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. $$ x^{5} $$

Short answer

The polynomial \(x^5\) can be expressed as a linear combination of Legendre polynomials: \( x^5 = \frac{8}{63}P_5(x) + \frac{2}{9}P_3(x) + \frac{10}{21}P_1(x) \).

Step-by-step solution

Identify the form of Legendre polynomials

Legendre polynomials, denoted as \(P_n(x)\), are a sequence of orthogonal polynomials which satisfy a specific recurrence relation. For this exercise, we use the first few Legendre polynomials. Here are the first six: \[ P_0(x) = 1 \]\[ P_1(x) = x \]\[ P_2(x) = \frac{1}{2} (3x^2 - 1) \]\[ P_3(x) = \frac{1}{2} (5x^3 - 3x) \]\[ P_4(x) = \frac{1}{8} (35x^4 - 30x^2 + 3) \]\[ P_5(x) = \frac{1}{8} (63x^5 - 70x^3 + 15x) \]

Express \(x^5\) using \(P_5(x)\)

Starting with the highest power of \(x\), we use the expression for \(P_5(x)\):\[ P_5(x) = \frac{1}{8} (63x^5 - 70x^3 + 15x) \]To isolate \(x^5\), multiply both sides by 8 and simplify:\[ 8P_5(x) = 63x^5 - 70x^3 + 15x \]Thus, \[ x^5 = \frac{8}{63}P_5(x) + \frac{70}{63}x^3 - \frac{15}{63}x \]

Express \(x^3\) using \(P_3(x)\)

Next, express \(x^3\) using \(P_3(x)\):\[ P_3(x) = \frac{1}{2} (5x^3 - 3x) \]To isolate \(x^3\), multiply both sides by 2 and simplify:\[ 2P_3(x) = 5x^3 - 3x \]So, \[ x^3 = \frac{2}{5}P_3(x) + \frac{3}{5}x \]

Substitute \(x^3\) back into the equation

Substitute the expression for \(x^3\) from Step 3 back into the equation from Step 2:\[ x^5 = \frac{8}{63}P_5(x) + \frac{70}{63} \left( \frac{2}{5}P_3(x) + \frac{3}{5}x \right) - \frac{15}{63}x \]Simplify the equation:\[ x^5 = \frac{8}{63}P_5(x) + \frac{140}{315}P_3(x) + \frac{210}{315}x - \frac{15}{63}x \]Combine like terms:\[ x^5 = \frac{8}{63}P_5(x) + \frac{2}{9}P_3(x) + \frac{10}{21}x \]

Simplify the final expression

Rewriting the simplified expression, we get:\[ x^5 = \frac{8}{63}P_5(x) + \frac{2}{9}P_3(x) + \frac{10}{21}P_1(x) \]Note that \(P_1(x) = x\) is used here for simplicity.

Key concepts

Orthogonal Polynomials

Orthogonal polynomials are a set of polynomials where each polynomial in the set is orthogonal to the others with respect to a certain inner product. This means that the integral of the product of any two different polynomials over a specific interval is zero.

For Legendre polynomials, indicated as \( P_n(x) \), orthogonality is achieved over the interval \( [-1, 1] \). This orthogonality property can simplify various problems in mathematical physics and numerical analysis since it ensures that these polynomials are linearly independent.
The term 'orthogonal' might remind you of perpendicular lines, and in a way, these polynomials are perpendicular in a higher-dimensional vector space under the inner product defined by:
\[ \int_{-1}^{1} P_m(x)P_n(x) dx = 0 \quad \text{for} \quad m eq n \]
When writing any polynomial as a combination of Legendre polynomials, we exploit their orthogonality property to isolate each term and solve for coefficients uniquely.

Linear Combination

A linear combination involves summing several terms, each multiplied by a constant coefficient. For polynomials, this looks like combining different polynomials, each multiplied by a specific coefficient, to represent another polynomial.

• If we have polynomials \( P_1(x), P_2(x), \ldots, P_n(x) \)
• We can create a linear combination like \( c_1P_1(x) + c_2P_2(x) + \ldots + c_nP_n(x) \)

We do this when expressing a polynomial like \( x^5 \) using Legendre polynomials.
Suppose we can write \( x^5 \) as a combination of Legendre polynomials as follows:
\[x^5 = a_5P_5(x) + a_4P_4(x) + a_3P_3(x) + a_2P_2(x) + a_1P_1(x) + a_0P_0(x)\]
By finding the right coefficients \( a_i \), we can express any polynomial as a unique combination, thanks to the orthogonality of Legendre polynomials.

Polynomials

Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of the variables.

For example, \( x^5 - 2x^3 + x + 1 \) is a polynomial.
Polynomials have degrees, which is the highest power of the variable in the polynomial. For instance, the polynomial mentioned has a degree of 5.

• Legendre polynomials \( P_n(x) \) are specific types of polynomials that emerge in many physical applications. They are defined via a recurrence relation and are especially useful because of their orthogonal property.
  
• For example:
  \[P_5(x) = \frac{1}{8} (63x^5 - 70x^3 + 15x)\]
  By understanding and using Legendre polynomials, we can simplify complex expressions into manageable sums of these orthogonal polynomials, enabling easier manipulation and problem-solving in various fields of study.