Difference between revisions of "Legendre P"
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(Created page with "The Legendre polynomials are defined by the recurrence $$P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}(x^2-1)^n; n=0,1,2,\ldots$$ $$\begin{array}{ll} P_0(x) &= 1 \\ P_1(x) &= x \...") |
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| − | The Legendre polynomials are defined by the recurrence | + | The Legendre polynomials are [[orthogonal polynomials]] defined by the recurrence |
$$P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}(x^2-1)^n; n=0,1,2,\ldots$$ | $$P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}(x^2-1)^n; n=0,1,2,\ldots$$ | ||
$$\begin{array}{ll} | $$\begin{array}{ll} | ||
Revision as of 20:37, 7 October 2014
The Legendre polynomials are orthogonal polynomials defined by the recurrence $$P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}(x^2-1)^n; n=0,1,2,\ldots$$ $$\begin{array}{ll} P_0(x) &= 1 \\ P_1(x) &= x \\ P_2(x) &= \dfrac{1}{2}(3x^2-1) \\ P_3(x) &= \dfrac{1}{2}(5x^3-3x) \\ \vdots \end{array}$$
