).

Hermite Polynomial Properties

Hermite polynomials are a set of orthogonal polynomials, widely used in probability, theoretical physics, and numerical analysis. The standard Hermite polynomial, \( H_n(x) \), can be defined using a recursion relation: \[ H_{n+1}(x) = 2xH_{n}(x) - 2nH_{n-1}(x) \] These polynomials have several important properties:

• They are orthogonal with respect to the weight \( e^{-x^2} \) on the whole real line.
• The derivatives of Hermite polynomials are related by \( H_{n}'(x) = 2nH_{n-1}(x) \), simplifying calculations.

The orthogonality property means that the integral of the product of different Hermite polynomials over the same interval equals zero. This is crucial for simplifying integrals involving Hermite polynomials. In our particular exercise, these properties allow us to reduce the size and complexity of the integrand efficiently during integration by parts. Hermite polynomials thus serve as a powerful tool for solving differential equations and computing integrals in various fields.

Gaussian Integral

The Gaussian integral is a fundamental result in mathematical analysis and is given by: \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \]This result forms the backbone for many integration techniques involving exponential functions. In the context of this exercise, it confirms that as \( e^{-x^2} \) approaches zero much faster than any polynomial growth at infinity, integrals involving this terms vanish at the limits. The Gaussian integral, when involving polynomials such as \( x^2 e^{-x^2} \), can be extended with the use of Gamma functions and special properties of even and odd functions. For example, integrals involving \( x^{2n} e^{-x^2} \) result in a product of factorials and constants involving powers of \( \pi \). In practical applications, the Gaussian integral is pivotal in probability theory, where it's used to calculate probabilities and expectations for normally distributed variables, and in physics, particularly in the study of wave functions and quantum mechanics.

Recurrence Relations

Recurrence relations are equations that recursively define sequences, meaning each term is defined in terms of previous terms. For Hermite polynomials, the relation: \[ H_{n+1}(x) = 2xH_{n}(x) - 2nH_{n-1}(x) \] is essential in understanding and computing these polynomials efficiently across various applications.

• These relations simplify calculations by reducing the need to differentiate or integrate complex expressions directly.
• They are particularly useful in numerical methods, allowing iterative computation of polynomials to desired orders.

In our specific example, the recurrence relation allows expressing higher-order Hermite polynomials in terms of lower-order ones, streamlining integration by parts operations. This is particularly crucial in simplifying both terms in an integral and in ensuring that solutions are manageable without excessive computation. Understanding these recursive structures is beneficial not only in pure mathematics but also in engineering and physics where such polynomials model real-world phenomena.