4\. Use the generating function for the Legendre polynomials to evaluate $$ \int_{-1}^{1} P_{l}(w) P_{l^{\prime}}(w) d w $$

Orthogonality is a core property of Legendre polynomials. It means that the integral of the product of two different polynomials over a certain interval is zero. This property is crucial because it simplifies many calculations by reducing cross-terms to zero.

For Legendre polynomials, this interval is \([-1, 1]\). Specifically: \[ \int_{-1}^{1} P_l(w) P_{l^{\prime}}(w) \, dw = 0 \] if \l \eq l\'. This is helpful because orthogonality enables us to isolate individual terms when solving complex integrals.

When \l=l\', the integral becomes: \[ \int_{-1}^{1} P_l(w)^2 \, dw = \frac{2}{2l + 1}. \] This result gives us clear, manageable outcomes for definite integrals involving these polynomials.

A generating function is a tool that compactly encodes an infinite sequence of polynomials. For Legendre polynomials, the generating function is: \[ \frac{1}{\text{\sqrt{1 - 2wx + x^2}}} = \sum_{l=0}^{\text{\infty}} P_l(w) x^l. \]

This function is extremely useful because it allows us to work with the entire set of Legendre polynomials simultaneously. By expanding this function, we can derive individual Legendre polynomials or understand relationships between them.

In the context of our integration problem, this generating function provides the basis for applying orthogonality and evaluating the integral.

Evaluating definite integrals involving Legendre polynomials can be simplified using their orthogonality. A definite integral has specified limits of integration, making it possible to calculate the area under a curve between two points.

In our exercise, we evaluate the integral \[ \int_{-1}^{1} P_l(w) P_{l^{\prime}}(w) \, dw. \]
By applying the orthogonality of Legendre polynomials, we find:

• If \l \eq l\', then \int_{-1}^{1} P_l(w) P_{l^{\prime}}(w) \, dw = 0.
• If \l = l\', then \int_{-1}^{1} P_l(w)^2 \, dw = \frac{2}{2l + 1}.

Understanding this allows us to solve the integral more easily, focusing only on the cases where \l \eq l\' or \l = l\'.