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. $$ 7 x^{4}-3 x+1 $$

Short answer

\[ 7x^4 - 3x + 1 = \frac{56}{35} P_4(x) + 3P_2(x) - 3P_1(x) + \frac{14}{35} \]

Step-by-step solution

- Identify the Legendre Polynomials

The Legendre polynomials are commonly denoted as \(P_n(x)\) for \(n=0,1,2,3,...\). The first few are: \[P_0(x) = 1 \] \[P_1(x) = x \] \[P_2(x) = \frac{1}{2} (3x^2 - 1) \] \[P_3(x) = \frac{1}{2} (5x^3 - 3x) \] \[P_4(x) = \frac{1}{8} (35x^4 - 30x^2 + 3) \]

- Start with the highest power of x

The given polynomial is \(7x^4 - 3x + 1\). Start with \(x^4\). \[ 7x^4 = 7 \left( \frac{8}{35} P_4(x) + \frac{30}{35} x^2 - \frac{3}{35} \right) \] This simplifies to: \[ 7x^4 = \frac{56}{35} P_4(x) + 6x^2 - \frac{21}{35} \]

- Combine the constant terms

From the previous step, combine the constants with \(+1\). \[ \frac{56}{35} P_4(x) + 6x^2 - \frac{21}{35} + 1 = \frac{56}{35} P_4(x) + 6x^2 + \frac{14}{35} \]

- Address the \(-3x\)

Now consider the \(-3x\) term in the original polynomial. \[ -3x = -3P_1(x) \]

- Combine all terms

Combine all the terms: \[ 7x^4 - 3x + 1 = \frac{56}{35} P_4(x) + 6x^2 + \frac{14}{35} - 3P_1(x) \]

- Express remaining x^2 term as Legendre polynomials

Express the remaining \(x^2\) term in terms of \(P_2(x)\). \[ 6x^2 = 3 \left( \frac{1}{2} (3x^2 - 1) + 1 \right) = 3P_2(x) + 3 \]

- Final combination

Combine all the terms into the final expression: \[ 7x^4 - 3x + 1 = \frac{56}{35} P_4(x) + 3P_2(x) + 3 + \frac{14}{35} - 3P_1(x) \] Simplify the constants: \[ 7x^4 - 3x + 1 = \frac{56}{35} P_4(x) + 3P_2(x) - 3P_1(x) + \frac{119}{35} - 3 = \frac{56}{35} P_4(x) + 3P_2(x) - 3P_1(x) + \frac{14}{35} \]

Key concepts

polynomial expansion

Polynomial expansion refers to expressing a polynomial as a linear combination of simpler polynomials. This can help in solving complex problems by breaking them down into simpler, more manageable parts.
Let's consider the polynomial given in the exercise: \(7x^4 - 3x + 1\).
Our task is to express this polynomial as a combination of Legendre polynomials. Legendre polynomials are a set of orthogonal polynomials that are very useful in many areas of physics and engineering.
By starting with the highest power of \(x\) and working our way down, we can rewrite our original polynomial in terms of Legendre polynomials.
This process is called polynomial expansion, and it simplifies computations and helps us better understand the properties of polynomials.

orthogonal polynomials

Orthogonal polynomials, like Legendre polynomials, are a class of polynomials that are orthogonal under a specific inner product. Orthogonality means that the inner product of two different polynomials is zero. This property is very useful for simplifying computations and expanding functions.
In the context of our problem, the Legendre polynomials are orthogonal with respect to the weight function \(w(x) = 1\) on the interval \([-1, 1]\).
This means that integrating the product of two different Legendre polynomials over this interval results in zero:
\[ \int\_{-1}^1 P\_m(x)P\_n(x) dx = 0 \quad \text{for} \; m \eq n \]
Because of this orthogonality, Legendre polynomials can be used to expand other polynomials, making it easier to work with them in various applications like solving differential equations or performing orthogonal transformations.

Legendre polynomial series

A Legendre polynomial series is a series expansion in terms of Legendre polynomials. The general form of a Legendre polynomial series is:
\[ f(x) = \sum\_{n=0}^\infty a\_n P\_n(x) \]
where \(a_n\) are coefficients and \(P_n(x)\) are the Legendre polynomials.
In our exercise, we are given the polynomial \(7x^4 - 3x + 1\). We need to express this in terms of Legendre polynomials.
The solution involves identifying the coefficients \(a_n\) for each Legendre polynomial in the expansion:
\[ 7x^4 - 3x + 1 = \frac{56}{35}P\_4(x) + 3P\_2(x) - 3P\_1(x) + \frac{14}{35} \]
By expressing our original polynomial as a series of Legendre polynomials, we gain several advantages, including the ability to simplify calculations and connect with other orthogonal polynomial methods.
The Legendre polynomial series is a powerful tool in mathematical analysis, helping us tackle complex problems with ease.