Polynomial computation

This applet does some polynomial manipulations. It can do most simple processes, like multiplication, differentiation, and integration. The first box generates a table of values of Pn, the Legendre Polynomials, defined recursively by

P n ⁡ ( x ) = { 1 n = 0 x n = 1 ( 2 ⁢ n - 1 ) ⁢ x ⁢ P n - 1 - ( n - 1 ) ⁢ P n - 2 n otherwise

There is also an option which, will give exactly the same output, using an alternative formula (which is a Rodrigues equation):

P n ⁡ ( x ) = 1 2n ⁢ n! ⁢ ⅆn ⅆxn ⁡ ( x 2 - 1 ) n

The second button is to show that (at least the first few) have orthogonality properties under this inner product:

⟨ Pi , Pj ⟩ = ∫ -1 1 Pi ⁢ Pj ⁢ ⅆ x

Finally, to demonstrate that (at least the first few) satisfy a differential equation, you can evaluate

( 1 - x 2 ) ⁢ P n ″ - 2 ⁢ x ⁢ P n ′ + n ⁢ ( n+1 ) ⁢ P n .

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