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== Homework 9 Collaboration Area == | == Homework 9 Collaboration Area == | ||
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| + | Here are some hints about the problems on Legendre Polynomials. | ||
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| + | See p. 180 for a list of the first few Legendre Polynomials. | ||
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| + | The even numbered Legendre Polynomials only involve even powers of x, | ||
| + | so they are even functions. | ||
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| + | The odd numbered Legendre Polynomials only involve odd powers of x, | ||
| + | so they are odd functions. | ||
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| + | The Legendre Polynomials are orthogonal on the interval [-1,1]. | ||
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| + | p. 209, 5. asks you to show that | ||
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| + | <math>P_n(\cos\theta)</math> | ||
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| + | are orthogonal on [0,pi] with respect to the weight function | ||
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| + | <math>\sin\theta,</math> | ||
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| + | i.e., to show that | ||
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| + | <math>\int_0^\pi P_n(\cos\theta)P_m(\cos\theta)\sin\theta\ d\theta=0</math> | ||
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| + | if <math>n\ne m</math>. | ||
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| + | The key here is to use the change of variables | ||
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| + | <math>x=\cos\theta</math> | ||
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| + | and convert the integral to one in x over the interval [-1,1], where you can use the orthogonality of the Legendre Polynomials. | ||
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[[2010 MA 527 Bell|Back to the MA 527 start page]] | [[2010 MA 527 Bell|Back to the MA 527 start page]] | ||
Revision as of 10:05, 22 October 2010
Homework 9 Collaboration Area
Here are some hints about the problems on Legendre Polynomials.
See p. 180 for a list of the first few Legendre Polynomials.
The even numbered Legendre Polynomials only involve even powers of x, so they are even functions.
The odd numbered Legendre Polynomials only involve odd powers of x, so they are odd functions.
The Legendre Polynomials are orthogonal on the interval [-1,1].
p. 209, 5. asks you to show that
$ P_n(\cos\theta) $
are orthogonal on [0,pi] with respect to the weight function
$ \sin\theta, $
i.e., to show that
$ \int_0^\pi P_n(\cos\theta)P_m(\cos\theta)\sin\theta\ d\theta=0 $
if $ n\ne m $.
The key here is to use the change of variables
$ x=\cos\theta $
and convert the integral to one in x over the interval [-1,1], where you can use the orthogonality of the Legendre Polynomials.
