Legendre polynomials/Catalogs: Difference between revisions

Latest revision as of 07:04, 9 September 2009

Catalogs ^[?]

An informational catalog, or several catalogs, about Legendre polynomials.

The first twelve Legendre polynomials are:

{\begin{aligned}P_{0}(x)&=1\\P_{1}(x)&=x\\P_{2}(x)&={\tfrac {1}{2}}(3x^{2}-1)\\P_{3}(x)&={\tfrac {1}{2}}(5x^{3}-3x)\\P_{4}(x)&={\tfrac {1}{8}}(35x^{4}-30x^{2}+3)\\P_{5}(x)&={\tfrac {1}{8}}(63x^{5}-70x^{3}+15x)\\P_{6}(x)&={\tfrac {1}{16}}(231x^{6}-315x^{4}+105x^{2}-5)\\P_{7}(x)&={\tfrac {1}{16}}(429x^{7}-693x^{5}+315x^{3}-35x)\\P_{8}(x)&={\tfrac {1}{128}}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35)\\P_{9}(x)&={\tfrac {1}{128}}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x)\\P_{10}(x)&={\tfrac {1}{256}}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)\\P_{11}(x)&={\tfrac {1}{256}}(88179x^{11}-230945x^{9}+218790x^{7}-90090x^{5}+15015x^{3}-693x)\\\end{aligned}}

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 {{subpages}}
+The first twelve Legendre polynomials are:
 :<math>
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 P_9(x) &= \tfrac{1}{128}(12155x^9-  25740x^7 + 18018x^5 -4620x^3 + 315x)\\
 P_{10}(x) &= \tfrac{1}{256}(46189 x^{10}-  109395x^8 + 90090x^6 - 30030x^4 + 3465x^2 - 63 )\\
+P_{11}(x) &= \tfrac{1}{256}( 88179x^{11}-  230945x^9 + 218790x^7 - 90090x^5 + 15015x^3 - 693x )\\
 \end{align}
 </math>