Difference between revisions of "Legendre P"

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=Properties=
 
=Properties=
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[[Relationship between Legendre polynomial and hypergeometric 2F1]]<br />
<strong>Theorem</strong>: The Legendre polynomials satisfy the differential equation
 
$$(1-x^2)P_{k}''(x)-2xP_k'(x)+k(k+1)P_k(x)=0.$$</strong>
 
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<strong>Proof: █
 
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{{:Orthogonal polynomials footer}}
<strong>Theorem:</strong> The following formula holds:
 
$$\dfrac{1}{\sqrt{1-2xt+t^2}} = \displaystyle\sum_{k=0}^{\infty} P_n(x)t^n.$$
 
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<strong>Proof:</strong> █
 
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[[Category:SpecialFunction]]
<strong>Theorem:</strong> ([[Rodrigues' formula]]) The following formula holds:
 
$$P_n(x)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n} [(x^2-1)^n].$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> (Orthogonality) The following formula holds:
 
$$\displaystyle\int_{-1}^1 P_m(x)P_n(x)dx = \dfrac{2}{2n+1} \delta_{mn},$$
 
where $P_n$ denotes [[Legendre P|Legendre polynomials]] and $\delta$ denotes the [[Dirac delta]].
 
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<strong>Proof: █
 
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Latest revision as of 01:43, 22 June 2016

The Legendre polynomials are orthogonal polynomials defined by the formula $$P_n(x) = \dfrac{1}{2^n} \displaystyle\sum_{k=0}^n {n \choose k}^2 (x-1)^{n-k}(x+1)^k.$$

$$\begin{array}{ll} P_0(x) &= 1 \\ P_1(x) &= x \\ P_2(x) &= \dfrac{1}{2}(3x^2-1) \\ P_3(x) &= \dfrac{1}{2}(5x^3-3x) \\ P_4(x) &= \dfrac{1}{8}(35x^4-30x^2+3) \\ P_5(x) &= \dfrac{1}{8}(63x^5-70x^3+15x) \\ \vdots \end{array}$$

Properties

Relationship between Legendre polynomial and hypergeometric 2F1

Orthogonal polynomials