Talk:Brillouin and Langevin functions

(Error?)

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.

Hmm...I'm pretty sure the functional form is correct. I added a reference: [1] --HappyCamper 22:14, 30 May 2007 (UTC)

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)

Expansion of the inverse Langevin function

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)

Thanks. Fixed. Bbanerje (talk) 21:41, 4 August 2010 (UTC)

Alternative expansion of the Langevin function

I found, by numerical evidence, an alternative continued-fraction-like expansion of the Langevin function:

$L(x)={\frac {x}{3+{\frac {x^{2}}{5+{\frac {x^{2}}{7+{\frac {x^{2}}{9+\ldots }}}}}}}}$

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 7^th 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 7^th 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)

Pade approximants have been shown to provide excellent approximations for the inverse Langevin function (A.Cohen, (1991) Acta Rheologica, 30, p. 270). Since Pade approximant theory grew out of work in continued fractions one should be able to prove that your fractional approximation is equivalent to some Pade approximant and the error in the approximation should also be possible to find out. A nice discussion on such matters can be found at [2]. Bbanerje (talk) 00:20, 6 August 2010 (UTC)

Bingo! Your link led me easily to the Lambert's continued fraction expansion of tanh(x), from which my approximation derives trivially. I'm adding that to the article. Thanks! --Edgar.bonet (talk) 16:18, 6 August 2010 (UTC)

Padé approximant of inverse Langevin

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

$L^{-1}(x)\approx x{\frac {3-x^{2}}{1-x^{2}}}$

carries an error $O(x^{3})$.

On the other hand, the Taylor expansion leads to

$L^{-1}(x)=3x{\frac {35-12x^{2}}{35-33x^{2}}}+O(x^{7})$.

-- Edgar.bonet (talk) 12:43, 12 March 2011 (UTC)

I checked the article. They derive your preferred equation, then numerically approximate the coefficients. The approximation is very very close and it makes sense for the applications that the author had in mind -- analyzing experimental data, manipulating mathematical models of chemistry processes, etc. The paper abstract starts, "Application of the methodology of Padé approximants to a Taylor expansion of the inverse Langevin function led to an accurate analytical expression". So they don't exactly say that the formula is itself the Padé approximant, only that it was derived using that methodology among other things. Perhaps you should change the wording. --Steve (talk) 04:31, 13 March 2011 (UTC)

OK, I think I see the point. Both the Taylor expansion and the proper Padé approximant are small-$x$ expansions, and are thus only valid for moderate values of $x$ (up to roughly 0.75 and 0.85 respectively). The Cohen approximation, on the other hand, carries a relative error < 5% on the whole interval (-1, 1), which makes it far more useful for when you want a formula valid for arbitrary $x$. I'll change the wording accordingly. --Edgar.bonet (talk) 11:21, 15 March 2011 (UTC)

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Brillouin function convention

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)

Let us define it as Kittel does then.--ReyHahn (talk) 19:14, 6 October 2020 (UTC)