Difference between revisions of "Legendre P"

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The Legendre polynomials are [[orthogonal polynomials]] defined by the recurrence
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The Legendre polynomials are [[orthogonal polynomials]] defined by the formula
$$P_n(x) = \dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}(x^2-1)^n; n=0,1,2,\ldots$$
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$$P_n(x) = \dfrac{1}{2^n} \displaystyle\sum_{k=0}^n {n \choose k}^2 (x-1)^{n-k}(x+1)^k.$$
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$$\begin{array}{ll}
 
$$\begin{array}{ll}
 
P_0(x) &= 1 \\
 
P_0(x) &= 1 \\
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P_2(x) &= \dfrac{1}{2}(3x^2-1) \\
 
P_2(x) &= \dfrac{1}{2}(3x^2-1) \\
 
P_3(x) &= \dfrac{1}{2}(5x^3-3x) \\
 
P_3(x) &= \dfrac{1}{2}(5x^3-3x) \\
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P_4(x) &= \dfrac{1}{8}(35x^4-30x^2+3) \\
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P_5(x) &= \dfrac{1}{8}(63x^5-70x^3+15x) \\
 
\vdots
 
\vdots
 
\end{array}$$
 
\end{array}$$
  
[[File:Legendrepolynomials.png|450px]]
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<div align="center">
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<gallery>
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File:Legendrepolynomials.png|Graph of $P_n$ on $[-4,4]$ for $n=0,1,2,3,4,5$.
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</gallery>
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</div>
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=Properties=
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[[Relationship between Legendre polynomial and hypergeometric 2F1]]<br />
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{{:Orthogonal polynomials footer}}
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[[Category:SpecialFunction]]

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