Question

A proof of the formula in (4.61) for the norms of the Legendre polynomials is based on the following steps. (a) First, prove that $\left\|R_{k, k}\right\|^2=(-1)^k(2 k) ! \int_{-1}^1\left(t^2-1\right)^k d t$ by a repeated integration by parts. (b) Second, prove that $\int_{-1}^1\left(t^2-1\right)^k d t=(-1)^k \frac{2^{2 k+1}(k !)^2}{(2 k+1) !}$ by using the change of variables $t=\cos \theta$ in the integral. The resulting trigonometric integral can be done by another repeated integration by parts. (c) Finally, use the Rodrigues formula to complete the proof.

    A proof of the formula in (4.61) for the norms of the Legendre polynomials is based on the following steps. (a) First, prove that $\left\|R_{k, k}\right\|^2=(-1)^k(2 k) ! \int_{-1}^1\left(t^2-1\right)^k d t$ by a repeated integration by parts. (b) Second, prove that $\int_{-1}^1\left(t^2-1\right)^k d t=(-1)^k \frac{2^{2 k+1}(k !)^2}{(2 k+1) !}$ by using the change of variables $t=\cos \theta$ in the integral. The resulting trigonometric integral can be done by another repeated integration by parts. (c) Finally, use the Rodrigues formula to complete the proof.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 4, Problem 9

Instant Answer

Step 1

- Start with the integral $\int_{-1}^1 (t^2-1)^k dt$. - Use integration by parts, where $u = (t^2-1)^k$ and $dv = dt$. Then, $du = k(t^2-1)^{k-1} \cdot 2t \, dt$ and $v = t$. - The integration by parts formula is $\int u \, dv = uv - \int v \, du$. - Applying Show more…

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A proof of the formula in (4.61) for the norms of the Legendre polynomials is based on the following steps. (a) First, prove that $\left\|R_{k, k}\right\|^2=(-1)^k(2 k) ! \int_{-1}^1\left(t^2-1\right)^k d t$ by a repeated integration by parts. (b) Second, prove that $\int_{-1}^1\left(t^2-1\right)^k d t=(-1)^k \frac{2^{2 k+1}(k !)^2}{(2 k+1) !}$ by using the change of variables $t=\cos \theta$ in the integral. The resulting trigonometric integral can be done by another repeated integration by parts. (c) Finally, use the Rodrigues formula to complete the proof.

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