Problem 71 - Calculus Single Variable

Question

Involve the factorial numbers \(n !\), which were introduced in Section \(1.4 .\) They can be defined by \(0 !=1\) and, for a positive integer \(n, n !=n(n-1) \ldots 3 \ldots 2 \cdot 1\). For each nonnegative integer \(n,\) the Hermite polynomial \(H_{\mathrm{n}}\) is defined by $$H_{n}(x)=(-1)^{n} e^{x^{2}} \frac{d^{n}}{d x^{n}} e^{-x^{2}}$$ Calculate \(H_{0}(x), H_{1}(x), H_{2}(x), H_{3}(x),\) and \(H_{4}(x) .\)

Step-by-Step Solution

Answer

The Hermite polynomials are: \( H_0(x) = 1 \), \( H_1(x) = 2x \), \( H_2(x) = 4x^2 - 2 \), \( H_3(x) = 8x^3 - 12x \), and \( H_4(x) = 16x^4 - 48x^2 + 12 \).

Step 1: Understanding Hermite Polynomial at n=0

For Hermite polynomial when \( n = 0 \), we have \( H_0(x) = (-1)^0 e^{x^2} \frac{d^0}{dx^0} e^{-x^2} \). Since the zeroth derivative of any function is the function itself, we have \( H_0(x) = e^{x^2} \cdot e^{-x^2} = 1 \).

Step 2: Finding Hermite Polynomial at n=1

For \( n = 1 \), the expression becomes \( H_1(x) = (-1)^1 e^{x^2} \frac{d}{dx} e^{-x^2} \). Calculate the derivative \( \frac{d}{dx} e^{-x^2} = -2x e^{-x^2} \). Thus, \( H_1(x) = - e^{x^2}(-2x e^{-x^2}) = 2x \).

Step 3: Calculating Hermite Polynomial at n=2

For \( n = 2 \), \( H_2(x) = (-1)^2 e^{x^2} \frac{d^2}{dx^2} e^{-x^2} \). First, find the first derivative as \(-2x e^{-x^2}\), and then differentiate again: \(-2 e^{-x^2} + 4x^2 e^{-x^2} \). Replace into the Hermite formula: \( H_2(x) = e^{x^2}(4x^2 e^{-x^2} - 2 e^{-x^2}) = 4x^2 - 2 \).

Step 4: Determining Hermite Polynomial at n=3

For \( n = 3 \), we have \( H_3(x) = (-1)^3 e^{x^2} \frac{d^3}{dx^3} e^{-x^2} \). Continue from \( \frac{d^2}{dx^2} e^{-x^2} = (-2 + 4x^2) e^{-x^2} \). The third derivative becomes \((-8x + 8x^3) e^{-x^2} \). Thus, \( H_3(x) = -e^{x^2}(-8xe^{-x^2} + 8x^3 e^{-x^2}) = -8x + 8x^3 \).

Step 5: Calculating Hermite Polynomial at n=4

For \( n = 4 \), the expression is \( H_4(x) = (-1)^4 e^{x^2} \frac{d^4}{dx^4} e^{-x^2} \). Continuing from \(-8x + 8x^3\) we differentiate again: \((-8 + 24x^2 - 16x^4)e^{-x^2} \). The result is \( e^{x^2}(16x^4 - 48x^2 + 12)e^{-x^2} = 16x^4 - 48x^2 + 12 \).

Key Concepts

Factorial Numbers

Factorial numbers are a fundamental concept in mathematics, particularly in combinatorics and algebra. They are denoted by the symbol "!" and represent the product of all positive integers up to a specified number. For instance, the factorial of a nonnegative integer \( n \), written as \( n! \), is calculated as \( n \times (n-1) \times (n-2) \ldots 3 \times 2 \times 1 \). This series of multiplications continues until reaching 1. There is a special case where the factorial of zero, \( 0! \), is defined as 1. This might seem unintuitive, but it's an essential convention for many mathematical purposes, ensuring formulas work correctly in various situations.

Factorials grow very quickly as \( n \) increases, which is why they are often used in permutations and combinations, as well as in series expansions such as those found in advanced calculus and algebra. In the context of Hermite polynomials, factorials help control the growth rate and ensure the correct mathematical structure when deriving expressions.

Differentiation

Differentiation is a core concept in calculus, and it refers to the process of finding the derivative of a function. The derivative represents the rate of change of a function concerning its variable, typically denoted as \( x \). In Hermite polynomials, differentiation is a key step because each polynomial involves taking multiple derivatives of an exponential function.

When you differentiate a function like \( e^{-x^2} \), you repeatedly apply differentiation rules to find higher-order derivatives. For example, the first derivative \( \frac{d}{dx} e^{-x^2} \) equals \(-2x e^{-x^2} \), using the chain rule. Continuing this process for second, third, and fourth derivatives involves careful differentiation, where each step builds upon the previous one. Differentiation in Hermite polynomials helps simplify the expressions and derive the exact polynomial forms.

• Derivatives are calculations of instantaneous rates of change.
• Used extensively in physics to understand motion and forces.
• Essential in engineering for optimizing designs and processes.

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation. They are foundational in algebra and have the general form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( x \) is the variable.

In the context of Hermite polynomials, these are specific polynomials defined recursively, using derivatives and exponential functions. The Hermite polynomials are especially important in physics, particularly in quantum mechanics, where they appear in the solutions of the quantum harmonic oscillator.

• Polynomials can represent real-life quantities, like distances, areas, and volumes.
• Their graphs produce smooth, continuous curves.
• Used in calculus to approximate complex functions.

Exponential Functions

Exponential functions are mathematical functions characterized by a constant base raised to a variable exponent. In most contexts, the base of the exponential is \( e \), an irrational and transcendental number approximately equal to 2.71828. These functions are significant because they model growth and decay processes, such as population growth, radioactive decay, and interest calculations.

In Hermite polynomials, exponential functions play a dual role. The function \( e^{x^2} \) and its inverse \( e^{-x^2} \) are crucial for defining the polynomials.
The exponential function in the Hermite polynomial formula ensures that at each differentiation stage, the function behaves predictably, helping achieve the desired polynomial structure.

• Exponential growth describes things increasing at a consistent percentage rate.
• Exponential decay describes processes where quantities decrease over time, like cooling or depreciation values.
• Essential for describing dynamics in natural, scientific, and finance scenarios.