AuZex Approximation

Fig.1. Plot of AuZex (thick curve) and LambertW (thin curve)

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

AuZex Approximation describes the approximations of AuZex, which is inverse function of SuZex and Abel function of zex.

The resulting complex double implementation is loaded as AuZex.cin

The explicit plot of function AuZex is shown in figure 1.

Its complex map is shown in figure 2.

The approximation is described in Chapter 11 of book «Superfunctions», 2020 .

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 approximation AuZt with \(N\!=\!32\) with 32 terms is denoted as \(\mathrm{AuZt}_{32}\).

The agreement is estimated as

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

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

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

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 AuZex.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.

Exotic iteration

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 \(L=\mathrm e\approx 2.718\)

The similarity is that the Regular iteration cannot be applied "as is"; the logarithmic terms appear in the expansion.

The described way to evaluate the superfunction, the abelfunction and non-integer iterates of the transfer function in such a case is denoted with term Exotic iteration.

References

Keywords

«Abel function», «Abelfunction», «AuZex», «AuZex.cin», «Abel function», «Exotic iteration», «LambertW», «Exotic iteration», «Mathematics of Computation», «Superfunctions», «Zex»,