Can someone please provides me with the recursion relations of Associated Legendre Polynomials when using Schmidt quasi-normalization? I need that in the context of Geomagnetism to obtain the Spherical Harmonics coefficients g and h.
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$\begingroup$ The answers here might help with this also, for future browsers: earthscience.stackexchange.com/q/22368/14123 $\endgroup$WJB– WJB2021-06-15 09:14:48 +00:00Commented Jun 15, 2021 at 9:14
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FYI, the usual reference relied on for the equations used in geomagnetism is:
- Langel, R. A. "Chapter four: Main field." Geomagnetism, edited by JA Jacobs (1987).
If it wasn't for lockdown I'd have copied from the book locked in my office to make sure I got it right...but until someone corrects me...
The Schmidt quasi/semi-normalisations for the associated Legendre polynomials are given by these recursions, in the form you'd actually use for calculations typically:
 "S_{n,0}=S_{n-1,0}\left ( \frac{2n-1}{n} \right )")
(%5Cdelta_m%5E1+1)%7D%7Bn+m%7D&space;%7D "S_{n,m}=S_{n,m-1} \sqrt{ \frac{(n-m+1)(\delta_m^1+1)}{n+m} }")
Note that if you use Schmidt normalisation of Legendre polynomials from a given software language to do calculations, you do not want to include the Condon-Shortley phase factor of %5E%7Bm%7D "(-1)^{m}")
Edit: Some links to open source geomagnetism compatible software for calculating Schmidt normalised associated Legendre polynomials.
Python: ChaosMagPy
Python/Fortran-95: SHTools
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$\begingroup$ That exactly what I was looking for, except I couldn't find the mentioned reference anywhere on the web, so thanks for the hint. However, I really need a confirmation about the formula correctness including the derivatives too. $\endgroup$Ayoub– Ayoub2020-08-24 16:06:54 +00:00Commented Aug 24, 2020 at 16:06
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$\begingroup$ I don't think there is a digital version of Langel (1987) anywhere online, but I just found I could view enough of the following reference through Google Books to confirm the equations: Wertz, J. R. "Spacecraft Attitude Determination and Control", (1978). Use the "Search inside" option and search for "Appendix H", you can see the first few pages about Schmidt normalisation. $\endgroup$WJB– WJB2020-08-24 16:27:20 +00:00Commented Aug 24, 2020 at 16:27
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$\begingroup$ That was useful, thanks again! So, the factors S are used with Gauss functions to obtain Schmidt functions, where: P_Schmidt = S * P_Gauss. But, it's still a long way to go with. I would love to compute directly the P_Schmidt functions since we know that there's recursion relations for them out there. $\endgroup$Ayoub– Ayoub2020-08-24 17:33:32 +00:00Commented Aug 24, 2020 at 17:33
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$\begingroup$ Are you actually after the final recursive formula for the Schmidt P_n^m(cos(theta)) functions themselves, or are you ultimately trying to write code to calculate these values for geomag problems? I can add links to open source code in a few languages much more easily than I can work back and type up the formula working from those codes right now! $\endgroup$WJB– WJB2020-08-24 19:15:15 +00:00Commented Aug 24, 2020 at 19:15
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$\begingroup$ Exactly I'm trying to implement a code to calculate those values, any open source code is welcomed! but I can't just take the code without the formula since I need to document that. $\endgroup$Ayoub– Ayoub2020-08-24 21:22:53 +00:00Commented Aug 24, 2020 at 21:22
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1$\begingroup$ As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. $\endgroup$2025-11-15 12:07:35 +00:00Commented Nov 15 at 12:07