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Brillouin formula is wrong in the argument of the trigonometrical function,please correct this. —The preceding unsigned comment was added by 83.138.224.86 ( talk • contribs) 22:55, 29 May 2007.
According to [B. D. Cullity, C. D. Graham-Introduction to magnetic materials-Wiley-IEEE Press] the expression of the function is $$B(J,x)=\frac{2J+1}{2J}\coth\left(\frac{2J+1}{2J}\right)x-\frac{1}{2J}\coth\left(\frac{1}{2J}x\right)$$ — Preceding unsigned comment added by 2A01:CB00:A04:B100:C18A:9D13:143F:8893 ( talk) 23:02, 17 February 2018 (UTC)
Hi!
There seems to be a typo on the expansion of the inverse Langevin function. The 7th order term reads -1539⁄875 x7. I believe the sign should be + instead. -- Edgar.bonet ( talk) 08:45, 4 August 2010 (UTC)
I found, by numerical evidence, an alternative continued-fraction-like expansion of the Langevin function:
Compared to the Taylor series, this approximation seems far easier to remember (at any order), more accurate and better behaved. I did a comparison of both at the same 7th order. At this order the Taylor approximation breaks at x ≈ 2.5, while the above only breaks at x ≈ 7. Within their range of validity, both have errors that scale like x9, but Taylor has a prefactor about 215 times bigger. Outside their range of validity, the Taylor series diverges wildly, while the expansion above only goes smoothly to zero.
The numerical evidence is quite strong. I explored the asymptotic behavior of the error at many orders. I even used some arbitrary-precision aritmetics to check it beyond the limits of double precision calculations. My only problem is: however strong, numerical evidence is not a real proof!
Now, my Wikipedia-related questions are: does anyone know whether the expansion above has ever been proved to converge to L(x)? Any reference? May it be easy to prove? I assume a proof is a prerequisite for the expansion to appear on the article.
On a side note: it may be worth mentioning that when x is small, the Taylor approximation (and the one above, if proved) is numerically more accurate than a direct evaluation of the actual analytical expression, because the later suffers from catastrophic cancellation. For double precision arithmetics and 7th order expansions, the Taylor series is better than the analytical expression for x ≲ 0.07, while the threshold is about 0.12 for the expansion above.
-- Edgar.bonet ( talk) 17:44, 5 August 2010 (UTC)
Can someone please check the formula for the Padé approximant of the inverse Langevin function? I have no acces to the reference and it looks to me like
carries an error .
On the other hand, the Taylor expansion leads to
-- Edgar.bonet ( talk) 12:43, 12 March 2011 (UTC)
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The Brillouin function definition refers to Kittel and Darby. Neither defines it the way it is presented here. Both have extra 1/J in the arguments of coth and extra J in x. It is written the same way in papers, see [3] for a random example (many more can be added if needed). Of course, it is equivalent at the end, but I frequently find the formula to contain errors – and this one of their sources as people copy and mix'n'match.
Since attempts to fix the definitions to those actually used in the references are met with immediate reverts, I suggest to remove the links to Kittel and Darby (and possibly other references) and replace them with references which define the function the way it is presented here. Now the links do not support the presented content; they contradict it. — Preceding unsigned comment added by 147.229.99.101 ( talk) 13:33, 6 October 2020 (UTC)
![]() | This article is rated Start-class on Wikipedia's
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Brillouin formula is wrong in the argument of the trigonometrical function,please correct this. —The preceding unsigned comment was added by 83.138.224.86 ( talk • contribs) 22:55, 29 May 2007.
According to [B. D. Cullity, C. D. Graham-Introduction to magnetic materials-Wiley-IEEE Press] the expression of the function is $$B(J,x)=\frac{2J+1}{2J}\coth\left(\frac{2J+1}{2J}\right)x-\frac{1}{2J}\coth\left(\frac{1}{2J}x\right)$$ — Preceding unsigned comment added by 2A01:CB00:A04:B100:C18A:9D13:143F:8893 ( talk) 23:02, 17 February 2018 (UTC)
Hi!
There seems to be a typo on the expansion of the inverse Langevin function. The 7th order term reads -1539⁄875 x7. I believe the sign should be + instead. -- Edgar.bonet ( talk) 08:45, 4 August 2010 (UTC)
I found, by numerical evidence, an alternative continued-fraction-like expansion of the Langevin function:
Compared to the Taylor series, this approximation seems far easier to remember (at any order), more accurate and better behaved. I did a comparison of both at the same 7th order. At this order the Taylor approximation breaks at x ≈ 2.5, while the above only breaks at x ≈ 7. Within their range of validity, both have errors that scale like x9, but Taylor has a prefactor about 215 times bigger. Outside their range of validity, the Taylor series diverges wildly, while the expansion above only goes smoothly to zero.
The numerical evidence is quite strong. I explored the asymptotic behavior of the error at many orders. I even used some arbitrary-precision aritmetics to check it beyond the limits of double precision calculations. My only problem is: however strong, numerical evidence is not a real proof!
Now, my Wikipedia-related questions are: does anyone know whether the expansion above has ever been proved to converge to L(x)? Any reference? May it be easy to prove? I assume a proof is a prerequisite for the expansion to appear on the article.
On a side note: it may be worth mentioning that when x is small, the Taylor approximation (and the one above, if proved) is numerically more accurate than a direct evaluation of the actual analytical expression, because the later suffers from catastrophic cancellation. For double precision arithmetics and 7th order expansions, the Taylor series is better than the analytical expression for x ≲ 0.07, while the threshold is about 0.12 for the expansion above.
-- Edgar.bonet ( talk) 17:44, 5 August 2010 (UTC)
Can someone please check the formula for the Padé approximant of the inverse Langevin function? I have no acces to the reference and it looks to me like
carries an error .
On the other hand, the Taylor expansion leads to
-- Edgar.bonet ( talk) 12:43, 12 March 2011 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on Brillouin and Langevin functions. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
This message was posted before February 2018.
After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than
regular verification using the archive tool instructions below. Editors
have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
RfC before doing mass systematic removals. This message is updated dynamically through the template {{
source check}}
(last update: 5 June 2024).
Cheers.— InternetArchiveBot ( Report bug) 21:55, 25 July 2017 (UTC)
The Brillouin function definition refers to Kittel and Darby. Neither defines it the way it is presented here. Both have extra 1/J in the arguments of coth and extra J in x. It is written the same way in papers, see [3] for a random example (many more can be added if needed). Of course, it is equivalent at the end, but I frequently find the formula to contain errors – and this one of their sources as people copy and mix'n'match.
Since attempts to fix the definitions to those actually used in the references are met with immediate reverts, I suggest to remove the links to Kittel and Darby (and possibly other references) and replace them with references which define the function the way it is presented here. Now the links do not support the presented content; they contradict it. — Preceding unsigned comment added by 147.229.99.101 ( talk) 13:33, 6 October 2020 (UTC)