inverse_langevin_approximation(
    y,
    m::InverseLangevinApproximations.AbstractInverseLangevinApproximation
) -> Any

Method for computing the inverse Langevin function.

• y: The value of the Langevin Function
• m: The approximation Method

From Ref. [1]:

\[\mathcal{L}^{-1}(y) \approx\begin{cases} 1.31446\tan{1.58986y}+0.91209y & |y|\leq 0.84136 \\ \frac{1}{\text{sign}(y)-y} & 0.84136 \leq |x| < 1.0 \\ \end{cases}\]

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-\frac{36}{35}y^2}{1-\frac{33}{35}y^2}\]

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-y^2}{1-y^2}\]

From Ref. [3]:

\[{\mathcal{L}^{- 1}}(y) \approx y\frac{{y^{2} - 3y + 3}}{{1 - y}}.\]

From Ref. [4]:

\[\mathcal{L}^{-1}(y) \approx y\frac{3-\frac{773}{768}y^2-\frac{1300}{1351}y^4+\frac{501}{340}y^6-\frac{678}{138}y^8}{(1-y)(1+\frac{866}{853}*y)}\]

From Ref. [5]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3 + \frac{297}{175}y^5 + \frac{1539}{875}y^7 + \frac{126117}{67375}y^9 + \frac{43733439}{21896875}y^{11} + \frac{231321177}{109484375}y^{13} + \frac{20495009043}{9306171875}y^{15} + \frac{1073585186448381}{476522530859375}y^{17} + \frac{4387445039583}{1944989921875}y^{19}\]

From Ref. [6]:

\[{\mathcal{L}^{-1}}\left( y \right)\approx y\frac{3-y^{2}}{1-y^{2}}-0.488y^{3.243}+3.311y^{4.789}\left( y-0.76 \right)\left( y-1 \right)\]

From Ref. [2]:

\[\mathcal{L}^{-1}(y) \approx y \frac{(3 - \frac{36}{35} y^2)}{(1 - \frac{33}{35} y^2)}\]

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-\frac{3}{5}y^2}\]

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-\frac{3}{5}y^2-\frac{36}{175}y^4}\]

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3\]

From Ref. [7]:

\[\mathcal{L}^{-1}(y) \approx 3y + \frac{9}{5}y^3 + \frac{297}{175}y^5\]

From Ref. [8]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-y^3}\]

From Ref. [9]:

\[\mathcal{L}^{-1}(y) \approx \frac{3 y}{(1 - (\frac{3}{5} y^2 + \frac{36}{175} y^4 + \frac{108}{875} y^6))}\]

From Ref. [10]:

\[\mathcal{L}^{-1}(y) \approx \frac{3y}{1-y^2}\]