To prove Rodrigues’s formula (52) for Legendre

Chapter 8, Problem 33E

(choose chapter or problem)

To prove Rodrigues’s formula (52) for Legendre polynomials, complete the following steps.

(a) Let \(v_{n}:=\left(d^{n} / d x^{n}\right)\left\{\left(x^{2}-1\right)^{n}\right\}\) and show that \)v_{n}(x)\) is a polynomial of degree \(n\) with the coefficient of \(x^{n}\) equal to \((2 n) ! / n !\).

(b) Use integration by parts \(n\) times to show that, for any polynomial \(q(x) \) of degree less than \(n\),

\(\int_{-1}^{1} v_{n}(x) q(x) d x=0\).

Hint: For example, when \(n=2\),

\(\int_{-1}^{1} \frac{d^{2}}{d x^{2}}\left\{\left(x^{2}-1\right)^{2}\right\} q(x) d x\)

\(=\left.q(x)\frac{d}{dx}\left\{\left(x^{2}-1\right)^{2}\right\}\right|_{-1}  ^{1}-\left.\left\{q^{\prime}(x)\left(x^{2}-1\right)^{2}\right\}\right|_{-1} ^{1}\)

\(+\int_{-1}^{1} q^{\prime \prime}(x)\left(x^{2}-1\right)^{2} d x\),

Since \(n=2 \), the degree of \(q(x)\) is at most 1, and so \(q^{\prime \prime}(x) \equiv 0\). Thus

\(\int_{-1}^{1} \frac{d^{2}}{d x^{2}}\left\{\left(x^{2}-1\right)^{2}\right\} q(x) d x=0\)

(c) Use the result of Problem 31(b) to conclude that \(P_{n}(x)=c v_{n}(x)\) and show that \(c=1 / 2^{n} n !\)

by comparing the coefficients of  \(x^{n} \operatorname{in} P_{n}(x) \text { and } v_{n}(x)\)

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Text Transcription:

vn:=(dn/dxn){(x2-1) n}  

vn(x)    

n    

xn  

(2n)!/n!  

n    

q(x)  

n

-11vn(x)q(x)dx=0    

n=2  

\int-11d2dx2{(x2-1)2}q(x)dx

=q(x)ddx{(x2-1) 2}|-11-{q'(x)(x2-1) 2}|-11      

+ \int-11q''(x)(x2-1)2dx  

n=2    

q(x)    

q''(x)0      

\int-11d2dx2{(x2-1)2}q(x)dx=0  

Pn(x)=cvn(x)      

c=1/2nn!    

 xn in Pn(x) and vn(x)