Question

Show that \(\int_{-1}^{1} x^{m} P_{l}(x) d x=0\) if \(m

Short answer

The integral evaluates to zero due to the orthogonality of Legendre polynomials when \( m < l \).

Step-by-step solution

- Understand Rodrigues' formula

Rodrigues' formula for Legendre polynomials is given by \[ P_l(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l. \] This will be used for further steps.

- Initial integral setup

The integral to evaluate is \[ I = \int_{-1}^{1} x^m P_l(x) \, dx. \]

- Apply integration by parts

Use integration by parts where \[ u = x^m \text{ and } dv = P_l(x) \, dx. \] Then, \[ du = m x^{m-1} \, dx \text{ and } v = \frac{1}{2^l l!} \int P_l(x) \, dx. \]

- Repeatedly differentiate and integrate

Notice that differentiating powers of \( x \) decreases the degree, while integrating the polynomial retains the highest degree term. After repeatedly applying integration by parts, a term with \( P_{l-1}(x) \) will appear.

- Examine boundary conditions

Since \( P_l(x) \) is a polynomial of degree \( l \), and the Legendre polynomial satisfies orthogonality, any polynomial of lower degree, here \( x^m \) with \( m < l \), will result in the integral equating to zero.

- Conclusion

Therefore, \[ \int_{-1}^{1} x^m P_l(x) \, dx = 0 \] if \( m < l \). The orthogonality condition of Legendre polynomials confirms this.

Key concepts

Rodrigues' formula

Rodrigues' formula is fundamental for understanding Legendre polynomials. It provides a way to generate these polynomials systematically. The formula states: \[ P_l(x) = \frac{1}{2^l l!} \frac{d^l}{dx^l} (x^2 - 1)^l \]Here,

• \(P_l(x)\) is the Legendre polynomial of degree \(l\)
• \(2^l\) indicates that the exponent corresponds to the polynomial’s degree
• \(l!\) is the factorial of \(l\)
• \(\frac{d^l}{dx^l}\) represents the \(l-\text{th}\) derivative with respect to \(x\)

This formula shows how to derive \(P_l(x)\) using the derivatives of \((x^2 - 1)^l\). Each Legendre polynomial can thus be expressed in terms of these derivatives, providing a powerful tool for further calculations.
By inserting into integrals or differential equations, Rodrigues' formula streamlines the process and lays the groundwork for more advanced mathematical operations.

Integration by Parts

Integration by parts is a key technique to solve integrals involving products of functions. The rule is generally articulated as: \[ \int u \, dv = uv - \int v \, du\]where

• \(u\) and its derivative \(du\) are parts of one function
• \(dv\) and its antiderivative \(v\) are parts of another function

Use this method to tackle the problem where \[ I = \int_{-1}^{1} x^m P_l(x) \, dx \]Assign \(u = x^m\) and \(dv = P_l(x) \, dx\). Differentiating and integrating these parts repeatedly, recalling that differentiating powers of \(x\) decreases the degree, leads to simpler functions.
This gradual breakdown generates new integrals easier to solve or reveals boundary terms that evaluate to zero due to specific limits.
This rule is crucial to simplify and evaluate the integral provided in the exercise.

Orthogonality

Orthogonality in the context of polynomials means that the integral of the product of two different polynomials over a specific interval is zero. For Legendre polynomials, this property is mathematically expressed as: \[ \int_{-1}^{1} P_l(x) P_m(x) \, dx = 0, \text{ for } l eq m \]In our exercise, it proves that \( \int_{-1}^{1} x^m P_l(x) \, dx = 0 \) if \( m < l \).
Since \(P_l(x)\) is a polynomial of degree \(l\), any polynomial of lesser degree, like \(x^m\) where \( m < l \), when multiplied and integrated over the same interval with \(P_l(x)\), results in zero.
This orthogonality is a major simplification when dealing with polynomial integrals and is widely used in solving differential equations, series expansions, and more.
These integral properties arise from the symmetry and definition of Legendre polynomials, making them invaluable in mathematical analysis.