Show that the Legendre equation can also be written as (1 x2) y__ _ = ( + 1) y. Then it
Chapter 5, Problem 22(choose chapter or problem)
Show that the Legendre equation can also be written as (1 x2) y__ _ = ( + 1) y. Then it follows that (1 x2) P_ n ( x)_ _ = n(n + 1) Pn( x) and (1 x2) P_ m ( x)_ _ = m(m + 1) Pm( x). By multiplying the first equation by Pm( x) and the second equation by Pn( x), integrating by parts, and then subtracting one equation from the other, show that _ 1 1 Pn( x) Pm( x)dx = 0 ifn _= m. This property of the Legendre polynomials is known as the orthogonality property. If m = n, it can be shown that the value of the preceding integral is 2/(2n + 1).
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