Question

Express each of the following polynomials as linear combinations of Legendre polynomials. Hint: Start with the highest power of $x$ and work down in finding the correct combination. $$x-x^3$$

Short answer

$$x-x^3=\frac{2}{5}(x-P_3(x))$$

Step-by-step solution

Identify Legendre Polynomials

Legendre polynomials are denoted as $P_n(x)$ and the first few are: $P_0(x)=1$, $P_1(x)=x$, $P_2(x)=\frac{1}{2}(3x^2-1)$, $P_3(x)=\frac{1}{2}(5x^3-3x)$.

Express the highest power of $x$ using Legendre Polynomials

Notice that $x^3$ is present in the polynomial. The Legendre polynomial $P_3(x)$ contains $x^3$. Rewrite $x^3$ as it appears in $P_3(x)$: $$x^3=\frac{2}{5}{P}_3(x)+\frac{3}{5}x$$

Substitute $x^3$ into the original polynomial

Substitute the expression for $x^3$ from $P_3(x)$ into the original polynomial: $$x-x^3=x-(\frac{2}{5}{P}_3(x)+\frac{3}{5}x)$$

Simplify the expression

Combine like terms and simplify the expression: $$x-x^3=x-\frac{2}{5}{P}_3(x)-\frac{3}{5}x$$$$x-x^3=(1-\frac{3}{5})x-\frac{2}{5}{P}_3(x)$$$$x-x^3=\frac{2}{5}x-\frac{2}{5}{P}_3(x)$$

Factor out common factors

Factor out $\frac{2}{5}$ from the expression: $$x-x^3=\frac{2}{5}(x-P_3(x))$$

Key concepts

Polynomial Expansion

Polynomial expansion is a critical way to express functions as sums of simpler polynomials. For example, in the exercise, we are expanding the polynomial expression $x-x^3$. To accomplish this, we break down the original polynomial into a linear combination of Legendre Polynomials.

To express a polynomial as a sum of simpler polynomials, start by identifying the highest power of $x$ and then work your way down. In our case, the original polynomial contains $x^3$. We match this with the corresponding term in the Legendre Polynomial $P_3(x)$, which contains $x^3$. Next, observe the expansion of $x$ and adjust the coefficients accordingly.

This method makes the process of polynomial expansion systematic and manageable. This way, any polynomial can be expressed as a combination of simpler, more fundamental polynomials.

Orthogonal Polynomials

Orthogonal polynomials, like Legendre Polynomials, are sets of polynomials that are orthogonal to each other with respect to some weight function over a specified interval. This orthogonality property simplifies many problems in mathematical physics and engineering, as they can be used to solve differential equations more easily.

In our exercise, we expanded the polynomial $x-x^3$ using Legendre Polynomials. The Legendre Polynomials are orthogonal on the interval $[-1,1]$ with a weight function of 1. This means:

• $P_n(x)$ have an essential property where the integral of the product of two different Legendre Polynomials over the interval $[-1,1]$ is zero.
• This property helps in simplifying the process of finding coefficients for the polynomial expansion.

Using orthogonal polynomials allows for efficient and straightforward computation in many computational methods, making them extremely useful in various fields of study.

Mathematical Methods

Understanding and applying mathematical methods is crucial for tackling problems involving polynomial expansions and differential equations. In our exercise, we employed a step-by-step method to decompose the given polynomial into a sum of Legendre Polynomials.

Here’s a brief recap of the mathematical methods used:

• **Identifying Legendre Polynomials**: We first listed the known Legendre Polynomials to match the terms of the given polynomial.
• **Substitution Method**: We substituted parts of the given polynomial ($x^3$) with corresponding Legendre Polynomial terms to simplify the expression.
• **Simplification**: Using algebraic manipulation to combine like terms and factor out common coefficients.

These mathematical methods allow for a clear, concise, and straightforward approach to breaking down complex polynomial functions into simpler, more understandable forms. By mastering these steps, students can effectively tackle similar problems and deepen their conceptual understanding.