Langevin Function L(x) Calculator

Evaluate the Langevin function L(x)=coth(x)-1/x, with the small-x approximation x/3, an inverse-Langevin estimate, and a curve plot.

Input

Enter x to evaluate the Langevin function L(x)=coth(x)-1/x.

Value of x

Any real number is allowed (e.g. 1).

Value of L(1)

0.3130352855

Small-x approximation x/3

0.3333333333

Inverse-Langevin estimate

1.0071199714

Input x

Curve of L(x)

The Langevin function is defined as L(x)=coth(x)-1/x, where coth is the hyperbolic cotangent cosh(x)/sinh(x). Its range is -1 to 1, approaching 1 as x grows large.
Near x=0 the calculator avoids cancellation by using the series expansion L(x)=x/3 - x^3/45 + 2 x^5/945 - ... The leading term gives the small-x approximation x/3.
The Langevin function is the classical limit of the Brillouin function as the quantum number J goes to infinity, appearing in statistical-mechanics models of paramagnetic magnetization and polymer-chain extension.
The inverse Langevin function L^(-1)(y) has no closed form, so a practical Pade-type approximation y(3 - y^2)/(1 - y^2) is used. The result shows that estimate evaluated at L(x), which recovers an approximation to the input x.

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