Talk:Associated Legendre function: Difference between revisions
imported>Paul Wormer (→Problem: new section) |
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--[[User:Paul Wormer|Paul Wormer]] 11:15, 6 September 2009 (UTC) | --[[User:Paul Wormer|Paul Wormer]] 11:15, 6 September 2009 (UTC) | ||
:I checked some and I believe that the equation must be | |||
::<math> | |||
\int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l} dx = -\frac{2l}{2l+1} \int\limits_{-1}^{1}\left( x^{2} -1\right) ^{l-1} dx | |||
</math> | |||
:I changed the text accordingly, but there must be another sign error because the final result now obtains (−1)<sup>''l''</sup>, which is not correct. --[[User:Paul Wormer|Paul Wormer]] 13:14, 6 September 2009 (UTC) |
Revision as of 07:14, 6 September 2009
Started from scratch, because (as so often) the Wikipedia article contains many duplications.--Paul Wormer 10:08, 22 August 2007 (CDT)
Added link to proof of orthogonality and derivation of normalization constant
I added a link to a proof of the first integral equation in the Orthogonality relations section Dan Nessett 16:39, 11 July 2009 (UTC)
Formattting of proof
I formatted the proof somewhat to my taste (which is mainly determined by a careful reading of Knuth's the TeXbook). Most of my changes are self-explanatory. I added the integration by parts formula, because it was not clear at all where the u and the v' came from. Further
--Paul Wormer 12:43, 5 September 2009 (UTC)
- Nice formating. Thanks for catching the problem. Also, I agree it is better to explicitly provide the integration by parts formula, rather than expecting the reader to already know it. Dan Nessett 00:49, 6 September 2009 (UTC)
Move of equation
I moved the dif. eq. from the proof page to the main article. I did this because the proof does not use the dif. eq., but uses the definition of the assleg's (which is given in the statement of the theorem). --Paul Wormer 09:24, 6 September 2009 (UTC)
Another (non-essential) error
I didn't like the reference to WP and tried my hand at the integral, which indeed is not difficult. Doing this I found another slip of the pen.--Paul Wormer 10:43, 6 September 2009 (UTC)
Problem
Let us apply the following equation (taken from Dan's proof and which holds for l ≥ 1) for l = 1:
Then is it true that
Integration of both sides gives
The problem is not trivial because the recursion ends with this integral. Dan, HELP!
--Paul Wormer 11:15, 6 September 2009 (UTC)
- I checked some and I believe that the equation must be
- I changed the text accordingly, but there must be another sign error because the final result now obtains (−1)l, which is not correct. --Paul Wormer 13:14, 6 September 2009 (UTC)
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