AuZex Approximation

Fig.1. Plot of AuZex (thick curve) in comparison to LambertW (thin curve)

Fig.2. Complex map, $u\!+\!\mathrm i = \mathrm {AuZex}(x\!+\!\mathrm i y)$

This article describes the approximations of AuZex, which is inverse function of SuZex and Abel function of zex. The explicit plot of function AuZex is shown in figure 1. Its complex map is shown in figure 2. This map should be compared to the similar maps for the approximations below.

Background

AuZex is Inverse function of SuZex, so, in wide ranges of values of $z$, the relations

(1) $ ~ ~ ~ \mathrm{SuZex}\Big( \mathrm{AuZex}(z) \Big) = z $

and

(2) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{SuZex}(z) \Big) = z $

should hold. In particular, $~\mathrm{AuZex}(1)=0$.

Also, AuZex satisfies the Abel equation

(3) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{zex}(z) \Big) = \mathrm{AuZex}(z) +1$

in this case, zex appears as transfer function, and AuZex is its Abel function. Iterations of equation (3) gives the relations

(4) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{zex}^n(z) \Big) = \mathrm{AuZex}(z) +n$

This can be re-writtern also as

(5) $ ~ ~ ~ \mathrm{AuZex}\Big( \mathrm{LambertW}^n(z) \Big) = \mathrm{AuZex}(z) - n$

as LambertW is inverse function of zex.

The relations above indicate ways to construct the efficient approximation of AuZex, covering the whole complex plane, and make the efficient (fast and precise) implementation.

Taylor expansion at unity

Coefficients in expansion (3)

$\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\! \displaystyle

\begin{array}{r|r} n & c_n~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ \hline 0 & 0.0000000000000000 \\ 1 & 1.4011764331478447\\ 2 & -1.2313176379841106\\ 3 & 1.1612567820116564\\ 4 & -1.1231269305776580\\ 5 & 1.0992876544297898\\ 6 & -1.0830479804216504\\ 7 & 1.0713113178859344\\ 8 & -1.0624516150969114 \end{array}

$

Some tens of coefficients of the Taylor expansion of function AuZex can be evaluated just inverting the Taylor expansion of $\mathrm{SuZex}(z)$ at $z\!=\!0$; that leads to the approximation

(3) $ ~ ~ ~ \displaystyle \mathrm{AuZex}(1+t)\approx\mathrm{AuZt}_N(1\!+\!t)=\sum_{n=1}^N c_n t^n$

Approximatoins for the first eight coefficients $c$ are shown in the table at right.

The Complex map of this approximation AuZt with $N\!=\!32$ is shown in Figure 3. The agreement

(4) $ ~ ~ ~ \displaystyle A(z)=-\lg\left( \frac {\left|\mathrm{SuZex}\Big(\mathrm{AuZt}_{32}(z)\Big)- z\right|} {\left|\mathrm{SuZex}\Big(\mathrm{AuZt}_{32}(z)\Big)\right|+|z|} \right) $

is shown in figure 4.

(sorry, the figures are not yet loaded)

Iterations of function LambertW applied to the argument of function AuZt give the approximation

(5) $ ~ ~ ~ \displaystyle \mathrm{AuZex}(z) \approx \mathrm{AuZt}_n( \mathrm{LambertW}^n(z))+n$

The complex maps of these approximations are shown in figure 5. While the efficient implementation for function LambertW is available, the iterations for integer $n$ in (5) cause no problems.

Asymptotic expansion

The approximations by (5) cover the most of the complex plane, except the region in vicinity of the origin of coordinates. For this region, the asymptotic expansion below van be used:

(6) $ ~ ~ ~ \displaystyle \mathrm{AuZex}(z) \approx \mathrm{AsZa}_N\Big(\mathrm{LambertW}^m(z)\Big)+x_1+m$

where

(7) $ ~ ~ ~ \displaystyle \mathrm{AsZa}_N(t)=\frac{-1}{t} + \frac{1}{2} \ln(t) + \sum_{n=1}^N ~ b_n\, t^n$

The coefficients $b$ of this expansion can be generated by Mathematica with code

So1[z_, a_] := Extract[Extract[Solve[z, a], 1], 1] 
zex[z_] = z Exp[z]
Clear[b];
g[n_,z_] = -1/z + Log[z]/2 + Sum[b[m] z^m, {m, 1, n}]
For[k=1,k<64,
b[k]=ReplaceAll[b[k],So1[Coefficient[Series[g[k,zex[z]]-g[k,z]-1,{z,0,k+1}],z^(k+1)]==0, b[k]]];
Print[b[k]]; k++]

Coefficients $b_n$ for $n\!=\!1..9$ are shown in the table

$\begin{array}{c|ccl} n & b_n & &{\rm ~approximation ~ of~ ~} b_n\\ \hline \\ 1 & -1/6 &\approx&-0.1666666666666666667\\ 2 & 1/16 &\approx& ~ ~ ~ 0.0625\\ 3 & -19/540&\approx& -0.0351851851851851852 \\ 4 & 1/48&\approx& ~ ~ ~ 0.0208333333333333333 \\ 5 & -41/4200&\approx& -0.0097619047619047619\\ 6 & 37/103680&\approx& ~ ~ ~ 0.00035686728395061728\\ 7 & 18349/3175200&\approx&~ ~ ~ 0.005778848576467624 \\ 8 & -443/80640 &\approx& -0.005493551587301587\\ 9 & 55721/21555072&\approx& -0.002585052835824441 \end{array} $

Implementation of AuZex

The approximations above cover the whole complex plane. On the base of these approximations, the complex double function AuZex is implemented. The implementation is available as AsZex.cin. This implementation provides of order of 15 correct decimal digits, and the errors are comparable to the rounding errors at the Complex double arithmetics.

Similar approach is used to implement two Abel functions of the exponential to to base $b=\exp^2(-1)=\exp(1/\mathrm e)\approx 1.444667861$ . In that case, the transfer function $\exp_b$ also has derivative unity at its fixed point $L=\mathrm e\approx 2.718$