Question

Use the generating function for the Legendre polynomials \(P_{n}(x)\) to show that $$ \int_{0}^{1} P_{2 n+1}(x) d x=(-1)^{n} \frac{(2 n) !}{2^{2 n+1} n !(n+1) !} $$ and that, except for the case \(n=0\), $$ \int_{0}^{1} P_{2 n}(x) d x=0 $$.

Short answer

For odd n, integrate using generating function for the polynomial indices and apply integral limit changes. For the even case use zero proof or evaluate by its integral limit.

Step-by-step solution

Understanding the Generating Function

The generating function for the Legendre polynomials is given by \[ \frac{1}{\root{1-2xt+t^2}} = \sum_{n=0}^{u} P_{n}(x) t^n. \]

Integral of Odd Indexed Legendre Polynomials

We need to show that \[ \int_{0}^{1} P_{2n+1}(x) dx = (-1)^n \frac{ (2n)! }{ 2^{2n+1} n! (n+1)! }. \] We need to determine the integral of the equation by first checking when given odd n. To do this perform that integral for n = 1, from the general generating function and apply on. In the result we need to see sign factor change based on the given (-1)n.

Integral of Even Indexed Legendre Polynomials

We need to show that \[ \int_{0}^{1} P_{2n}(x) dx = 0, \] for the given even values of n. For this apply the even values only on the generating function and show how the output for anything other than zero cases are canceled or summed up.

Key concepts

Generating Function for Legendre Polynomials

The generating function is a powerful tool in solving problems related to Legendre polynomials. For Legendre polynomials, the generating function is defined as \[ \frac{1}{\sqrt{1-2xt+t^2}} = \sum_{n=0}^{\infty} P_n(x) t^n. \] Here, \(P_n(x)\) represents the Legendre polynomials, and the term on the left is expressed as a series in terms of \(t\). This function helps create each polynomial in the series rapidly by manipulating the series' terms. This same generating function can be used to derive properties, asymptotic behaviors, and integrals of Legendre polynomials.

Odd Indexed Legendre Polynomials

When dealing with odd indexed Legendre polynomials, the formula to keep in mind is \[ \int_{0}^{1} P_{2n+1}(x) dx = (-1)^n \frac{ (2n)! }{ 2^{2n+1} n! (n+1)! }. \] To derive this, one essentially uses the generating function for each term. By selecting odd indices \((2n+1)\), one can separate out contributions from terms in the expansion of the generating function. You'll notice a leveling-out pattern where terms return specific values based on their symmetry properties. The key takeaway here is that the integral changes sign based on the integer \(n\), indicated by the \((-1)^n\) factor. This outcome says much about the behavior of these polynomials under integration.

Even Indexed Legendre Polynomials

For even indexed Legendre polynomials, and specifically their integrals, the result simplifies dramatically. Here, the integral of the polynomials \(P_{2n}(x)\) over the interval from 0 to 1 is given as \[ \int_{0}^{1} P_{2n}(x) dx = 0, \] except for the special case of \(n=0\). This stems from the symmetry and orthogonality of Legendre polynomials. When \(n\) is even, the polynomials exhibit specific symmetric properties causing their integral over a symmetric interval \([0,1]\) to cancel out. This result highlights an essential characteristic of even-indexed Legendre polynomials concerning their integrals.

Integral of Legendre Polynomials

A crucial aspect of Legendre polynomials is their integral properties. The tasks involve integrating polynomials of specific indices, leveraging the generating function to simplify the process. For odd-indexed polynomials, the integral results in a non-zero value involving factorial terms, as shown in the formula \[ \int_{0}^{1} P_{2n+1}(x) dx = (-1)^n \frac{ (2n)! }{ 2^{2n+1} n! (n+1)! }. \] Conversely, when dealing with even-indexed polynomials, their integrals over the interval result in zero, excluding the case where \(n=0\): \[ \int_{0}^{1} P_{2n}(x) dx = 0. \] This distinction is essential to note and comes directly from the properties and symmetries of these polynomials. Understanding these integrals provides insight into the behavior and practical applications of Legendre polynomials.