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. $$ 3 x^{2}+x-1 $$

Short answer

The polynomial is expressed as the linear combination: \( 2P_2(x) + P_1(x) - P_0(x) \).

Step-by-step solution

Identify the Legendre Polynomials

The first few Legendre polynomials are: \( P_0(x) = 1 \)\( P_1(x) = x \)\( P_2(x) = \frac{1}{2}(3x^2 - 1) \)

Match the Highest Power

Observe that the highest power in the given polynomial is \(x^2\). Match it with the highest power in the Legendre polynomials:\( P_2(x) = \frac{1}{2}(3x^2 - 1) \).We need to match the \(3x^2\) term. Notice: \( 3x^2 = 3 \cdot \frac{1}{2}(3x^2 - 1) + \frac{3}{2} \)Thus, we can say: \( 3x^2 = 2P_2(x) + \frac{3}{2} \)

Simplify the Matching Coefficients

We obtained \( 3x^2 = 2P_2(x) + \frac{3}{2} \).Now subtract \( \frac{3}{2} \):\( 3x^2 - \frac{3}{2} = 2P_2(x) \)

Combine Terms to Form the Polynomial

Next, we include the remaining terms of the polynomial. The terms left are \(x - 1\). Rewriting:\[3x^2 + x - 1 = 2P_2(x) + x - 1 + \frac{3}{2} - \frac{3}{2}\]

Express Remaining Terms as Legendre Polynomials

We notice the polynomial can now be expressed with remaining Legendre polynomials:So, \( x \) remains as \( P_1(x) \) and \(- 1\) can be rewritten as:\( -1 = -1P_0(x) \)Thus the final expression becomes:\[ 3x^2 + x - 1 = 2P_2(x) + P_1(x) - P_0(x) \]

Key concepts

Legendre polynomials

Legendre polynomials are a sequence of orthogonal polynomials that solve certain types of differential equations. These polynomials are crucial in physics and engineering, especially in contexts involving spherical coordinates. Each Legendre polynomial is denoted by \( P_n(x) \), where \( n \) is the degree. For example:

• \( P_0(x) = 1 \)
• \( P_1(x) = x \)
• \( P_2(x) = \frac{1}{2}(3x^2 - 1) \)

Notice how each polynomial includes terms of lower degrees? This hierarchy helps express other polynomials as a combination of Legendre polynomials.

Polynomial expression

A polynomial expression is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. For example, in the polynomial \( 3x^2 + x - 1 \), the terms are:

• \( 3x^2 \) (second-degree term)
• \( x \) (first-degree term)
• \( -1 \) (constant term)

To express a given polynomial as a linear combination of Legendre polynomials, match each term in the target polynomial to forms in available Legendre polynomials.

Orthogonal polynomials

Orthogonal polynomials, like Legendre polynomials, are polynomials that remain orthogonal with respect to a specific weight function over a given interval. Orthogonality implies that the integral of the product of two different polynomials over a certain interval is zero: \[\[\begin{equation}otag \int_{-1}^{1} P_m (x) P_n (x) dx = 0 \end{equation}\]\] for \( m eq n \). Since Legendre polynomials are orthogonal, they simplify solving problems involving expansions of functions, making them quite useful in approximation techniques and simplifying complex problems into manageable forms.

Mathematical methods

Mathematical methods involving Legendre polynomials often include expressing functions as series, solving differential equations, or optimizing certain expressions. For instance, in our problem:

To express \( 3x^2 + x - 1 \) as a linear combination of Legendre polynomials:

• Identify Legendre polynomials up to the necessary degree.
• Match the highest power term to the corresponding Legendre polynomial.
• Simplify and match lower power terms iteratively.

This approach breaks down the problem into easier, digestible steps, ensuring accuracy and understanding. Applying these methods gives the essential skills for tackling broader mathematical problems.