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For some weeks have been studying Legendre polynomial as a solution to this equation. $$ (1-x^2)\frac{d^2}{dx^2}f(x)-2x\frac{d}{dx}f(x)+n(n+1)f(x)=0.$$

I've found them very interesting to learn from purely mathematical perspective but I haven't come across their specific use. Why are they so important and any example suggesting the use of Legendre polynomial would be appreciated.

Thank you.

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    See [this](http://dlmf.nist.gov/14.30.iv) and [this](http://dlmf.nist.gov/14.31).2012-08-01
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    And also [this](http://wolfram.com/xid/0e7qj6m0-j4xu0s).2012-08-01
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    As I've already alluded to, a good portion of the importance of the Legendre functions is when they are in the form of "spherical harmonics". Try searching for `spherical harmonics applications` in your favorite search engine; it should turn up a number of interesting results.2012-08-01

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