On Schröder-Type Series Expansions for the Lambert W Function

Abstract

Schröder based series for the principal and negative one branches of the Lambert W function are defined; the series are generic and are in terms of an initial, arbitrary, approximating function. Upper and lower bounds for the initial approximating functions, consistent with convergence, are determined. Approximations for both branches of the Lambert W function are proposed which have modest relative error bounds over their domains of definition and which are suitable as initial approximation functions for a convergent Schröder series. For the principal branch, a proposed approximation yields, for a series of order 128, a relative error bound below 10⁻¹³⁶. For the negative one branch, a proposed approximation yields, for a series of order 128, a relative error bound below 10⁻¹⁴³. Applications of the approximations for the principal and negative one branches include new approximations for the Lambert W function, analytical approximations for the integral of the Lambert W function, upper and lower bounded functions for the Lambert W function, approximations for the power of the Lambert W function and approximations to the solution of the equations c^c = y and C^C = e^v, respectively, for c and C.

1. Introduction

The Lambert W function defined by $w=h^{-1}\left(z\right)$, $z=h\left(w\right)=w{e}^w$ has been studied since 1758, and a brief historical overview of the research associated with this function is provided, for example, by Mező [1]. The Lambert W function finds widespread application, mainly for the principal branch and the real case, e.g., Refs. [2,3]. The negative one branch has received less attention, but applications have been detailed and include, for example, Ref. [4], Table 1, Ref. [5], Equations (7) and (27), Ref. [6], Equation (11), and Ref. [7], e.g., Equation (15). The nature of $h$ and $h^{-1}$ are shown in Figure 1 for the real case.

Figure 1. Graphs of $z=h\left(w\right)=w{e}^w,$ and its inverse $w=h^{-1}\left(z\right)$, for the principal and negative one branches. The real case is assumed.

As is common with inverse functions, an explicit analytical expression for the Lambert W function does not exist. Custom approximations for the principal branch and the real case are well documented, e.g., Refs. [3,8,9,10,11], and custom approximations for the negative one branch include those documented by [11,12,13,14].

Of interest are convergence series for the Lambert W function. The simplest approach to defining series is to utilize a Taylor series for $h\left(w\right)=w{e}^w$ around a set point and then use series inversion to obtain a series approximation for the Lambert W function. For the principal branch, such a series based on the origin is well established, e.g., Ref. [2], Equation (3.1). Use of the branch point $(-1/e,-1)$ is also utilized to establish a series for both the principal and negative one branches, e.g., Ref. [15], Equation (38). As is usual with Taylor series, the region of convergence is limited.

A fundamental relationship (see Equation (62) in [15]) for the Lambert W function was established by de Bruin [16], and this has underpinned asymptotic approximations for both the principal branch and the negative one branch, e.g., Ref. [2], Equations (4.18) and (4.19); Ref. [3], Equation (38); Ref. [15], Section 4.3. For the principal branch, the series is convergent for $\left[e,\infty \right)$; for the negative one branch, the series is convergent over the domain $\left[-1/e,0\right)$ but has a poor rate of convergence close to $-1/e$. Corless [15], Section 4.4f, utilizes the fundamental relationship to define a general series form (Equation (72)) and provides examples. These have a finite region of convergence.

There has been related research into establishing series approximations to the Lambert W function with an exponential argument (e.g., Ref. [15], Equation (48); [13], Equation (16)). Barry [13] details a series for the negative one branch, which has a finite region of convergence but is useful, when used with continued fraction form and general iteration forms, to define custom approximations which have low relative error bounds over the domain $[-1/e,0)$.

Corcino [17] details continued fraction expansions for the principal branch of the Lambert W function. The published results indicate a modest region of convergence.

In general, published series approximations have a limited region of convergence. To establish series that are convergent over the domain of definition for the principal and negative one branches of the Lambert W function, an approach with potential is to note that finding an inverse function value is essentially a root problem. Root values can be approximated by utilizing Schröder approximations of the first kind, e.g., Ref. [18]. Such approximations are the basis for establishing Schröder-based series, e.g., Ref. [19], for inverse functions based on an initial arbitrary approximating function. With an appropriate choice of the initial approximating function, a Schröder-based series will be convergent over the domain of definition. Howard [19] details explicit low-order (orders one to four) Schröder-based approximations for the principal branch of the Lambert W function, but a general form has yet to be detailed.

In this paper, an explicit general form for a Schröder series for the Lambert W function is defined that is valid for both the principal and negative one branches. The series is based on an initial approximating function. Upper and lower bounds for the initial approximating function are determined such that the associated Schröder series is convergent. These bounds are numerically confirmed.

New approximations, valid over the domains $(-1/e,0)$ and $(-1/e,\infty )$, respectively, for the negative one and principal branches, are proposed. The approximations lead to series, of order 128, having relative error bounds well below 10⁻¹³⁰. Some applications of the approximations are detailed.

1.1. Structure of Paper

An overview of published series approximations for the principal and negative one branches of the Lambert W function are detailed in Section 2. In Section 3, the general form for Schröder series approximations are defined and these are in terms of an initial approximating function. Initial approximating functions for the negative one and principal branches are proposed, respectively, in Section 4 and Section 5. Results, based on the proposed initial approximations, are detailed in Section 6. The basis for establishing the region of convergence for a Schröder series is detailed in Section 7. Approximate bounds for convergence are detailed in Section 8. Several applications of the results are detailed in Section 9 and conclusions are noted in Section 10.

1.2. Assumptions and Notation

In terms of notation, $h^{-1}$ is used for both the principal and negative one branches of the Lambert W function; the context makes it clear which branch is being considered. An initial approximation to $h^{-1}$ is denoted $h_0^{-1}$. The underlying function of $w{e}^w$ is denoted $h$.

For an arbitrary function $f$, defined over the interval $[\alpha ,\beta ]$, an approximating function $f_A$ has a relative error, at a point $x_1$, defined according to $\mathrm{re}\left(x_1\right)=1-f_A\left(x_1\right)/f\left(x_1\right)$. The relative error bound for the approximating function, over the interval $[\alpha ,\beta ]$, is defined according to

$${\mathrm{re}}_B=\max \left\{\left|\mathrm{re}\left(x_1\right)\right|: x_1\in \left[\alpha ,\beta \right]\right\}.$$

(1)

The notation $f^{\left(k\right)}$, as well as the operator notation $D^{\left(k\right)}\left[f\right]$, is used for the $k\mathrm{th}$ derivative of a function.

Mathematica has been used to facilitate analysis and to obtain numerical results. In general, relative error results associated with approximations have been obtained by sampling specified intervals, in either a linear or logarithmic manner, as appropriate, with $1000$ points.

2. Published Series Approximations

The following sub-sections provide an overview of published series approximations for the Lambert W function.

2.1. Taylor Series

A $k\mathrm{th}$ order Taylor series, based at the origin, yields the following series for the principal branch of the Lambert W function (e.g., Ref. [2], Equation (3.1)):

$$\begin{array}{l}{h}_{T,k}^{-1}\left(z\right)={\displaystyle \sum _{i=1}^k}{\displaystyle \frac{{(-1)}^{i-1}{z}^i}{i!}}\\ =z-z^2+{\displaystyle \frac{3z^3}{2}}-{\displaystyle \frac{8z^4}{3}}+{\displaystyle \frac{125z^5}{24}}-{\displaystyle \frac{54z^6}{5}}+{\displaystyle \frac{16807z^7}{720}}-{\displaystyle \frac{16384z^8}{315}}+\cdots +{\displaystyle \frac{{(-1)}^{k-1}{z}^k}{k!}}\end{array}$$

(2)

The relative errors associated with Taylor series, for orders $1, 2, 3, 4, 8, 16, 32, 64$, are shown in Figure 2. As usual with Taylor-series-based approximations, the region of convergence is limited and, for the principal branch of the Lambert W function, this is the interval $(-1/e,1/e)$.

Figure 2. Graphs of the relative errors, over the interval $[-1/e,2/e]$, in Taylor series approximations, of order $k$, $k\in \{1, 2, 3, 4, 8, 16, 32, 64\}$, for the principal branch and as defined by (2).

Taylor Series at the Branch Point for the Negative One and Principal Branches

A Taylor series approximation for $z=h\left(w\right)=w{e}^w$, based at the point $-1$, yields the approximation

$$z\approx {\displaystyle \frac{-1}{e}}+{\displaystyle \frac{{(w+1)}^2}{2e}}+{\displaystyle \frac{{(w+1)}^3}{3e}}+{\displaystyle \frac{{(w+1)}^4}{8e}}+{\displaystyle \frac{{(w+1)}^5}{30e}}+\cdots$$

(3)

A second-order approximation of $z={\displaystyle \frac{-1}{e}}+{\displaystyle \frac{{(w+1)}^2}{2e}}$ yields the following approximations for the principal and negative one branches:

$$h^{-1}\left(z\right)\approx \left\{\begin{array}{l}-1-\sqrt{2}\sqrt{1+ez}, w\le -1,{\displaystyle \frac{-1}{e}}\le z<0, \mathrm{negative} \mathrm{one} \mathrm{branch} \\ -1+\sqrt{2}\sqrt{1+ez}, w\ge -1,z\ge {\displaystyle \frac{-1}{e}}, \mathrm{principal} \mathrm{branch}\end{array}\right.$$

(4)

Higher order series can be established, via series inversion, for the principal and negative one branches (e.g., Ref. [15], Equation (38); [3], Equation (37)) according to

$$h^{-1}\left(z\right)\approx \left\{ \begin{array}{l}\begin{array}{c}-1-\sqrt{2}\sqrt{1+ez}-{\displaystyle \frac{2\left(1+ez\right)}{3}}-{\displaystyle \frac{11·2^{3/2}{\left(1+ez\right)}^{3/2}}{72}}-{\displaystyle \frac{43{·4\left(1+ez\right)}^2}{540}}-\\ \begin{array}{cc}{\displaystyle \frac{769·2^{5/2}{\left(1+ez\right)}^{5/2}}{17280}}-\cdots ,& {\displaystyle \frac{-1}{e}}\le z<0, \mathrm{negative} \mathrm{one} \mathrm{branch}\end{array}\end{array} \\ \begin{array}{c}-1+\sqrt{2}\sqrt{1+ez}-{\displaystyle \frac{2\left(1+ez\right)}{3}}+{\displaystyle \frac{11·2^{3/2}{\left(1+ez\right)}^{3/2}}{72}}-{\displaystyle \frac{43·4{\left(1+ez\right)}^2}{540}}+\\ \begin{array}{cc}{\displaystyle \frac{769·2^{5/2}{\left(1+ez\right)}^{5/2}}{17280}}-\cdots ,& z\ge {\displaystyle \frac{-1}{e}}, \mathrm{principal} \mathrm{branch}\end{array}\end{array}\end{array}\right.$$

(5)

For a fixed order of approximation, such approximations exhibit an increasing relative error as the argument $z$ moves away from the point $z=-1/e$ and approaches zero.

2.2. Asymptotic Series

An asymptotic approach (e.g., Ref. [2], Equations (4.18) and (4.19); Ref. [3], Equation (38)) yields the $k\mathrm{th}$ order approximation for the principal branch defined according to

$$\begin{array}{l}{h}_0^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}[\ln \left(z)\right],\\ h_k^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}[\ln \left(z)\right]+{\displaystyle \sum _{u=1}^k}{\displaystyle \sum _{i=0}^{u-1}}{\displaystyle \frac{{\left(-1\right)}^i}{\left(u-1\right)!}} ·\left|\left[\begin{array}{c}u\\ i+1\end{array}\right]\right|· {\displaystyle \frac{{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^{u-i}}{{\ln \left(z\right)}^u}}, k\in \left\{\mathrm{1,2},\cdots \right\},\end{array}$$

(6)

where $\left[\begin{array}{c}u\\ i+1\end{array}\right]$ is a Stirling number of the first kind. The series is convergent for $z>e$. An explicit expansion for the Lambert W function, valid for $z>e$, is

$$\begin{array}{l}{h}^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}[\ln \left(z)\right]+{\displaystyle \frac{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{\ln \left(z\right)}}+{\displaystyle \frac{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{{\ln \left(z\right)}^2}}·\left[-1+{\displaystyle \frac{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{2}}\right]+\\ \begin{array}{l} {\displaystyle \frac{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{{\ln \left(z\right)}^3}}·\left[1-{\displaystyle \frac{3\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{2}}+{\displaystyle \frac{{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^2}{3}}\right]+\\ \begin{array}{l} {\displaystyle \frac{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{{\ln \left(z\right)}^4}}·\left[-1+3\mathrm{l}\mathrm{n}[\ln \left(z)\right]-{\displaystyle \frac{{11\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^2}{6}}+{\displaystyle \frac{{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^3}{4}}\right]+\\ {\displaystyle \frac{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}{{\ln \left(z\right)}^5}}·\left[1-5\mathrm{l}\mathrm{n}[\ln \left(z)\right]+{\displaystyle \frac{{35\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^2}{6}}-{\displaystyle \frac{{25\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^3}{12}}+{\displaystyle \frac{{\mathrm{l}\mathrm{n}[\ln \left(z)\right]}^4}{5}}\right]+\cdots \end{array}\end{array}\end{array}$$

(7)

The relative errors in the approximations, of orders $0, 1, 2, 3, 4, 8$, are shown Figure 3.

Figure 3. Graphs of the relative errors in the approximation, defined by (6), for the principal branch and for orders $k\in \{0, 1, 2, 3, 4, 8\}$.

Negative One Branch

For the negative one branch, the $k\mathrm{th}$ order approximation is defined according to

$$\begin{array}{l}{h}_0^{-1}\left(z\right)=\ln \left(-z\right)-\mathrm{l}\mathrm{n}[-\ln \left(-z)\right],\\ h_k^{-1}\left(z\right)=\ln \left(-z\right)-\mathrm{l}\mathrm{n}[-\ln \left(-z)\right]+{\displaystyle \sum _{u=1}^k}{\displaystyle \sum _{i=0}^{u-1}}{\displaystyle \frac{{\left(-1\right)}^i}{\left(u-1\right)!}} ·\left|\left[\begin{array}{c}u\\ i+1\end{array}\right]\right|· {\displaystyle \frac{{\mathrm{l}\mathrm{n}[-\ln \left(-z)\right]}^{u-i}}{{\ln \left(-z\right)}^u}}, \end{array}$$

(8)

for $k\in \{1, 2, \cdots \}$. The relative errors in the approximations, of orders $k\in \{0, 1, 2, 3 ,4, 8\}$, are shown in Figure 4, and the approximations converge over the interval $[-1/e,0)$.

Figure 4. Graphs of the relative errors in the approximation, defined by (8), for the negative one branch and for orders $k\in \{0, 1, 2, 3, 4, 8\}$.

2.3. Series Based on Exponential Arguments

Corless et al. [15] detail a general form for series for the Lambert W function, which, in general, has finite regions of convergence. One specific series ([15], Equation (31)), based on a Taylor series for $h^{-1}\left(e^t\right)$ around the point $t=1$, is

$$h^{-1}\left(e^t\right)=1+{\displaystyle \frac{t-1}{2}}+{\displaystyle \frac{{(t-1)}^2}{16}}-{\displaystyle \frac{{\left(t-1\right)}^3}{192}}-{\displaystyle \frac{{\left(t-1\right)}^4}{3072}}+{\displaystyle \frac{{13\left(t-1\right)}^5}{61440}}+\cdots , t\in \mathit{R}.$$

(9)

The transformation $z=e^t$, i.e., $t=\mathrm{l}\mathrm{n}\left(z\right)$, $z>0$, yields the following series for the principal branch

$$\begin{array}{c}{h}^{-1}\left(z\right)\approx 1+{\displaystyle \frac{\ln \left(z\right)-1}{2}}+{\displaystyle \frac{{\left(\ln \left(z\right)-1\right)}^2}{16}}-{\displaystyle \frac{{\left(\ln \left(z\right)-1\right)}^3}{192}}-{\displaystyle \frac{{\left(\ln \left(z\right)-1\right)}^4}{3072}}+\\ {\displaystyle \frac{{13\left(\ln \left(z\right)-1\right)}^5}{61440}}+\cdots , z>0.\end{array}$$

(10)

This series has a finite region of convergence around the point $e$ and diverges as $z\to 0$ and as $z\to \infty$.

A related series for the negative one branch is defined by Barry et al. [13], Equation (16), and is defined according to

$$h^{-1}\left(-e^{-1-t}\right)\approx -1-\sqrt{2}\sqrt{t}-{\displaystyle \frac{2t}{3}}-{\displaystyle \frac{t^{3/2}}{9\sqrt{2}}}+{\displaystyle \frac{2t^2}{135}}-{\displaystyle \frac{t^{5/2}}{540\sqrt{2}}}-{\displaystyle \frac{4t^3}{8505}}+\cdots , t\ge 0.$$

(11)

The transformation $z=-e^{1-t}$, i.e., $t=-1-\mathrm{l}\mathrm{n}(-z)$, yields the following series for the negative one branch:

$$\begin{array}{c}{h}^{-1}\left(z\right)\approx -1-\sigma -{\displaystyle \frac{{\sigma }^2}{3}}-{\displaystyle \frac{{\sigma }^3}{36}}+{\displaystyle \frac{{\sigma }^4}{270}}-{\displaystyle \frac{{\sigma }^5}{4320}}-{\displaystyle \frac{{\sigma }^6}{17010}}+{\displaystyle \frac{139{\sigma }^7}{5443200}}+{\displaystyle \frac{{109\sigma }^8}{11340}}+\cdots ,\\ zϵ\left[-1/e,0\right),\end{array}$$

(12)

where $\sigma =\sqrt{2}\sqrt{-1-\ln \left(-z\right)}.$ The relative errors for approximations of orders one to eight and based on this series are shown in Figure 5. Whilst the series is clearly not convergent as $z\to 0^{-}$, it is the basis for accurate approximations for the negative one branch (see Equations (18)–(20), (22)–(26) and (28) in [13]).

Figure 5. Graphs of the relative errors in approximation of orders one to eight and, as defined by (12), for the negative one branch.

2.4. Continued Fraction Expansion

Corcino [17] details continued fraction expansions for the principal branch of the Lambert W function. The published results indicate a modest region of convergence. One expansion is

$$h^{-1}\left(z\right)={\displaystyle \frac{z}{1+\frac{z}{1+\frac{z}{2+\frac{5z}{3+\frac{17z}{10+\frac{133z}{17+\frac{1927z}{190+\frac{13582711z}{94423+\cdots }}}}}}}}}$$

(13)

with the first, second, ... order approximations defined according to

$$\begin{array}{ccc}{h}_1^{-1}\left(z\right)={\displaystyle \frac{z}{1+z/1}},& h_2^{-1}\left(z\right)={\displaystyle \frac{z}{1+\frac{z}{1+z/2}}},& \cdots \end{array}$$

(14)

The relative errors over the interval $[-1/e,3/2]$, associated with such approximations, are shown in Figure 6. The approximations have an increasing relative error as $z$ increases.

Figure 6. Graph of the relative errors in the approximations, defined by (13), for the principal branch and for orders one to seven.

2.5. Iterative Approximations

The classical iteration method for the Lambert W function is based on the logarithmic form of $z=w{e}^w$ leading to (e.g., Ref. [9], Equations (8)–(10)):

$$\left\{\begin{array}{l}\begin{array}{l}\ln \left(-z\right)=w+\ln \left(-w\right), \mathrm{negative} \mathrm{one} \mathrm{branch}, w<-1,z\in (-1/e,0)\\ \mathrm{principal} \mathrm{branch}, w\ge -1,z\in [-1/e,0)\end{array}\\ \ln \left(z\right)=w+\mathrm{l}\mathrm{n}\left(w\right), \mathrm{principal} \mathrm{branch}, w\ge 0,z\ge 0\end{array}\right.$$

(15)

With an initial approximation of $h_0^{-1}$, improved approximations for the Lambert W function are

$$\left\{\begin{array}{l}\begin{array}{l}{h}_1^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[-h_0^{-1}\left(z\right)\right], \mathrm{negative} \mathrm{one} \mathrm{branch}, w<-1,z\in (-1/e,0)\\ \mathrm{principal} \mathrm{branch}, w\ge -1,z\in [-1/e,0)\end{array}\\ h_1^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}\left[h_0^{-1}\left(z\right)\right], \mathrm{principal} \mathrm{branch}, w\ge 0,z\ge 0\end{array}\right.$$

(16)

and iteration can be utilized to establish approximations with a lower relative error. Naturally, the improvement in accuracy with iteration depends on the initial approximation. A better approach is to utilize the iterative relationship to solve for the error, and the result is an iteration formula with improved accuracy, e.g., Ref. [9], Equation (11).

In general, recursive relationships can be utilized to establish approximations, and Lóczi [11] provides a good overview. Iacone [9] and Howard [10] detail useful recursively defined approximations. In general, recursive relationships lead to rapidly convergent approximations, but explicit analytical approximations are usually complex and problematic.

3. Schröder-Based Series

An approach for approximating the root of a function $f$ is to directly utilize the inverse function $f^{-1}$ with the result being Schröder’s approximations of the first kind, e.g., Refs. [18,20]. Essentially, finding the inverse function $f^{-1}$ to a given function $f$ is a root problem, and Schröder’s approximations of the first kind can be utilized to define Schröder-based series for an inverse function, e.g., Ref. [19], and convergent analytical approximations for both the negative one branch and the principal branch of the Lambert W function. Howard [19] details approximations (but not an explicit series form) for the principal branch of the Lambert W function, which are valid over the interval $[0,\infty )$.

Consider the illustration for the negative one branch, shown in Figure 7, with an approximation $h_0^{-1}$ to $h^{-1}$ and $h_0$ to $h$. A similar illustration can be constructed for the principal branch. The analytical form for $h^{-1}$ is unknown, but derivative values of all orders can be established at a set known point leading to the Schröder series for $h^{-1}$. For the point $(z_0,w_0)$, $z_0=w_0{e}^{w_0}$, the Schröder series, $h_S^{-1}$, for $h^{-1}$ can be defined, e.g., Ref. [19], according to

$$\begin{array}{c}{h}_S^{-1}\left(z\right)=h^{-1}\left(z_0\right)+\left(z-z_0\right)D\left[h^{-1}\left(z_0\right)\right]+{\displaystyle \frac{{(z-z_0)}^2{D}^{\left(2\right)}\left[h^{-1}\left(z_0\right)\right]}{2}}+\cdots +\\ {\displaystyle \frac{{(z-z_0)}^k{D}^{\left(k\right)}\left[h^{-1}\left(z_0\right)\right]}{k!}}+\cdots \end{array}$$

(17)

Figure 7. Illustration for the negative one branch where the function $h_0^{-1}$ is an approximation to $h^{-1}$. The goal is to determine $w$ given a set value of $z$ where $z=w{e}^w$. Left: A Taylor series approximation to $w{e}^w$ at the point $\left(w_0,z_0\right)$ can be used to establish an approximation $w_1$ to $w$ where the level $z$ is reached. Right: A Schröder approximation to $h^{-1}\left(z\right)$, based on the point $\left(z_0,w_0\right)$, defines the approximation to $w$ of $w_1$. The definitions for $w_0$ and $z_0$ are as follows: For $z$ fixed, the value $w_0=h_0^{-1}\left(z\right)$ is defined. It is the case that $z=h_0\left(w_0\right)$. The point $z_0$ is defined according to $z_0=h\left(w_0\right)=h_0^{-1}\left(z\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[h_0^{-1}\right(z\left)\right]$ and it then follows that $w_0=h^{-1}\left(z_0\right)$. The fundamental result $w_0=h^{-1}\left(z_0\right)=h_0^{-1}\left(z\right)$ then follows.

Theorem 1.

Schröder-Based Approximation for Lambert W Function. For the principal branch, assume a suitable approximation $h_0^{-1}$ with $h_0^{-1}(-1/e)=-1$ and a point $z_0\in \left[-1/e,\infty \right.)$. For the negative one branch, assume a suitable approximation $h_0^{-1}$ with $h_0^{-1}(-1/e)=-1$ and a point $z_0\in \left[-1/e,0\right.)$. A Schröder series of order $i$, based on the point $z_0=h_0^{-1}\left(z\right)exp\left[h_0^{-1}\right(z\left)\right]$, for either of the two cases, is defined according to

$$\begin{array}{c}{h}_{S,i}^{-1}\left(z\right)=h_0^{-1}\left(z\right)+{\displaystyle \sum _{k=1}^i}{\displaystyle \frac{{(-1)}^{k-1}{\left[z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right]}^k}{{\left[1+h_0^{-1}\left(z\right)\right]}^{2k-1}{e}^{k{h}_0^{-1}\left(z\right)}}}·{\displaystyle \frac{1}{k!}}·{\displaystyle \sum _{j=0}^{k-1}}{c}_{k,j}{\left[h_0^{-1}\left(z\right)\right]}^j\\ =h_0^{-1}\left(z\right)+{\displaystyle \sum _{k=1}^i}{\displaystyle \frac{{\left[z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right]}^k}{{\left[1+h_0^{-1}\left(z\right)\right]}^{2k-1}{e}^{k{h}_0^{-1}\left(z\right)}}}·{\displaystyle \frac{1}{k!}}·p\left[k,h_0^{-1}\left(z\right)\right]\end{array}$$

(18)

Here, the coefficients $c_{k,j}$, $k\in \{1,2,3,\cdots \}$, $j\in \{0,1,2,\cdots ,k-1\}$ are based on a function $p$ and are defined according to

$$\begin{array}{l}p\left[k,w\right]=\left(1+w\right){\displaystyle \frac{\partial p\left[k-1,w\right]}{\partial w}}-\left[\left(k-1\right)w+3k-4\right]p\left[k-1,w\right],\\ p\left[1,w\right]=1,\end{array}$$

(19)

$$\begin{array}{l}{c}_{k,j}={\displaystyle \frac{1}{j!}}·\left|{\left.{\displaystyle \frac{{\partial }^{j}p\left[k,w\right]}{\partial w^j}}\right|}_{w\to 0}\right|, k\in \left\{\mathrm{2,3},4,\cdots \right\}, j\in \left\{\mathrm{0,1},2,\cdots ,k-1\right\},\\ c_{1,0}=1.\end{array}$$

(20)

Proof of Theorem 1.

Given an initial approximation of $h_0^{-1}\left(z\right)$, and the general form for a Schröder series as defined by (17), it follows that an explicit expression for the $k\mathrm{th}$ derivative ${\left.D^{\left(k\right)}\left[h^{-1}\left(z\right)\right]\right|}_{z=z_0}$, where $z_0={\mathrm{h}}_0^{-1}\left(\mathrm{z}\right){\mathrm{e}}^{{\mathrm{h}}_0^{-1}\left(\mathrm{z}\right)}$, is required. Use of the inverse function theorem can be directly used to establish the derivatives, e.g.,

$${\left.{\displaystyle \frac{d\left[h^{-1}\left(z\right)\right]}{dz}}\right|}_{z=z_0}={\displaystyle \frac{1}{{\left.\frac{d\left[h\left(w\right)\right]}{dw}\right|}_{w=w_0,w_0=h^{-1}\left(z_0\right)}}}={\displaystyle \frac{1}{(1+w_0)e^{w_0}}}={\displaystyle \frac{1}{z_0+e^{h^{-1}\left(z_0\right)}}}$$

(21)

Further differentiation (e.g., Ref. [2], Equations (3.3) and (3.4); Ref. [21], Equation (1)) yields the explicit result

$$D^{\left(k\right)}\left[h^{-1}\left(z\right)\right]={\displaystyle \frac{{(-1)}^{k-1}}{{\left[1+h^{-1}\left(z\right)\right]}^{2k-1}{e}^{k{h}^{-1}\left(z\right)}}}·\sum _{j=0}^{k-1}{c}_{k,j}{\left[h^{-1}\left(z\right)\right]}^j$$

(22)

where the coefficients $c_{k,j}$ are defined by (19) and (20)—see [15], Equation (8); Ref. [21], Equation (1.2); Ref. [22], Example 4.3 and https://oeis.org/A042977 (accessed on 10 May 2025). Here, the coefficients, $c_{k,j}$, are defined as being positive, and this is achieved by utilizing the alternating negative one sign via the term ${(-1)}^{k-1}$. It remains to establish ${\left.D^{\left(k\right)}\left[h^{-1}\left(z\right)\right]\right|}_{z=z_0}$, and this is based on the fundamental result $h^{-1}\left(z_0\right)={\mathrm{h}}_0^{-1}\left(\mathrm{z}\right)$, which arises from the definition of $z_0={\mathrm{h}}_0^{-1}\left(\mathrm{z}\right){\mathrm{e}}^{{\mathrm{h}}_0^{-1}\left(\mathrm{z}\right)}$. The details leading to this result are stated in the caption of Figure 7. The required result of

$${\left.D^{\left(k\right)}\left[h^{-1}\left(z\right)\right]\right|}_{z=z_0}={\displaystyle \frac{{(-1)}^{k-1}}{{\left[1+h_0^{-1}\left(z\right)\right]}^{2k-1}{e}^{k{h}_0^{-1}\left(z\right)}}}·\sum _{j=0}^{k-1}{c}_{k,j}{\left[h_0^{-1}\left(z\right)\right]}^j$$

(23)

then follows. □

3.1. Notes

Specific coefficient values for $c_{k,j}$ are detailed in Table 1. It is the case that $c_{k,0}=k^{k-1}$, $c_{k,k-1}=\left(k-1\right)!$ and it can be shown that $c_{k,1}=\left(3k-1\right)k^{k-1}-{(k+1)}^k$ (an alternative expression is detailed in [21], Theorem 2.1).

Table 1. Tabulation of coefficient values $c_{k,j}$ for $k\in \{\mathrm{1,2},\cdots ,8\}$, $j\in \{\mathrm{0,1},2,\cdots ,k-1\}$.

k	j = 0	j = 1	j = 2	j = 3	j = 4	j = 5	j = 6	j = 7
1	1
2	2	1
3	9	8	2
4	64	79	36	6
5	625	974	622	192	24
6	7776	14,543	11,758	5126	1200	120
7	117,649	255,828	248,250	137,512	45,756	8640	720
8	2,097,152	5,187,775	5,846,760	3,892,430	1,651,480	445,572	70,560	5040

Schröder series are independent of the nature of the continuity/differentiability of the approximating function $h_0^{-1}$; this function simply specifies an initial approximation value for $h^{-1}\left(z\right)$, $z$ set.

3.2. Explicit Expressions

An explicit expansion for a Schröder series is

$$\begin{array}{l}\begin{array}{l}\begin{array}{l}{h}_S^{-1}\left(z\right)=h_0^{-1}\left(z\right)+{\displaystyle \frac{\mathrm{z}-h_0^{-1}\left(z\right){\mathrm{e}}^{h_0^{-1}\left(z\right)}}{\left[1+h_0^{-1}\left(z\right)\right]{\mathrm{e}}^{h_0^{-1}\left(z\right)}}}-{\displaystyle \frac{{\left[\mathrm{z}-h_0^{-1}\left(z\right){\mathrm{e}}^{h_0^{-1}\left(z\right)}\right]}^2}{{2\left[1+h_0^{-1}\left(z\right)\right]}^3{\mathrm{e}}^{2h_0^{-1}\left(z\right)}}}·\left[2+h_0^{-1}\left(z\right)\right]+\\ {\displaystyle \frac{{[\mathrm{z}-h_0^{-1}(z\left){\mathrm{e}}^{h_0^{-1}\left(z\right)}\right]}^3}{{6\left[1+h_0^{-1}\left(z\right)\right]}^5{\mathrm{e}}^{3h_0^{-1}\left(z\right)}}}·\left[9+{8h}_0^{-1}\left(z\right)+{2\left[h_0^{-1}\left(z\right)\right]}^2\right]-\end{array}\\ {\displaystyle \frac{{\left[\mathrm{z}-h_0^{-1}\left(z\right){\mathrm{e}}^{h_0^{-1}\left(z\right)}\right]}^4}{{24\left[1+h_0^{-1}\left(z\right)\right]}^7{\mathrm{e}}^{4h_0^{-1}\left(z\right)}}}·\left[64+{79h}_0^{-1}\left(z\right)+36{\left[h_0^{-1}\left(z\right)\right]}^2+6{\left[h_0^{-1}\left(z\right)\right]}^3\right]+\end{array}\\ {\displaystyle \frac{{\left[\mathrm{z}-h_0^{-1}\left(z\right){\mathrm{e}}^{h_0^{-1}\left(z\right)}\right]}^5}{{120\left[1+h_0^{-1}\left(z\right)\right]}^9{\mathrm{e}}^{5h_0^{-1}\left(z\right)}}}·\left[\begin{array}{r}625+{974h}_0^{-1}\left(z\right)+622{\left[h_0^{-1}\left(z\right)\right]}^2+\\ 192{\left[h_0^{-1}\left(z\right)\right]}^3+24{\left[h_0^{-1}\left(z\right)\right]}^4\end{array}\right]-\cdots \end{array}$$

(24)

A first-order Schröder series approximation simplifies to the following form:

$$h_1^{-1}\left(z\right)={\displaystyle \frac{{\left[h_0^{-1}\left(z\right)\right]}^2+z·\mathrm{e}\mathrm{x}\mathrm{p}[-h_0^{-1}(z\left)\right]}{1+{\mathrm{h}}_0^{-1}\left(\mathrm{z}\right)}}$$

(25)

3.3. Special Approximation Form

Custom approximations for the principal and negative one branches are often in the form $h_0^{-1}\left(z\right)=\ln \left(z\right)-\ln \left[A\left(z\right)\right]$, $z>0$, $A\left(z\right)>0$, or the form $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[-A\left(z\right)\right],$ $z\in [-1/e,0)$, $A\left(z\right)<0$, for a defined function $A.$ It is of interest to establish a general form for a Schröder series based on such an approximation.

Theorem 2.

Schröder Series: Special Approximation Form. For the case of an initial approximation of the form $h_0^{-1}\left(z\right)=\ln \left(\pm z\right)-\ln \left[\pm A\left(z\right)\right]$, a Schröder series of order $i$ is defined according to

$$\begin{gathered} h_{S,i}^{-1}\left(z\right)=\mathit{ln}\left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]+\end{array}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ \sum _{k=1}^i{\displaystyle \frac{{(-1)}^{k-1}{\left[A\left(z\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^k}{{\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^{2k-1}}}·{\displaystyle \frac{1}{k!}}·\sum _{j=0}^{k-1}{c}_{k,j}{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^j \end{gathered}$$

(26)

with the positive sign for the case of $z,A\left(z\right)>0$; the negative sign for the case of $A\left(z\right)<0,$ $z\in [-1/e,0)$.

Proof of Theorem 2.

For the general case of $h_0^{-1}\left(z\right)=\ln \left[z/A\left(z\right)\right]$, $z,A\left(z\right)>0$ or $z,A\left(z\right)<0$, it is the case that

$$\begin{array}{cc}\begin{array}{cc}{e}^{h_0^{-1}\left(z\right)}={\displaystyle \frac{z}{A\left(z\right)}},& e^{{kh}_0^{-1}\left(z\right)}={\displaystyle \frac{z^k}{A^k\left(z\right)}},\end{array}& k\in \{1,2,\end{array}\cdots \}.$$

(27)

It then follows, from the general form for a Schröder series of order $i$, as specified by (18), that

$$\begin{array}{ll}{h}_{S,i}^{-1}\left(z\right)& =h_0^{-1}\left(z\right)+{\displaystyle \sum _{k=1}^i}{\displaystyle \frac{{(-1)}^{k-1}{\left[z-\frac{z}{A\left(z\right)}·\ln \begin{array}{c}\left[\frac{z}{A\left(z\right)}\right]\end{array}\right]}^k}{{\left[1+\ln \begin{array}{c}\left[\frac{z}{A\left(z\right)}\right]\end{array}\right]}^{2k-1}·\frac{z^k}{A^k\left(z\right)}}}·{\displaystyle \frac{1}{k!}}·{\displaystyle \sum _{j=0}^{k-1}}{c}_{k,j}{\ln \begin{array}{c}\left[{\displaystyle \frac{z}{A\left(z\right)}}\right]\end{array}}^j\\ & =h_0^{-1}\left(z\right)+{\displaystyle \sum _{k=1}^i}{\displaystyle \frac{{(-1)}^{k-1}{A^k\left(z\right)\left[1-\frac{1}{A\left(z\right)}·\ln \begin{array}{c}\left[\frac{z}{A\left(z\right)}\right]\end{array}\right]}^k}{{\left[1+\ln \begin{array}{c}\left[\frac{z}{A\left(z\right)}\right]\end{array}\right]}^{2k-1}}}·{\displaystyle \frac{1}{k!}}·{\displaystyle \sum _{j=0}^{k-1}}{c}_{k,j}{\ln \begin{array}{c}\left[{\displaystyle \frac{z}{A\left(z\right)}}\right]\end{array}}^j\\ & =\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]+\end{array}\\ & {\displaystyle \sum _{k=1}^i}{\displaystyle \frac{{(-1)}^{k-1}{\left[A\left(z\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^k}{{\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^{2k-1}}}·{\displaystyle \frac{1}{k!}}·{\displaystyle \sum _{j=0}^{k-1}}{c}_{k,j}{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^j\end{array}$$

(28)

with the positive sign for the case of $z,A\left(z\right)>0$; a negative sign for the case of $A\left(z\right)<0$, $z\in [-1/e,0)$. □

Explicit Expression

An explicit expansion for this custom form of the Schröder series is

$$\begin{array}{l}\begin{array}{l}\begin{array}{l}{h}_S^{-1}\left(z\right)=\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}+{\displaystyle \frac{\mathrm{A}\left(\mathrm{z}\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}}{\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}}-\\ {\displaystyle \frac{{\left[\mathrm{A}\left(\mathrm{z}\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^2}{{2\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^3}}·\left[2+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]+\\ {\displaystyle \frac{{[\mathrm{A}\left(\mathrm{z}\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}]}^3}{{3!\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^5}}·\left[\begin{array}{c}9+8[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}]+\\ {2\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^2\end{array}\right]-\end{array}\\ {\displaystyle \frac{{\left[\mathrm{A}\left(\mathrm{z}\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^4}{{4!\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^7}}·\left[\begin{array}{r}64+79[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}]+\\ 36{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^2+\\ 6{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^3\end{array}\right]+\end{array}\\ {\displaystyle \frac{{\left[\mathrm{A}\left(\mathrm{z}\right)-\ln \left(\pm z\right)+\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^5}{{5!\left[1+\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^9}}·\left[\begin{array}{r}\begin{array}{r}625+974[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}]+\\ 622{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^2+\end{array}\\ \begin{array}{r}192{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^3+\\ 24{\left[\ln \left(\pm z\right)-\ln \begin{array}{c}\left[\pm A\left(z\right)\right]\end{array}\right]}^4\end{array}\end{array}\right]-\cdots \end{array}$$

(29)

4. Initial Approximations for Negative One Branch

To establish a Schröder series for the Lambert W function, consistent with Theorem 1 or Theorem 2, a suitable initial approximation, which, ideally, is valid over the completed domain $\left[-1/e,0\right)$ for the negative one branch and $\left[-1/e,\infty \right)$ for the principal branch is required. The goal is for simple approximations with high accuracy over the complete domain. The requirements of simplicity and accuracy are usually incompatible, and a trade-off is usual. The negative one branch is considered in this section; the principal branch is in the following section.

Theorem 3.

Initial Approximations: Negative One Branch. Two initial approximations for the negative one branch, that are valid for the domain $\left[-1/e,0\right)$, are

$$h_0^{-1}\left(z\right)=\left(1+ez\right)\ln \left(-z\right)-\left[1+\sqrt{2}\sqrt{1+ez}\right]$$

(30)

$$h_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$$

(31)

The relative error bounds associated with these approximations, respectively, are  $9.03\times {10}^{-2}$ and $8.95\times {10}^{-2}$. An improved approximation is

$$h_1^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]$$

(32)

 which has a relative error bound of $1.17\times {10}^{-2}$.

Proof of Theorem 3.

Consider a Taylor series approximation for $h\left(w\right)=w{e}^w$, based on the point $-1$, as defined by (3), and the resultant approximation, defined in (4), for the negative one branch of ${\mathrm{h}}_0^{-1}\left(\mathrm{z}\right)=-1-\sqrt{2}\sqrt{1+\mathrm{e}\mathrm{z}}$. The relative error associated with this approximation is shown in Figure 8, and such an approximation has an increasing relative error consistent with ${\mathrm{h}}_0^{-1}\left(0\right)=-1-\sqrt{2}$, as $z\to 0^{-}$. In contrast, the approximation $h^{-1}\left(z\right)\approx \mathrm{l}\mathrm{n}(-z)$, arising from the relationship defined in (15) of $\ln \left(-z\right)=w+\mathrm{l}\mathrm{n}(-z)$, is an approximation with a decreasing relative error magnitude as $z\to 0^{-}$ (see Figure 8). These two approximations can be combined via the use of the demarcation function $1+ez$ for $\mathrm{l}\mathrm{n}(-z)$, leading to the composite approximation defined according to

$$h_0^{-1}\left(z\right)=-1-\sqrt{2}\sqrt{1+ez}+\left(1+ez\right)\ln \left(-z\right)$$

(33)

Figure 8. Graphs of the relative errors in approximations for the negative one branch.

The second approximation, defined in (31), arises from this equation by noting that $\ln \left(-z\right)$ equals $-1$ when $z=-1/e$ and by excluding the demarcation function $1+ez$.

The third approximation, defined in (32), arises from a first-order iteration, based on (16), of the second approximation defined by $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$. □

4.1. Notes

The graphs of the relative errors in the approximations detailed in Theorem 3 are shown in Figure 8. The second approximation (31) is simpler than the first, has a slightly lower relative error bound, and is a preferable approximation. The broad requirements of a modest relative error bound over the domain of approximation and with a form consistent with modest or low complexity are satisfied by the second and third approximations, with the latter being preferable because of its significantly lower relative error bound.

4.2. Published Approximations

Only published approximations that are valid over the domain for the negative one branch of $[-1/e,0)$ are considered. First, similar approximations to those detailed in (30) and (31) have been proposed by Chatzigeorgiou [14], Theorem 1:

$$\begin{array}{l}{h}_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{-1-\ln (-z)}\\ h_0^{-1}\left(z\right)={\displaystyle \frac{2}{3}}\ln \left(-z\right)-\sqrt{2}\sqrt{-1-\ln \left(-z\right)}-{\displaystyle \frac{1}{3}}\end{array}$$

(34)

The first approximation has a relative error bounds of $0.137$ for the interval $[-1/e,0)$; the second has a slowly increasing relative error as $z\to 0^{-}$ and a relative error bound of $0.161$ for $[-1/e,{10}^{-12})$.

A better approximation, but with greater complexity, has been detailed by Lóczi ([11], remark 2.19):

$$h_0^{-1}\left(\mathrm{z}\right)={\displaystyle \frac{ez\mathrm{l}\mathrm{n}[1-\sqrt{1+ez}]\left[\ln \left[1+ez-\sqrt{1+ez}\right]-\ln \left[\ln \left[1-\sqrt{1+ez}\right]\right] \right]}{1+ez-\sqrt{1+ez}+ez\mathrm{l}\mathrm{n}[1-\sqrt{1+ez}]}}$$

(35)

which has a relative error bound of $7.24\times {10}^{-3}$.

Based on the series defined in (12), Barry et al. [13] (see Equation (18)) proposed the approximation

$$h_0^{-1}\left(z\right)=-1-{\displaystyle \frac{{\sigma }^2}{2}}-{\displaystyle \frac{\sigma }{1+\frac{\sigma }{6}}}=\ln \left(-z\right)-{\displaystyle \frac{\sqrt{2}\sqrt{-1-\mathrm{l}\mathrm{n}(-\mathrm{z})}}{1+\frac{\sqrt{2}\sqrt{-1-\mathrm{l}\mathrm{n}(-\mathrm{z})}}{6}}}$$

(36)

which has a relative error bound of $4.61\times {10}^{-3}$. By replacing the factor $6$ by $6.3$, the relative bound is reduced to $2.90\times {10}^{-3}$. Iteration based on (15) yields the approximation (Equation (22) in [13])

$$h_1^{-1}\left(\mathrm{z}\right)=\ln \left(-z\right)-\ln \left[{\displaystyle \frac{\sqrt{2}\sqrt{-1-\mathrm{l}\mathrm{n}(-\mathrm{z})}}{1+\frac{\sqrt{2}\sqrt{-1-\mathrm{l}\mathrm{n}(-\mathrm{z})}}{6}}}-\mathrm{l}\mathrm{n}(-z)\right]$$

(37)

which has a relative error bound of $3.05\times {10}^{-4}$. Barry et al. [13] detail further approximations (see Equations (24)–(26) and (28)) with lower relative error bounds but with increasing complexity. An earlier approximation, with a relative error bound of $2.53\times {10}^{-4}$, was published by Barry et al. [12] (Equation (A5)) in 1993 and has been referenced in later papers, e.g., Ref. [4], Equation (5); Ref. [23], Equation (3) and [8], Equation (11).

5. Initial Approximations for Principal Branch

Establishing approximations over the domain $\left[-1/e,\infty \right)$ for the principal branch is facilitated by simple affine transformations. With $z=h\left(w\right)=w{e}^w$, $w=h^{-1}\left(z\right)$, $\mathrm{w}\in [-1,\infty )$ and $z\in \left[-1/e,\infty \right)$, consider the definitions $z_1=h_1\left(w_1\right)$ and $w_1=h_1^{-1}\left(z_1\right)$, where

$$\begin{array}{cc}{z}_1=z+{\displaystyle \frac{1}{e}},& z_1\in \left[0,\infty \right),\\ w_1=w+1,& w_1\in \left[0,\infty \right).\end{array}$$

(38)

It then follows that

$$\begin{array}{l}{z}_1=h_1\left(w_1\right)={\displaystyle \frac{1}{e}}+h\left(w_1-1\right)={\displaystyle \frac{1}{e}}+\left(w_1-1\right)e^{w_1-1},\\ w_1=h_1^{-1}\left(z_1\right)=1+h^{-1}\left(z\right)=1+h^{-1}\left(z_1-{\displaystyle \frac{1}{e}}\right),\end{array}$$

(39)

and graphs of $h\left(w\right)$, $h_1\left(w_1\right)$, $h^{-1}\left(z\right)$ and $h_1^{-1}\left(z_1\right)$ are shown in Figure 9. It then directly follows that

$$w=h^{-1}\left(z\right)=h_1^{-1}\left(z_1\right)-1=h_1^{-1}\left[z+{\displaystyle \frac{1}{e}}\right]-1.$$

(40)

The following derivative values can then be established (see Appendix A):

$$\begin{array}{cc}{\left.D\left[h_1\left(w_1\right)\right]\right|}_{w_1=1}=1,& {\left.D\left[h_1^{-1}\left(z_1\right)\right]\right|}_{z_1=1/e}=1.\end{array}$$

(41)

Figure 9. Graphs of $h\left(w\right)$, $h_1\left(w_1\right)$, $h^{-1}\left(z\right)$ and $h_1^{-1}\left(z_1\right)$.

5.1. Initial Approximation

Consider the transformed function $z_1=h_1\left(w_1\right)$, defined by (39), and the series expansion at the origin for ${w_1=h}_1^{-1}\left(z_1\right).$ as detailed in Appendix A and specified by (A4):

$$h_1^{-1}\left(z_1\right)=\sqrt{2e{z}_1}-{\displaystyle \frac{2e{z}_1}{3}}+{\displaystyle \frac{11e^{3/2}{z}_1^{3/2}}{18\sqrt{2}}}-{\displaystyle \frac{43e^2{z}_1^2}{135}}+\cdots , z_1>0.$$

(42)

The requirement, for the case of $z_1\gg 1$, is for a logarithmic variation, and as $\ln \left(1+x\right)\approx x$ for $\left|x\right|\ll 1$, it follows that an initial approximation of the form

$$h_{1,0}^{-1}\left(z_1\right)=\ln \left[1+\sqrt{2e{z}_1}\right],$$

(43)

can be considered which will have a low relative error close to the origin. As a form of $\mathrm{l}\mathrm{n}\left(z_1\right)$ is required for $z_1\gg 1$, the approximation can be modified according to

$$h_{1,0}^{-1}\left(z_1\right)=\ln \left[1+\sqrt{2e{z}_1}\left[1+{\alpha }_1\sqrt{z_1}\right]\right],$$

(44)

for an optimally chosen constant ${\alpha }_1$.

At the point $z_1=1/e$, corresponding to $z=0$, the requirement is for $h_{1,0}^{-1}\left(1/e\right)=1$ and, as specified by (41), ${\left.D\left[h_{1,0}^{-1}\left(z_1\right)\right]\right|}_{z_1=1/e}=1$. To satisfy these two constraints, a second constant needs to be incorporated into the approximation form. Consistent with the asymptotic series defined in (7) and the nature of variation for large arguments, a term of the form with a double logarithm nature is appropriate, leading to the approximation form

$$h_{1,0}^{-1}\left(z_1\right)=\ln \left[1+\sqrt{2e{z}_1}\left[1+{\alpha }_1\sqrt{z_1}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}(1+{\alpha }_3{z}_1)\right]\right].$$

(45)

Here, an additional variable ${\alpha }_3$ has been included. The two constraints of $h_{1,0}^{-1}\left(1/e\right)=1$ and ${\left.D\left[h_{1,0}^{-1}\left(z_1\right)\right]\right|}_{z_1=1/e}=1$ are satisfied when

$${\alpha }_1={\displaystyle \frac{\sqrt{\frac{e}{2}}·\left[2{\alpha }_3\left[1+\sqrt{2}-e\right]+\left(2-\sqrt{2}\right)\left({\alpha }_3+e\right)\mathrm{l}\mathrm{n}({\alpha }_3+e)\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\right({\alpha }_3+e\left)\right]\right]}{-2{\alpha }_3+2({\alpha }_3+e)\mathrm{l}\mathrm{n}({\alpha }_3+e)\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\right({\alpha }_3+e\left)\right]}}$$

(46)

$${\alpha }_2={\displaystyle \frac{-(4+\sqrt{2}-2e)\left({\alpha }_3+e\right)\mathrm{l}\mathrm{n}({\alpha }_3+e)}{-2{\alpha }_3+2({\alpha }_3+e)\mathrm{l}\mathrm{n}({\alpha }_3+e)\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\right({\alpha }_3+e\left)\right]}}$$

(47)

The associated approximation to $h^{-1}$, based on the relationship $h^{-1}\left(z\right)=h_1^{-1}\left(z+1/e\right)-1$, is

$$h_0^{-1}\left(z\right)=\ln \left[1+\sqrt{2}\sqrt{1+ez}\left[1+{\displaystyle \frac{{\alpha }_1}{\sqrt{e}}}\sqrt{1+ez}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}\left[1+{\displaystyle \frac{{\alpha }_3}{e}}(1+ez)\right]\right]\right]-1.$$

(48)

The coefficient ${\alpha }_3$ can be chosen to minimize the relative error bound in $h_0^{-1}$ over the interval $\left[-1/e,\infty \right).$ The optimum value is ${\alpha }_3\approx 22/100$ leading to ${\alpha }_1\approx 0.176242$ and ${\alpha }_2\approx 2.04009$. The relative error bound associated with $h_0^{-1}$, over the interval $\left[-1/e,\infty \right)$, is $0.0556$. The relative error bound, over the interval $\left[-1/e,1/e\right],$ is $3.59\times {10}^{-3}$. A graph of the relative error is shown in Figure 10.

Figure 10. Graphs of the relative errors in the approximation $h_0^{-1}\left(z\right)$ defined by (48), for the optimized case of ${\alpha }_3=22/100$, and (50) for the case of ${\alpha }_3=1/2.$

Improved Approximation

To establish an approximation with a lower relative error bound, a first-order iterative approximation, based on $h_{1,0}^{-1}\left(z_1\right)$ as specified by (45), yields the approximation (see Appendix A, (A7)) of

$$h_{\mathrm{1,1}}^{-1}\left(z_1\right)=\left\{\begin{array}{l}\begin{array}{c}1+\ln \left[{\displaystyle \frac{z_1-1/e}{\ln \left[1+\sqrt{2e{z}_1}\left[1+{\alpha }_1\sqrt{z_1}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}(1+{\alpha }_3{z}_1)\right]\right]-1}}\right]\\ z_1\ne 1/e,\end{array} \\ 1, z_1=1/e\end{array}\right.$$

(49)

The associated approximation for $h^{-1}$, based on the relationship $h^{-1}\left(z\right)=h_1^{-1}\left(z+1/e\right)-1$, is

$$h_0^{-1}\left(z\right)=\left\{\begin{array}{l}\begin{array}{c}\ln \left[{\displaystyle \frac{z}{\ln \left[1+\sqrt{2}\sqrt{1+ez}\left[1+\frac{{\alpha }_1}{\sqrt{e}}\sqrt{1+ez}\right]+{\alpha }_2\ln \left[1+\ln \left[1+\frac{{\alpha }_3}{e}(1+ez)\right]\right]\right]-1}}\right]\\ z\ne 0\end{array} \\ 0, z=0\end{array}\right.$$

(50)

The constraint of $\underset{z\to 0}{\lim }{h}_0^{-1}\left(z\right)=0$ is satisfied when ${\alpha }_1$ and ${\alpha }_2$ are as specified in (46) and (47). The coefficient ${\alpha }_3$ can be chosen to minimize the relative error bound over the interval $\left(-1/e,\infty \right)$. The optimum value for ${\alpha }_3$ is close $0.46$ with a relative error bound for $h_0^{-1}$, over the interval $\left[-1/e,\infty \right)$, of $9.95\times {10}^{-3}$. A pragmatic value is ${\alpha }_3=1/2$ with a relative error bound of $1.06\times {10}^{-2}$. With such a value, the coefficients ${\alpha }_1$ and ${\alpha }_2$ become

$${\alpha }_1={\displaystyle \frac{\sqrt{\frac{e}{2}}·\left[1+\sqrt{2}-e+\left(2-\sqrt{2}\right)\left[e+\frac{1}{2}\right]\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]\right]\right]}{-1+(1+2e)\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]\right]}}$$

(51)

$${\alpha }_2={\displaystyle \frac{-\left(4+\sqrt{2}-2e\right)\left[e+\frac{1}{2}\right]\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]}{-1+(1+2e)\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\left[e+\frac{1}{2}\right]\right]}}$$

(52)

with approximate values ${\alpha }_1\approx 0.266493$ and ${\alpha }_2\approx 0.483789$.

The relative error in this approximation, with ${\alpha }_3=1/2$, is shown in Figure 10. Note, however, that the constraint of ${\left.D\left[h_0^{-1}\left(z\right)\right]\right|}_{z=0}=1$ is not satisfied. To satisfy such a constraint, an additional term in the original approximation, as specified by (45), is required. This results in a more complex approximation and complex expressions for the coefficients.

5.2. Published Results

Many approximations for the principal branch have been proposed, but only a few have low relative errors over the complete domain of $\left[-1/e,\infty \right)$. Barry et al. [4], Equation (15), proposed the approximation

$$h_0^{-1}\left(z\right)=\left[1+\delta \right]\ln \left[{\displaystyle \frac{6}{5}}·{\displaystyle \frac{z}{\ln \left[\frac{12}{5}·\frac{z}{\mathrm{l}\mathrm{n}(1+12z/5)}\right]}}\right]-\delta \ln \left[{\displaystyle \frac{2z}{\mathrm{l}\mathrm{n}(1+2z)}}\right]$$

(53)

$\delta =0.4586887$, which has a relative error bound, for $z>0$, of $1.96\times {10}^{-3}$ but a high relative error as $z\to -1/e$. An improved approximation has been proposed by Iacono and Boyd ([9], Equations (19) and (20)):

$$\begin{array}{l}{h}_0^{-1}\left(z\right)=-1+a·\ln \left[{\displaystyle \frac{1+b\sqrt{1+ez}}{1+c·\ln \left[1+\sqrt{1+ez}\right]}}\right],\\ \begin{array}{cc}& \begin{array}{cc}c={\displaystyle \frac{e^{1/a}-1-\sqrt{2}/a}{1-\mathrm{l}\mathrm{n}\left(2\right)e^{1/a}}},& b={\displaystyle \frac{\sqrt{2}}{a}}\end{array}+c,\end{array}\end{array}$$

(54)

which yields a relative error bound for $z>-1/e$ of $4.53\times {10}^{-3}$ (the bound occurs at $z$ of the order of ${10}^{12}$) for the case of $a$ optimally chosen as $a=2.036$. The approximation is sharp at $z=-1/e$.

6. Schröder-Based Approximations

Schröder approximations for the negative one branch, and the principal branch, are detailed in the following two subsections.

6.1. Negative One Branch

For the negative one branch, consider the initial approximation, $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]$, defined in (32). The first order Schröder approximation, as defined by (25), is

$$h_{S,1}^{-1}\left(z\right)={\displaystyle \frac{-\sqrt{2}\sqrt{1+ez}+\ln \left(-z\right)+{\left[\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]\right]}^2}{1+\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]}}$$

(55)

and has a relative error bound, over the domain of $[-1/e,0)$, of $4.00\times {10}^{-4}$. This approximation is of similar complexity to the approximation proposed by Lóczi ([11], remark 2.19), stated in (35), which has a higher relative error bound of $7.24\times {10}^{-3}$. The relative error bound is comparable with the relative error bound of $3.05\times {10}^{-4}$ for the approximation proposed by Barry et al. [13] (Equation (22)) and stated in (37).

A second order Schröder approximation arising from (24) is

$$\begin{gathered} h_{S,2}^{-1}\left(z\right)={\displaystyle \frac{-\sqrt{2}\sqrt{1+ez}+\ln \left(-z\right)+{\left[\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\right]}^2}{1+\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]}}- \\ {\displaystyle \frac{\left[2+\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\right]{\left[\sqrt{2}\sqrt{1+ez}-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\right]}^2}{{2\left[1+\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\right]}^3}} \end{gathered}$$

(56)

and has a relative error bound, over the domain of $[-1/e,0)$, of $2.05\times {10}^{-5}$.

A fifth-order series approximation, as defined by (29) with $A\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$, is

$$\begin{array}{l}\begin{array}{l}\begin{array}{l}{h}_{S,5}^{-1}\left(z\right)=\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}+{\displaystyle \frac{-\sqrt{2}\sqrt{1+ez}+\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}}{1+\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}}}-\\ {\displaystyle \frac{{\left[-\sqrt{2}\sqrt{1+ez}+\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^2}{{2\left[1+\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^3}}·\left[2+\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]+\\ {\displaystyle \frac{{[-\sqrt{2}\sqrt{1+ez}+\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}]}^3}{{3!\left[1+\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^5}}·\left[\begin{array}{c}9+8[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}]+\\ {2\left[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^2\end{array}\right]-\end{array}\\ {\displaystyle \frac{{\left[-\sqrt{2}\sqrt{1+ez}+\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^4}{{4!\left[1+\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^7}}·\left[\begin{array}{r}64+79[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}]+\\ 36{\left[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^2+\\ 6{\left[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^3\end{array}\right]+\end{array}\\ {\displaystyle \frac{{\left[-\sqrt{2}\sqrt{1+ez}+\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^5}{{5!\left[1+\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^9}}·\left[\begin{array}{r}\begin{array}{r}625+974[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}]+\\ 622{\left[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^2+\end{array}\\ \begin{array}{r}192{\left[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^3+\\ 24{\left[\ln \left(-z\right)-\ln \begin{array}{c}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\end{array}\right]}^4\end{array}\end{array}\right]\end{array}$$

(57)

and has a relative error bound, over the domain of $[-1/e,0)$, of $5.27\times {10}^{-9}$.

The graphs of the relative errors of first to fourth-order Schröder approximations, based on $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]$, are shown in Figure 11.

Figure 11. Graphs of the relative errors in first to fourth order Schröder approximations for the negative one branch based on an initial approximation defined by $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]$.

The improvement in the relative error bound, as the order of the series increases, is detailed in Table 2 for the two initial approximations defined by (31) and (32). Note the poor convergence of Schröder approximations based on (31).

Table 2. The improvement, with the order of approximation, in the relative error bounds for the negative one branch associated with the two initial approximations defined by (31) and (32).

Order of Approx.	Equation (31): Relative Error Bound for the Interval $(-1/\mathit{e},{10}^{-60})$	Equation (31): Relative Error Bound for the Interval $(-1/\mathit{e},{10}^{-12})$	Equation (32): Relative Error Bound for the Interval $(-1/\mathit{e},0)$
initial approx.: $h_0^{-1}\left(z\right)$	$8.95\times {10}^{-2}$	$8.95\times {10}^{-2}$	$1.17\times {10}^{-2}$
first order: $h_{S,1}^{-1}\left(z\right)$	$3.81\times {10}^{-2}$	$3.81\times {10}^{-2}$	$4.00\times {10}^{-4}$
second order: $h_{S,2}^{-1}\left(z\right)$	$2.42\times {10}^{-2}$	$2.42\times {10}^{-2}$	$2.05\times {10}^{-5}$
third order: $h_{S,3}^{-1}\left(z\right)$	$1.78\times {10}^{-2}$	$1.72\times {10}^{-2}$	$1.22\times {10}^{-6}$
fourth order: $h_{S,4}^{-1}\left(z\right)$	$1.40\times {10}^{-2}$	$1.27\times {10}^{-2}$	$7.82\times {10}^{-8}$
eighth order: $h_{S,8}^{-1}\left(z\right)$	$7.66\times {10}^{-3}$	$4.50\times {10}^{-3}$	$1.89\times {10}^{-12}$
16th order: $h_{S,16}^{-1}\left(z\right)$	$3.99\times {10}^{-3}$	$8.03\times {10}^{-4}$	$2.01\times {10}^{-21}$
32nd order: $h_{S,32}^{-1}\left(z\right)$	$1.62\times {10}^{-3}$	$3.96\times {10}^{-5}$	$4.34\times {10}^{-39}$
64th order: $h_{S,64}^{-1}\left(z\right)$	$3.79\times {10}^{-4}$	$1.61\times {10}^{-7}$	$3.97\times {10}^{-74}$
128th order: $h_{S,128}^{-1}\left(z\right)$	$3.20\times {10}^{-5}$	$4.74\times {10}^{-12}$	$6.57\times {10}^{-144}$

6.2. Principal Branch

For the principal branch, consider two approximations: the first defined in (48) with ${\alpha }_3=22/100$ and with ${\alpha }_1$ and ${\alpha }_2$ defined, respectively, in (46) and (47), and the second defined in (50) with ${\alpha }_3=1/2$ and with ${\alpha }_1$ and ${\alpha }_2$ and defined, respectively, in (51) and (52).

A first-order Schröder approximation, based on (50), has the explicit analytical form

$$h_{S,1}^{-1}\left(z\right)={\displaystyle \frac{-1+\ln \left[A\left(z\right)\right]+\ln {\left[\frac{z}{\ln \left[A\left(z\right)\right]-1}\right]}^2}{1+\ln \left[\frac{z}{\ln \left[A\left(z\right)\right]-1}\right]}}$$

(58)

where

$$A\left(z\right)=1+\sqrt{2}\sqrt{1+ez}+{\alpha }_1\sqrt{{\displaystyle \frac{2}{e}}}·\left(1+ez\right)+{\alpha }_2\ln \left[1+\ln \left[1+{\displaystyle \frac{1}{2e}}·\left(1+ez\right)\right]\right],$$

(59)

and a relative error bound for the interval $[-1/e,\infty )$, of $4.13\times {10}^{-4}$. The relative error bound for the interval $[-1/e,1/e]$, is $1.42\times {10}^{-5}$. The relative error bound of $4.13\times {10}^{-4}$ is an order of magnitude lower than the relative error bound of $4.53\times {10}^{-3}$ that is associated with the approximation proposed by Iacono and Boyd ([9], Equations (19) and (20)), and stated in (54). This approximation, however, has a greater degree of complexity.

The relative error bounds for higher order Schröder approximations, based on the approximations defined by (48) and (50), are tabulated in Table 3. For a set order, the relative error bound for the Schröder approximation based on (48) decreases relatively rapidly for the interval $[-1/e,1/e]$ but is slowly increasing as the upper bound of the interval being considered is increased.

Table 3. The improvement, with the order of approximation, in the relative error bounds for the principal branch based on the initial approximations defined by (48) and (50).

Order of Approx.	Equation (48). Relative Error Bound for the Interval $\left(-1/\mathit{e},1/\mathit{e}\right)$.	Equation (48). Relative Error Bound for the Interval $\left(-1/\mathit{e},{10}^{12}\right)$.	Equation (50). Relative Error Bound for the Interval $\left(-1/\mathit{e},1/\mathit{e}\right)$.	Equation (50). Relative Error Bound for the Interval $\left(-1/\mathit{e},\mathit{\infty }\right)$.
initial approx.: $h_0^{-1}\left(z\right)$	$3.60\times {10}^{-3}$	$5.56\times {10}^{-2}$	$7.57\times {10}^{-3}$	$1.06\times {10}^{-2}$
first order: $h_{S,1}^{-1}\left(z\right)$	$3.22\times {10}^{-6}$	$2.42\times {10}^{-2}$	$1.42\times {10}^{-5}$	$4.13\times {10}^{-3}$
second order: $h_{S,2}^{-1}\left(z\right)$	$4.21\times {10}^{-9}$	$1.33\times {10}^{-2}$	$3.89\times {10}^{-8}$	$2.38\times {10}^{-5}$
third order: $h_{S,3}^{-1}\left(z\right)$	$6.45\times {10}^{-12}$	$7.96\times {10}^{-3}$	$1.25\times {10}^{-10}$	$1.59\times {10}^{-6}$
fourth order: $h_{S,4}^{-1}\left(z\right)$	$1.08\times {10}^{-14}$	$4.97\times {10}^{-3}$	$4.39\times {10}^{-13}$	$1.15\times {10}^{-7}$
eighth order: $h_{S,8}^{-1}\left(z\right)$	$1.30\times {10}^{-25}$	$9.36\times {10}^{-4}$	$1.02\times {10}^{-22}$	$4.42\times {10}^{-12}$
16th order: $h_{S,16}^{-1}\left(z\right)$	$1.53\times {10}^{-46}$	$4.99\times {10}^{-5}$	$1.18\times {10}^{-41}$	$1.20\times {10}^{-20}$
32nd order: $h_{S,32}^{-1}\left(z\right)$	$1.09\times {10}^{-87}$	$2.33\times {10}^{-7}$	$3.83\times {10}^{-79}$	$1.69\times {10}^{-37}$
64th order: $h_{S,64}^{-1}\left(z\right)$	$1.51\times {10}^{-169}$	$9.00\times {10}^{-12}$	$1.04\times {10}^{-153}$	$6.59\times {10}^{-71}$
128th order: $h_{S,128}^{-1}\left(z\right)$		$2.49\times {10}^{-20}$		$2.00\times {10}^{-137}$

7. Convergence Analysis

Consider a Schröder series of order $i$, based on an initial approximation $h_0^{-1}$, and defined by Theorem 1. Convergence is dependent, in part, on the nature of the summation

$$p\left(k,h_0^{-1}\right)={\left(-1\right)}^{k-1}\sum _{j=0}^{k-1}{c}_{k,j}{\left[h_0^{-1}\right]}^j$$

(60)

where $p$ is defined by (19) and the coefficients $c_{k,j}$ are positive numbers. The following approximation underpins establishing bounds for convergence:

Theorem 4.

Approximation for Summation. The following approximation, related to the sum $p$, is valid for large $k$:

$$\left|{\displaystyle \frac{p\left(k+1,h_0^{-1}\right)}{p\left(k,h_0^{-1}\right)}}\right|\approx \left\{\begin{array}{c}k\left[3+h_0^{-1}\right],\hspace{1em} h_0^{-1}>-1, principal branch\hfill \\ {\displaystyle \frac{k\left|h_0^{-1}\right|}{c_A\left(h_0^{-1}\right)}}, h_0^{-1}<-1, negative one branch\end{array}\right.$$

(61)

Here, $c_A$ is defined according to

$$\begin{array}{r}{\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\right)=\left[0.314+0.171h_0^{-1}+0.37{\left[h_0^{-1}\right]}^2\right]u\left(2.7+h_0^{-1}\right)+ \\ \left[1.02+{\displaystyle \frac{18.6}{1-h_0^{-1}}}+{\displaystyle \frac{18.3h_0^{-1}}{{(1-h_0^{-1})}^2}}\right]\left[1-u(2.7+h_0^{-1})\right]\end{array}$$

(62)

where $u$ is the unit step function.

Proof of Theorem 4.

The proof is detailed in Appendix B.

Sufficient Condition for Convergence

The approximation specified in Theorem 4 underpins the following sufficient condition for convergence, which is based on the ratio test.

Theorem 5.

Sufficient condition for convergence. Consistent with the ratio test, a sufficient condition for convergence, at a point $z$, of a Schröder series defined in Theorem 1 and based on an approximation $h_0^{-1}$, is for the existence of a constant $k_o>0$ such that $\forall k>k_o$:

$$\begin{array}{cc}{\displaystyle \frac{\left|z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right|}{{\left[1+h_0^{-1}\left(z\right)\right]}^2{e}^{h_0^{-1}\left(z\right)}}}·\left[3+h_0^{-1}\left(z\right)\right]<1+{\displaystyle \frac{1}{k}}, & h_0^{-1}\left(z\right)\ge -1, \mathit{principal} \mathit{branch}\\ {\displaystyle \frac{\left|z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right|}{{\left[1+h_0^{-1}\left(z\right)\right]}^2{e}^{h_0^{-1}\left(z\right)}}}·{\displaystyle \frac{\left|h_0^{-1}\left(z\right)\right|}{c_A\left(h_0^{-1}\left(z\right)\right)}}<1+{\displaystyle \frac{1}{k}},& h_0^{-1}\left(z\right)<-1, \mathit{negative} \mathit{one} \mathit{branch}\end{array}$$

(63)

Proof of Theorem 5.

With the $k\mathrm{t}\mathrm{h}$ term of a Schröder approximation, denoted $a_k$ and defined (see Theorem 1) according to

$$a_k(z)={\displaystyle \frac{{(-1)}^{k-1}{\left[z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right]}^k}{{\left[1+h_0^{-1}\left(z\right)\right]}^{2k-1}{e}^{k{h}_0^{-1}\left(z\right)}}}·{\displaystyle \frac{1}{k!}}·\sum _{j=0}^{k-1}{c}_{k,j}{\left[h_0^{-1}\left(z\right)\right]}^j$$

(64)

it follows that $a_{k+1}\left(z\right)/a_k\left(z\right)$ has the magnitude

$$\begin{array}{l}\left|{\displaystyle \frac{a_{k+1}\left(z\right)}{a_k\left(z\right)}}\right|={\displaystyle \frac{\left|z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right|}{{\left[1+h_0^{-1}\left(z\right)\right]}^2{e}^{h_0^{-1}\left(z\right)}}}·{\displaystyle \frac{1}{k+1}}·\left|{\displaystyle \frac{p\left(k+1,h_0^{-1}\left(z\right)\right)}{p\left(k,h_0^{-1}\left(z\right)\right)}}\right|\\ \approx {\displaystyle \frac{\left|z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right|}{{\left[1+h_0^{-1}\left(z\right)\right]}^2{e}^{h_0^{-1}\left(z\right)}}}·{\displaystyle \frac{k}{k+1}}·\left\{\begin{array}{c}3+h_0^{-1}\left(z\right),\hspace{1em} h_0^{-1}>-1, \mathrm{principal} \mathrm{branch}\hfill \\ {\displaystyle \frac{\left|h_0^{-1}\left(z\right)\right|}{{\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\left(z\right)\right)}}, h_0^{-1}<-1, \mathrm{negative} \mathrm{one} \mathrm{branch}\end{array}\right.\end{array}$$

(65)

where the last result arises from the approximation for ${\displaystyle \frac{p\left(k+1,h_0^{-1}\right)}{p\left(k,h_0^{-1}\right)}}$ detailed in (61).

Consistent with the ratio test, a sufficient condition for convergence, at a point $z$, of a Schröder series defined in Theorem 1 and based on an approximation $h_0^{-1}$, is for

$${\displaystyle \frac{\left|z-h_0^{-1}{e}^{h_0^{-1}}\right|}{{\left[1+h_0^{-1}\right]}^2{e}^{h_0^{-1}}}}·{\displaystyle \frac{k}{k+1}}·\left\{\begin{array}{c}3+h_0^{-1} \\ {\displaystyle \frac{\left|h_0^{-1}\right|}{{\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\right)}} \end{array}<1\right.$$

(66)

as $k\to \infty$, i.e., the existence of a constant $k_o>0$ such that for all $k>k_o$

$${\displaystyle \frac{\left|z-h_0^{-1}{e}^{h_0^{-1}}\right|}{{\left[1+h_0^{-1}\right]}^2{e}^{h_0^{-1}}}}·\left\{\begin{array}{c}3+h_0^{-1} \\ {\displaystyle \frac{\left|h_0^{-1}\right|}{{\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\right)}} \end{array}<1+{\displaystyle \frac{1}{k}}\right.$$

(67)

□

8. Bounds on Approximation for Convergence

The region of convergence, at a point $z$, for a Schröder series defined in Theorem 1 and based on $h_0^{-1}$, is defined according to

$$h_{0,\min }^{-1}\left(z\right)<h_0^{-1}\left(z\right)<h_{0,\max }^{-1}\left(z\right)$$

(68)

where for $z$ fixed, $h_{0,\min }^{-1}$ and $h_{0,\max }^{-1}$ can be approximated by considering the ratio test result defined in Theorem 5, i.e.,

$${\left.{\displaystyle \frac{\left|z-h_0^{-1}{e}^{h_0^{-1}}\right|}{{\left[1+h_0^{-1}\right]}^2{e}^{h_0^{-1}}}}·\left\{\begin{array}{c}3+h_0^{-1} \\ {\displaystyle \frac{\left|h_0^{-1}\right|}{{\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\right)}} \end{array}\right.\right|}_{h_0^{-1}\in \left\{h_{0,\min }^{-1}\left(z\right),h_{0,\max }^{-1}\left(z\right)\right\}}=1$$

(69)

where the upper/lower specifications are associated, respectively, with the principal and negative one branches.

The bounds $h_{0,\min }^{-1}$ and $h_{0,\max }^{-1}$ define bounds on the relative error, at a point $z$, for an approximation $h_0^{-1}\left(z\right)$ to yield a convergent Schröder series. The associated relative error bounds are

$$\begin{array}{cc}{\mathrm{re}}_{\mathrm{lower}}\left(z\right)=1-{\displaystyle \frac{h_{0,\min }^{-1}\left(z\right)}{h^{-1}\left(z\right)}},& {\mathrm{re}}_{\mathrm{upper}}\left(z\right)=1-{\displaystyle \frac{h_{0,\max }^{-1}\left(z\right)}{h^{-1}\left(z\right)}}.\end{array}$$

(70)

8.1. Bounds for Convergence for Negative One Branch

For the negative one branch, numerically solving (69) yields the upper and lower bounds, $h_{0,\min }^{-1}$ and $h_{0,\max }^{-1}$, defining the region of convergence and these are shown in Figure 12. The relative errors associated with these bounds are shown in Figure 13. For the case of $z\to 0^{-}$, the relative errors associated with the upper and lower bounds are more useful for assessing a given approximation $h_0^{-1}$.

Figure 12. Negative one branch: graphs of the upper and lower bounds for convergence.

Figure 13. Negative one branch: graphs of the relative errors associated with the upper and lower bounds $h_{0,\min }^{-1}$ and $h_{0,\max }^{-1}$.

The discontinuity in the upper bound $h_{0,\max }^{-1}\left(z\right)$, which is evident in Figure 12 and Figure 13, is a result of the nature of

$$\left|{\displaystyle \frac{a_{k+1}\left(z\right)}{a_k\left(z\right)}}\right|={\displaystyle \frac{\left|z-h_0^{-1}\left(z\right)e^{h_0^{-1}\left(z\right)}\right|}{{\left[1+h_0^{-1}\left(z\right)\right]}^2{e}^{h_0^{-1}\left(z\right)}}}·{\displaystyle \frac{1}{k+1}}·\left|{\displaystyle \frac{p\left(k+1,h_0^{-1}\left(z\right)\right)}{p\left(k,h_0^{-1}\left(z\right)\right)}}\right|$$

(71)

which can be approximated for $k\gg 1$, consistent with (65), according to

$$r_A\left(h_0^{-1}\right)={\displaystyle \frac{\left|z-h_0^{-1}{e}^{h_0^{-1}}\right|}{{\left[1+h_0^{-1}\right]}^2{e}^{h_0^{-1}}}}·{\displaystyle \frac{\left|h_0^{-1}\right|}{{\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\right)}}.$$

(72)

The ratio $\left|a_{k+1}\left(z\right)/a_k\left(z\right)\right|$ and $r_A\left(h_0^{-1}\right)$ are shown in Figure 14 for the case of $z=-{10}^{-4}$ and $k=128$. For these fixed values, note the plateau nature of $\left|a_{k+1}\left(z\right)/a_k\left(z\right)\right|$, around the level of unity, as $h_0^{-1}$ varies over a broad range. The nature of this plateau level changes as $z$ varies. This change, along with the finite resolution used and the small error associated with the approximation ${\mathrm{c}}_{\mathrm{A}}\left(h_0^{-1}\right)$, leads to the discontinuity in $h_{0,\max }^{-1}\left(z\right)$ as $z$ varies.

Figure 14. Graphs of the ratio $a_{k+1}\left(z\right)/a_k\left(z\right)$ and $r_A\left(h_0^{-1}\right)$, as $h_0^{-1}$ varies, for the case of $z=-{10}^{-4}$ and $k=128$. Note: $h^{-1}\left(-{10}^{-4}\right)\approx -11.67$.

8.2. Bounds for Convergence for Principal Branch

For the principal branch, the upper and lower bounds $h_{0,\min }^{-1}$ and $h_{0,\max }^{-1}$, defining the region of convergence are shown in Figure 15. The relative errors associated with these bounds are shown in Figure 16. Note, for the interval $\left[-1/e,0\right]$, that the bound levels are more useful than the relative error in the bounds. For large values of $z$, e.g., $z\in [10,\infty )$, the relative error associated with the upper and lower bounds is useful for assessing the convergence/divergence of a given approximation $h_0^{-1}$ at a fixed point of $z$.

Figure 15. Principal branch: graphs of the upper and lower bounds for convergence.

Figure 16. Principal branch: graphs of the relative errors associated with the upper and lower bounds $h_{0,\min }^{-1}$ and $h_{0,\max }^{-1}$.

8.3. Confirmation of Bounds: Negative One Branch

For the negative one branch, consider the approximations $h_0^{-1}\left(z\right)=-1-\sqrt{2}\sqrt{1+ez}$, $h_0^{-1}\left(z\right)=\ln \left(-z\right)$ and $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$. The relative errors in these approximations are shown in Figure 17 and Figure 18, along with the relative error bounds associated with convergence. Note, for the approximation, $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$, that the relative error is positive for $z>-0.15$ and, hence, the upper relative error bound indicates convergence for $z\in \left(-\mathrm{0.15,0}\right)$. Whilst the magnitude of the relative error in this approximation crosses the lower relative error bound around the point $z=-1.5\times {10}^{-3}$—see Figure 18—the approximation is convergent for $z\in \left[-1/e,0\right)$.

Figure 17. Negative one branch: graphs of the relative errors in the approximations $-1-\sqrt{2}\sqrt{1+ez}$, $\mathrm{l}\mathrm{n}(-z)$ and $\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$ along with the upper and lower relative error bounds associated with convergence.

Figure 18. Negative one branch: graphs of the relative errors in the approximations $-1-\sqrt{2}\sqrt{1+ez}$ and $\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$ along with the upper and lower relative error bounds associated with convergence. For the second approximation, the upper relative error bound is the relevant bound as $z\to 0^{-}$.

Simulation Results

First, consider the approximation $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$ for the negative one branch. The relative errors in various order Schröder approximations, based on this function, are shown in Figure 19. Convergence is indicated over the domain $z\in \left(-1/e,0\right)$, and this is consistent with the relative error bounds for convergence shown in Figure 17 and Figure 18.

Figure 19. Negative one branch. Graphs of the relative error in Schröder approximations for orders $k\in \left\{0, 1, 2, 4, 8, 16, 32\right\}$, based on $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\sqrt{2}\sqrt{1+ez}$.

Second, consider the approximation $h_0^{-1}\left(z\right)=\ln \left(-z\right)$ and the relative error results shown in Figure 20, which indicate convergence for $z\in (-\mathrm{0.105,0})$. This is broadly consistent with the relative error shown in Figure 17, where the relative error crosses the upper relative error bound at a value slightly greater than $z=-0.11$.

Figure 20. Negative one branch. Graphs of the relative errors in Schröder approximations for orders $k\in \left\{0, 1, 2, 4, 8, 16, 32, 64, 128\right\}$, based on $h_0^{-1}\left(z\right)=\ln \left(-z\right)$.

Third, consider the approximation $h_0^{-1}\left(z\right)=-1-\sqrt{2}\sqrt{1+ez}$ and the relative error results shown in Figure 21, which indicate convergence for $z\in \left(-1/e,-0.145\right)$ (approximate bound) and divergence for $z\in \left(-\mathrm{0.145,0}\right)$. The relative error result shown in Figure 17 for $-1-\sqrt{2}\sqrt{1+ez}$ is broadly consistent with these results where convergence is indicated for the interval $z\in \left(-1/e,-0.14\right)$.

Figure 21. Negative one branch. Graphs of the relative errors in Schröder approximations for orders $k\in \left\{0, 1, 2, 4, 8, 16, 32, 64, 128\right\}$, based on $h_0^{-1}\left(z\right)=-1-\sqrt{2}\sqrt{1+ez}$.

8.4. Confirmation of Bounds: Principal Branch

Consider the approximation $h_0^{-1}\left(z\right)=-1+\mathrm{l}\mathrm{n}[1+\sqrt{2}\sqrt{1+ez}]$ for the principal branch. The relative error bound associated with Schröder approximations of various orders, and based on this function, are shown in the left part of Figure 22. Convergence is indicated for $0<z<2.8$, and this is broadly consistent with the relative error bounds for convergence shown on the right of this figure, as the relative error in the approximation crosses the lower relative error bound for convergence at a point around $z=2$ (the lower bound is the relevant bound as the error in $h_0^{-1}\left(z\right)$ is positive for $z>0$). The lower relative error bound is based on the approximation detailed in Theorem 4, and the assumption in this approximation is that $0<r\left(w\right)\ll 1$ (see (A14)). For the case of $z$ equal to two, $r\left(w\right)$ is close to 0.12.

Figure 22. Results associated with the approximation $h_0^{-1}\left(z\right)=-1+\mathrm{l}\mathrm{n}[1+\sqrt{2}\sqrt{1+ez}]$ for the principal branch. Left: Graphs of the relative errors in Schröder approximations for orders $k\in \left\{0, 1, 2, 4, 8, 16, 32, 64\right\}$. Right: Graphs of the relative error in the approximation along with the upper and lower relative error bounds for convergence.

9. Applications

The following sub-sections detail several applications of Schröder approximations for the negative one and principal branches of the Lambert W function.

9.1. Approximations for Principal Branch

A first-order Schröder approximation, based on (50), is defined in (58) and has a relative error bound of $4.13\times {10}^{-4}$ for the interval $[-1/e,\infty )$. Simpler Schröder based approximations can be defined which are convergent over a restricted domain. Consider an approximation for the interval $\left[0,1/e\right]$, based on the affine approximation

$$\begin{array}{cc}{h}_0^{-1}\left(z\right)=k_{o}z,& k_o=e{h}^{-1}\left(1/e\right)\approx {\displaystyle \frac{757}{1000}}.\end{array}$$

(73)

The first order Schröder approximation (see (25)), based on this approximation, is

$$\begin{array}{cc}{h}_{S,1}^{-1}\left(z\right)={\displaystyle \frac{z\left[k_o^{2}z+e^{-k_{o}z}\right]}{1+k_{o}z}},& k_o={\displaystyle \frac{757}{1000}},\end{array}$$

(74)

and has a relative error bound of $2.59\times {10}^{-3}$ for the interval $\left[0,1/e\right]$.

The second order Schröder approximation is

$$h_{S,2}^{-1}\left(z\right)={\displaystyle \frac{z\left[k_o^{2}z+e^{-k_{o}z}\right]}{1+k_{o}z}}-{\displaystyle \frac{z^2\left[2+k_{o}z\right]{\left[k_o-e^{-k_{o}z}\right]}^2}{{2\left(1+k_{o}z\right)}^3}},$$

(75)

and has a relative error bound of $6.27\times {10}^{-5}$ for the interval $\left[0,1/e\right]$. Such an approximation is superior to Taylor series approximations (see (2) and Figure 2) in terms of a relative error bound over the interval $\left[0,1/e\right]$. Graphs of the relative errors in first to third-order Schröder approximations are shown in Figure 23.

Figure 23. Graphs of the relative errors in Schröder approximations for the principal branch, of order $k$, $k\in \{1, 2, 3\}$ and based on the initial approximation $h_0^{-1}\left(z\right)=k_{o}z$, $k_o=757/1000.$

Approximation for Principal Branch and Interval [−1/e,1/e]

A first-order Schröder approximation for the principal branch, based on $h_0^{-1}\left(z\right)$ as defined by (48) with ${\alpha }_3=1/2$, is

$$\begin{array}{l}{h}_{S,1}^{-1}\left(z\right)=\\ {\displaystyle \frac{\begin{array}{l}ez+\left[1+\sqrt{2}\sqrt{1+ez}\left[1+\frac{{\alpha }_1}{\sqrt{e}}\sqrt{1+ez}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}\left[1+\frac{1}{2e}(1+ez)\right]\right]\right]·\\ {\left[-1+\ln \left[1+\sqrt{2}\sqrt{1+ez}\left[1+\frac{{\alpha }_1}{\sqrt{e}}\sqrt{1+ez}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}\left[1+\frac{1}{2e}(1+ez)\right]\right]\right]\right]}^2\end{array}}{\begin{array}{c}\left[1+\sqrt{2}\sqrt{1+ez}\left[1+\frac{{\alpha }_1}{\sqrt{e}}\sqrt{1+ez}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}\left[1+\frac{1}{2e}(1+ez)\right]\right]\right]·\\ \ln \left[1+\sqrt{2}\sqrt{1+ez}\left[1+\frac{{\alpha }_1}{\sqrt{e}}\sqrt{1+ez}\right]+{\alpha }_2\ln \left[1+\mathrm{l}\mathrm{n}\left[1+\frac{1}{2e}(1+ez)\right]\right]\right]\end{array}}} \end{array}$$

(76)

where ${\alpha }_1,{\alpha }_2$ are, respectively, defined in (46) and (47). The relative error bound in this approximation, over the interval $\left[-1/e,1/e\right]$, is $1.10\times {10}^{-6}$. Graphs of the relative error in the original approximation (48) and this approximation are shown in Figure 24. This approximation is superior to Taylor series approximations (see (2) and Figure 2) in terms of a relative error bound over the interval $\left[-1/e,1/e\right]$ and, in particular, close to the point $-1/e$.

Figure 24. Graphs of the relative errors in the approximation defined by (48) for the case of ${\alpha }_3=1/2$, and the associated first-order Schröder approximation defined by (76).

9.2. Upper/Lower Bounds

There is interest in upper and lower bounds for the principal and negative one branches of the Lambert W function. For the negative one branch, Chatzigeorgiou [14] details simple upper and lower bounded functions, that are defined in (34), which have relatively poor relative error bounds. Any of the approximations detailed above can be utilized ([24], Lemma 1) to establish upper and lower bounded functions for the negative one or principal branch of the Lambert W with better accuracy. For example, for the negative one branch, the approximation defined by (32), with an approximate relative error bound of $1.17\times {10}^{-2}$, leads to

$$\begin{array}{r}{\displaystyle \frac{\ln \left(-z\right)-\mathrm{l}\mathrm{n}\left[\sqrt{2}\sqrt{1+ez}-\mathrm{l}\mathrm{n}(-z)\right]}{1-{ϵ}_B}}<h^{-1}\left(z\right)<\\ {\displaystyle \frac{\ln \left(-z\right)-\mathrm{l}\mathrm{n}\left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]}{1+{ϵ}_B}}\end{array}$$

(77)

where ${ϵ}_B=118/10000$. The relative error bound for the lower and upper bounded functions, respectively, are $2.15\times {10}^{-2}$ and $2.33\times {10}^{-2}$.

A Schröder approximation of order $k$ can be utilized to specify upper and lower bounds, of arbitrary accuracy. For example, with ${ϵ}_k$ specified in Table 2 or Table 3 (and accounting for the rounding in the stated relative error bounds), it follows that

$$\begin{array}{cc}{\displaystyle \frac{h_{S,k}^{-1}\left(z\right)}{1-{ϵ}_k}}\le h^{-1}\left(z\right)\le {\displaystyle \frac{h_{S,k}^{-1}\left(z\right)}{1+{ϵ}_k}},& z\in \left[-1/e,0\right], \mathrm{principal} \mathrm{or} \mathrm{negative} \mathrm{one} \mathrm{branch} \\ {\displaystyle \frac{h_{S,k}^{-1}\left(z\right)}{1+{ϵ}_k}}\le h^{-1}\left(z\right)\le {\displaystyle \frac{h_{S,k}^{-1}\left(z\right)}{1-{ϵ}_k}},& z\in \left[0,\infty \right), \mathrm{principal} \mathrm{branch}\hfill \end{array}$$

(78)

9.3. Integral Approximation

The integral result for the negative one branch of the Lambert W function:

$$\begin{array}{r}{I}_{-1}\left(z\right)={\int }_{-1/e}^z{h}^{-1}\left(\lambda \right)d\lambda ={\displaystyle \frac{-3}{e}}-z+z{h}^{-1}\left(z\right)+{\displaystyle \frac{z}{h^{-1}\left(z\right)}}\\ ={\displaystyle \frac{-3}{e}}-z+{\displaystyle \frac{z\left[1+{\left[h^{-1}\left(z\right)\right]}^2\right]}{h^{-1}\left(z\right)}}\end{array}$$

(79)

$-1/e<z<0$, can be utilized to establish approximations for $I_{-1}\left(z\right)$. For example, the approximation $h_0^{-1}\left(z\right)=\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]$, as stated in (32), yields the approximation to $I_{-1}\left(z\right)$ of

$$I_{-1}\left(z\right)\approx {\displaystyle \frac{-3}{e}}-z+{\displaystyle \frac{z\left[1+{\left[\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]\right]}^2\right]}{\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]}}$$

(80)

This approximation has a poor relative error bound of $6.50\times {10}^{-2}$ for the interval $\left(-1/e,0\right)$.

The first order Schröder approximation defined by (55) leads to the following approximation for $I_{-1}\left(z\right)$:

$$\begin{array}{r}{I}_{-1}\left(z\right)\approx {\displaystyle \frac{-3}{e}}-z+{\displaystyle \frac{z\left[-\sqrt{2}\sqrt{1+\mathit{ez}}+\ln \left(-z\right)+{\left[\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]\right]}^2\right]}{1+\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln \left(-z\right)\right]}}+ \\ {\displaystyle \frac{z\left[1+\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]\right]}{-\sqrt{2}\sqrt{1+ez}+\mathrm{l}\mathrm{n}(-z)+{\left[\ln \left(-z\right)-\ln \left[\sqrt{2}\sqrt{1+ez}-\ln (-z)\right]\right]}^2}}\end{array}$$

(81)

which has a relative error bound of $5.33\times {10}^{-4}$ for the interval $\left(-1/e,0\right)$. Naturally, approximations for $h^{-1}$ with a lower relative error bound lead to more accurate approximations for $I_{-1}$.

9.4. Principal Branch: Asymptotic-Based Approximation for z > 1

There has been interest, e.g., Ref. [15], in asymptotic approximations for the principal branch of the Lambert W function leading to the approximation stated in (6). Consider the case of an initial approximation $h_0^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\right(z\left)\right]$ for the principal branch. It then follows, from the special form for a Schröder series of order $i$, as specified in Theorem 2, that

$$\begin{array}{l}{h}_{\mathrm{S},\mathrm{i}}^{-1}\left(z\right)=\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]+\\ {\displaystyle \sum _{k=1}^i}{\displaystyle \frac{{(-1)}^{k-1}{\left[\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\right(z)\right]}^k}{{\left[1+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^{2k-1}}}·{\displaystyle \frac{1}{k!}}·{\displaystyle \sum _{j=0}^{k-1}}{c}_{k,j}{\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^j\end{array}$$

(82)

An explicit expression is

$$\begin{array}{l}\begin{array}{l}\begin{array}{l}{h}_S^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}\left[\mathrm{l}\mathrm{n}\right(z\left)\right]+{\displaystyle \frac{\ln \left[\ln \left(z\right)\right]}{1+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]}}-\\ {\displaystyle \frac{{\left[\ln \left[\ln \left(z\right)\right]\right]}^2}{{2\left[1+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^3}}·\left[2+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]+\\ {\displaystyle \frac{{\left[\ln \left[\ln \left(z\right)\right]\right]}^3}{{3!\left[1+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^5}}·\left[\begin{array}{c}9+8[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]]+\\ {2\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^2\end{array}\right]-\end{array}\\ {\displaystyle \frac{{\left[\ln \left[\ln \left(z\right)\right]\right]}^4}{{4!\left[1+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^7}}·\left[\begin{array}{r}64+79[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]]+\\ 36{\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^2+\\ 6{\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^3\end{array}\right]+\end{array}\\ {\displaystyle \frac{{\left[\ln \left[\ln \left(z\right)\right]\right]}^5}{{5!\left[1+\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^9}}·\left[\begin{array}{r}\begin{array}{r}625+974[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]]+\\ 622{\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^2+\end{array}\\ \begin{array}{r}192{\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^3+\\ 24{\left[\ln \left(z\right)-\ln \left[\ln \left(z\right)\right]\right]}^4\end{array}\end{array}\right]-\cdots \end{array}$$

(83)

Graphs of the relative errors, associated with various orders of this approximation, are shown in Figure 25. A comparison of the results in this Figure with the results shown in Figure 3, associated with the asymptotic approximation defined by (7), indicates that the Schröder approximation has a greater range of convergence with a lower bound of one rather than $e$. For $z\gg 1$, the rate of convergence, with approximation order, is not as fast as the asymptotic approximation defined by (7). This is a consequence of the asymptotic approximation defined by (7) being the result of the successive approximation of an error term (see, for example, the analysis detailed in [2]).

Figure 25. Graphs of the relative errors in Schröder approximations for the principal branch, based on $h_0^{-1}\left(z\right)=\ln \left(z\right)-\mathrm{l}\mathrm{n}[\mathrm{l}\mathrm{n}(z\left)\right]$ and as detailed in (82). The relative errors are for the approximations of order $k$, $k\in \{1, 2, 3, 4, 8\}$.

9.5. Approximations for a Power of the Lambert W Function

The basis for establishing approximations for a power of the Lambert W function is the definition $z=w{e}^w$ which implies $w=z{e}^{-w}$ and, thus, $w^k={\left[h^{-1}\left(z\right)\right]}^k=z^k{e}^{-k{h}^{-1}\left(z\right)}$. It then follows that an $i\mathrm{t}\mathrm{h}$ order Schröder approximation, $h_{S,i}^{-1}$, leads to an approximation for ${\left[h^{-1}\left(z\right)\right]}^k$ according to

$${\left[h^{-1}\left(z\right)\right]}^k\approx z^k{e}^{-k{h}_{S,i}^{-1}\left(z\right)}$$

(84)

For example, for the principal branch and using the first order Schröder approximation detailed in (58) (with ${\alpha }_1$ and ${\alpha }_2$ defined, respectively, in (51) and (52)), an approximation to ${\left[h^{-1}\left(z\right)\right]}^k$ is

$$\begin{array}{l}{h}_{1,k}^{-1}\left(z\right)=z^k\exp \left[{\displaystyle \frac{-k\left[-1+\ln \left[A\left(z\right)\right]+\ln {\left[\frac{z}{\ln \left[A\left(z\right)\right]-1}\right]}^2\right]}{1+\ln \left[\frac{z}{\ln \left[A\left(z\right)\right]-1}\right]}}\right]\\ A\left(z\right)=1+\sqrt{2}\sqrt{1+ez}+{\alpha }_1\sqrt{{\displaystyle \frac{2}{e}}}·\left(1+ez\right)+{\alpha }_2\ln \left[1+\ln \left[1+{\displaystyle \frac{1}{2e}}·\left(1+ez\right)\right]\right]\end{array}$$

(85)

and has a relative error bound over the interval $\left[-1/e,\infty \right)$ of $3.64\times {10}^{-3}$ for $k=1$, $7.28\times {10}^{-3}$ for $k=2$ and $1.09\times {10}^{-2}$ for $k=3$. For the case of $k=4$, a fourth-order Schröder approximation has a relative error bound over the interval $\left[-1/e,\infty \right)$ of $4.48\times {10}^{-6}$.

9.6. Approximation to Solutions of c^c = y and C^C = e^v

The solutions to the equations $C^C=e^v$ and $c^c=y$, respectively, for $C$ and $c$, involve the Lambert W function according to

$$C\left(v\right)=\left\{\begin{array}{cc}{\displaystyle \frac{v}{h^{-1}\left(v\right)}},& v\in \left[-1/e,0\right), \mathrm{negative} \mathrm{one} \mathrm{branch}\\ {\displaystyle \frac{v}{h^{-1}\left(v\right)}},& v\in \left[-1/e,\infty \right), \mathrm{principal} \mathrm{branch} \end{array}\right.$$

(86)

$$c\left(y\right)=\left\{\begin{array}{cc}{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(y\right)}{h^{-1}⌈\mathrm{l}\mathrm{n}\left(y\right)⌉}},& y\in \left[e^{-1/e},1\right),\ln \left(y\right)\in \left[-1/e,0\right), \mathrm{negative} \mathrm{one} \mathrm{branch}\\ {\displaystyle \frac{\mathrm{l}\mathrm{n}\left(y\right)}{h^{-1}\left[\mathrm{l}\mathrm{n}\left(y\right)\right]}},& y\in \left[e^{-1/e},\infty \right), \mathrm{principal} \mathrm{branch} \hfill \end{array}\right.$$

(87)

The nature of $c^c$ is shown in Figure 26. The graphs of $c$ and $C$ are shown in Figure 27.

Figure 26. Graph of $c^c$. For $c=1/e$, $c^c=e^{-1/e}\approx 0.6922$.

Figure 27. Left: graph of $C\left(v\right)$, $v\in [-1/e,0)$ for the negative one branch and $v\in [-1/e,\infty )$ for the principal branch. Right: graph of $c\left(y\right)$,$y\in \left[e^{-1/e},1\right)$ for the negative one branch and $y\in \left[e^{-1/e},\infty \right)$ for the principal branch.

For the negative one branch, for example, approximations for $h^{-1}$ provide a basis for specifying approximations to $c\left(y\right)$ and $C\left(v\right)$ for $y\in \left[e^{-1/e},1\right)$, $v\in \left[-1/e,0\right)$. For example, the approximation specified by (32) yields

$$C\left(v\right)={\displaystyle \frac{v}{\ln \left(-v\right)-\ln \left[\sqrt{2}\sqrt{1+ev}-\ln \left(-v\right)\right]}}, v\in [-1/e,0),$$

(88)

$$c\left(y\right)={\displaystyle \frac{\ln (y)}{\ln [-\ln \left(y\right)]-\ln \left[\sqrt{2}\sqrt{1+e\ln \left(y\right)}-\ln [-\ln \left(y\right)]\right]}}, y\in [e^{-1/e},1),$$

(89)

and both approximations have a relative error bound of $1.19\times {10}^{-2}$.

The first order Schröder approximation specified in (55) leads to the approximations:

$$\begin{array}{l}C\left(v\right)={\displaystyle \frac{v\left[1+\ln \left(-v\right)-\ln \left[\sqrt{2}\sqrt{1+ev}-\ln \left(-v\right)\right]\right]}{-\sqrt{2}\sqrt{1+ev}+\ln \left(-v\right)+{\left[\ln \left(-v\right)-\ln \left[\sqrt{2}\sqrt{1+ev}-\ln \left(-v\right)\right]\right]}^2}} \\ v\in [-1/e,0),\end{array}$$

(90)

$$\begin{array}{l}c\left(y\right)=\\ {\displaystyle \frac{\mathrm{l}\mathrm{n}(y)\left[1+\ln \left[-\mathrm{l}\mathrm{n}\left(y\right)\right]-\ln \left[\sqrt{2}\sqrt{1+e\ln \left(y\right)}-\ln \left[-\mathrm{l}\mathrm{n}\left(y\right)\right]\right]\right]}{-\sqrt{2}\sqrt{1+e\mathrm{l}\mathrm{n}\left(y\right)}+\ln \left[-\mathrm{l}\mathrm{n}\left(y\right)\right]+{\left[\ln \left[-\mathrm{l}\mathrm{n}\left(y\right)\right]-\ln \left[\sqrt{2}\sqrt{1+e\mathrm{l}\mathrm{n}\left(y\right)}-\ln \left[-\mathrm{l}\mathrm{n}\left(y\right)\right]\right]\right]}^2}}\end{array}$$

(91)

$y\in \left[e^{-1/e},1\right)$ where both approximations have a relative error bound of $4.00\times {10}^{-4}$.

10. Conclusions

Schröder based series for the principal and negative one branches of the Lambert W function were defined; the series are generic and are in terms of an initial, arbitrary approximation function. Schröder series for the special but common case, where the initial approximating function is of the form $h_0^{-1}\left(z\right)=\ln \left(\pm z\right)-\ln \left[\pm A\left(z\right)\right]$, were detailed. Approximations for both branches of the Lambert W function were proposed which have modest relative error bounds over their domains of definition and which are suitable as initial approximation functions for a convergent Schröder series. For the principal branch, a proposed approximation yields, for a series of order 128, a relative error bound below ${10}^{-136}$. For the negative one branch, a proposed approximation yields, for a series of order 128, a relative error bound below ${10}^{-143}$.

For both the principal and negative one branches, sufficient conditions for convergence of a Schröder series, based on the ratio test, were established. Approximate bounds for an initial approximating function were determined consistent with the region of convergence for the associated Schröder series. The regions of convergence for both branches were confirmed by considering initial approximating functions, with relative errors that change from being low to high (or high to low), and their associated Schröder series, which have both a region of convergence and a region of divergence.

Applications of the approximations for the principal and negative one branches were detailed and included new approximations for the Lambert W function, analytical approximations for the integral of the Lambert W function, upper and lower bounded functions for the Lambert W function, approximations to the power of the Lambert W function and approximations to the solutions of the equations $c^c=y$ and $C^C=e^v$, respectively, for $c$ and $C$.

Future Research

As the nature of the convergence of a Schröder series approximation for the Lambert W function depends on the accuracy of the initial approximation, there is interest in establishing initial approximations with low complexity and a low relative error bound over the domain of definition for both the principal and negative one branches. The challenge is to find approximations that satisfy the constraints of an infinite rate of change at the branch point $\left(-1/e,-1\right)$ and logarithmic type change as $z\to 0^{-}$ for the negative one branch and as $z\to \infty$ for the principal branch. In addition, for the principal branch, there is an additional requirement to satisfy the constraints at the origin of a zero value and a unity rate of change.

Appendix A.1. Derivative Values

As stated in (39),$z_1=h_1\left(w_1\right)={\displaystyle \frac{1}{e}}+\left(w_1-1\right)e^{w_1-1}$, and it then follows that $D\left[h_1\left(w_1\right)\right]={\displaystyle \frac{w_1{e}^{w_1}}{e}}$. Thus, ${\left.D\left[h_1\left(w_1\right)\right]\right|}_{w_{1=1}}=1$ and

$${\left.D\left[h_1^{-1}\left(z_1\right)\right]\right|}_{z_{1=1/e}}={\displaystyle \frac{1}{{\left.D\left[h_1\left(w_1\right)\right]\right|}_{w_{1=1}}}}=1.$$

(A1)

Appendix A.2. Approximation at Origin

Use of the standard series for the exponential function in the relationship defined by (39),

$$z_1=h_1\left(w_1\right)={\displaystyle \frac{1}{e}}\left[1+\left(w_1-1\right)e^{w_1}\right],$$

(A2)

leads to the series expansion

$$\begin{array}{l}{z}_1={\displaystyle \frac{1}{e}}\left[1+\left(w_1-1\right)\left[1+w_1+{\displaystyle \frac{w_1^2}{2}}+{\displaystyle \frac{w_1^3}{6}}+\cdots \right]\right]\\ ={\displaystyle \frac{w_1^2}{e}}\left[{\displaystyle \frac{1}{2}}+{\displaystyle \frac{w_1}{3}}+{\displaystyle \frac{w_1^2}{8}}+{\displaystyle \frac{w_1^3}{30}}+{\displaystyle \frac{w_1^4}{144}}+\cdots +w_1^{k-2}\left[{\displaystyle \frac{1}{\left(k-1\right)!}}-{\displaystyle \frac{1}{k!}}\right]+\cdots \right].\end{array}$$

(A3)

Series inversion yields

$$w_1\approx \sqrt{2e{z}_1}-{\displaystyle \frac{2e{z}_1}{3}}+{\displaystyle \frac{11e^{3/2}{z}_1^{3/2}}{18\sqrt{2}}}-{\displaystyle \frac{43e^2{z}_1^2}{135}}+{\displaystyle \frac{769e^{5/2}{z}_1^{5/2}}{2160\sqrt{2}}}-\cdots .$$

(A4)

Appendix A.3. Improved Approximation via Iteration

The relationship $z_1={\displaystyle \frac{1}{e}}+\left(w_1-1\right)e^{w_1-1}$ leads to

$$\left\{\begin{array}{ll}\ln \left[\left|z_1-{\displaystyle \frac{1}{e}}\right|\right]=w_1-1+\ln \left[\left|w_1-1\right|\right],& z_1\ne {\displaystyle \frac{1}{e}},w_1\ne 1\\ w_1=1,& z_1={\displaystyle \frac{1}{e}}\end{array}\right.$$

(A5)

and the relationship

$$w_1=1+\ln \left[\left|z_1-{\displaystyle \frac{1}{e}}\right|\right]-\ln \left[\left|w_1-1\right|\right], z_1\ne {\displaystyle \frac{1}{e}},w_1\ne 1.$$

(A6)

Given $z_1$, and an initial approximation to $w_1$ of $h_{1,0}^{-1}\left(z_1\right)$, it follows that a better approximation to $w_1$, potentially, is

$$w_1\approx \left\{\begin{array}{l}1+\ln \left[\left|z_1-{\displaystyle \frac{1}{e}}\right|\right]-\ln \left[\left|h_{1,0}^{-1}\left(z_1\right)-1\right|\right]=1+\ln \left[{\displaystyle \frac{z_1-\frac{1}{e}}{h_{1,0}^{-1}\left(z_1\right)-1}}\right], z_1\ne 1/e\\ 1, z_1=1/e\end{array}\right.$$

(A7)

assuming $z_1-1/e$ and $h_{1,0}^{-1}\left(z_1\right)-1$ have the same sign. The requirement is for $h_{1,0}^{-1}\left(1/e\right)=1$ and for $h_{1,0}^{-1}\left(z\right)$ to be monotonically increasing with $z$.

Appendix B.1. Principal Branch

For the principal branch, the values shown in Figure A1 and Figure A2 indicate, for $w$ fixed, that the ratio $r(k,w)$ becomes independent of $k$ for $k$ large and varies inversely with $w$, for the case of $w\gg 1$. The two sub-sections, Appendix B.3 and Appendix B.4, provide further evidence for the independence of $r(k,w)$ on $k$ for $k$ large.

For the principal branch and for $k$ large, and with $r(k,w)$ written as $r\left(w\right)$, it follows that

$$\left(1+w\right){\displaystyle \frac{\partial p\left[k,w\right]}{\partial w}}\approx r\left(w\right)\left[kw+3(k+1)-4\right]p\left[k,w\right]$$

(A12)

and from (A8)

$$p\left[k+1,w\right]\approx -\left[1-r\left(w\right)\right]\left[kw+3k-1\right]p\left[k,w\right].$$

(A13)

Consistent with the results shown in Figure A1 and Figure A2, it is the case that $0<r\left(w\right)\ll 1$ and the required approximation, for $k$ large, of

$${\displaystyle \frac{p\left[k+1,w\right]}{p\left[k,w\right]}}\approx -k(3+w)$$

(A14)

follows.

Appendix B.2. Negative One Branch

For the negative one branch and consistent with the results shown in Figure A3, the approximation specified in (A14) holds for $w<-10$ but not for values of $w$ in the interval $(-10,-1.1)$ and with the largest values for the ratio $r(k,w)$ occurring for $w=-3$.

Consider the magnitude of the normalized ratio ${\displaystyle \frac{p\left[k+1,w\right]}{p\left[k,w\right]}}·{\displaystyle \frac{1}{kw}}$ shown in Figure A4, which is largely independent of $k$ for $k$ large. The variation in this ratio, with $w$, can be compensated for by a function $c\left(w\right)$ defined according to

$$c\left(w\right)=k\left|w\right|·\left|{\displaystyle \frac{p\left[k,w\right]}{p\left[k+1,w\right]}}\right|$$

(A15)

The nature of the function $c$ is illustrated in Figure A5, and this function can be approximated, in a piecewise manner, by a function $c_A$ defined according to

$$\begin{array}{r}{\mathrm{c}}_{\mathrm{A}}\left(w\right)=\left[0.314+0.171w+0.37w^2\right]u\left(2.7+w\right)+ \\ \left[1.02+{\displaystyle \frac{18.6}{1-w}}+{\displaystyle \frac{18.3w}{{(1-w)}^2}}\right]\left[1-u(2.7+w)\right]\end{array}$$

(A16)

The relative error in this approximation is shown in Figure A5. It then follows, for $k$ large, that

$$\left|{\displaystyle \frac{p\left[k+1,w\right]}{p\left[k,w\right]}}\right|\approx {\displaystyle \frac{k\left|w\right|}{c_A\left(w\right)}}$$

(A17)

which is the second required approximation.

Figure A4. Graph of the magnitude of the ratio ${\displaystyle \frac{p\left[k+1,w\right]}{p\left[k,w\right]}}·{\displaystyle \frac{1}{kw}}$ for $k\in \{2, 4, 8, 16, 32, 64, 128, 256, 512\}$ and for set negative values of $w$—values consistent with the negative one branch.

Figure A5. Left: graph of the compensation function $c\left(w\right)$ for the case of $k=512$. Right: relative error in the approximation $c_A\left(w\right)$ to $c\left(w\right)$.

Appendix B.3. Independence of r(k,w) on k for the Case of w = 0

Consider the approximation, defined in (A11), for the case of $w=0$ and for $k$ large:

$$r\left(k,0\right)\approx {\displaystyle \frac{\frac{\partial p\left[k,0\right]}{\partial w}}{p\left(k,0\right)}}·{\displaystyle \frac{1}{3k}}={\displaystyle \frac{c_{k,1}}{c_{k,0}}}·{\displaystyle \frac{1}{3k}}$$

(A18)

using the form for $p[k,w]$ define in (A9). With $c\left(k,0\right)=k^{k-1},$ $c\left(k,1\right)={(3k-1)k}^{k-1}-{(k+1)}^k$ (see Section 3.1), it follows that

$$r\left(k,0\right)\approx {\displaystyle \frac{1}{3k}}·{\displaystyle \frac{{\left(3k-1\right)k}^{k-1}-{\left(k+1\right)}^k}{k^{k-1}}}=1-{\displaystyle \frac{1}{3k}}-{\displaystyle \frac{1}{3}}{\left[1+{\displaystyle \frac{1}{k}}\right]}^k$$

(A19)

As $\underset{k\to \infty }{\lim }{\left(1+{\displaystyle \frac{1}{k}}\right)}^k=e$, it follows that

$$\underset{k\to \infty }{\lim }r\left(k,0\right)\approx 1-{\displaystyle \frac{e}{3}}\approx 0.0939$$

(A20)

and this result is consistent with simulation results as given in Figure A1 and Figure A2.

Appendix B.4. Independence of r(k,w) on k for w Fixed, w, k Large

Consider the definition of $p[k,w]$ defined by (19), i.e.,

$$p\left[k,w\right]=\left(1+w\right){\displaystyle \frac{\partial p\left[k-1,w\right]}{\partial w}}-\left[\left(k-1\right)w+3k-4\right]p\left[k-1,w\right]$$

(A21)

which implies

$$\begin{array}{l}{\displaystyle \frac{\partial p\left[k,w\right]}{\partial w}}={\displaystyle \frac{\partial p\left[k-1,w\right]}{\partial w}}+\left(1+w\right){\displaystyle \frac{{\partial }^{2}p\left[k-1,w\right]}{\partial w^2}}-\\ \left(k-1\right)p\left[k-1,w\right]-\left[\left(k-1\right)w+3k-4\right]{\displaystyle \frac{\partial p\left[k-1,w\right]}{\partial w}}.\end{array}$$

(A22)

Thus:

$$\begin{array}{l}{\displaystyle \frac{\frac{\partial p\left[k,w\right]}{\partial w}}{p[k,w]}}·{\displaystyle \frac{1}{k}}=\\ {\displaystyle \frac{\left(1+w\right)p^{\left(2\right)}\left[k-1,w\right]+\left[1-\left[\left(k-1\right)w+3k-4\right]\right]p^{\left(1\right)}\left[k-1,w\right]-(k-1)p[k-1,w]}{k\left[\left(1+w\right)p^{\left(1\right)}\left[k-1,w\right]-\left[\left(k-1\right)w+3k-4\right]p[k-1,w]\right]}}\end{array}$$

(A23)

Consider the case of $k,w\gg 1$:

$$\begin{array}{r}{\displaystyle \frac{\frac{\partial p\left[k,w\right]}{\partial w}}{p[k,w]}}·{\displaystyle \frac{1}{k}}\approx {\displaystyle \frac{w{p}^{\left(2\right)}\left[k-1,w\right]-\left[kw+3k\right]p^{\left(1\right)}\left[k-1,w\right]-kp\left[k-1,w\right]}{k\left[w{p}^{\left(1\right)}\left[k-1,w\right]-\left[kw+3k\right]p\left[k-1,w\right]\right]}} \\ =·{\displaystyle \frac{{\frac{1}{k}·p}^{\left(2\right)}\left[k-1,w\right]-\left[1+\frac{3}{w}\right]p^{\left(1\right)}\left[k-1,w\right]-\frac{1}{w}·p\left[k-1,w\right]}{p^{\left(1\right)}\left[k-1,w\right]-k\left[1+\frac{3}{w}\right]p\left[k-1,w\right]}}\end{array}$$

(A24)

Simulation results indicate that ${{\displaystyle \frac{1}{k}}·p}^{\left(2\right)}\left[k-1,w\right]\ll p^{\left(1\right)}\left[k-1,w\right]$ when $k,w\gg 1$. It then follows, for the case of $k,w\gg 1$, that

$${\displaystyle \frac{\frac{\partial p\left[k,w\right]}{\partial w}}{p[k,w]}}·{\displaystyle \frac{1}{k}}\approx {\displaystyle \frac{-\frac{\partial p\left[k-1,w\right]}{\partial w}}{\frac{\partial p\left[k-1,w\right]}{\partial w}-kp\left[k-1,w\right]}}\approx {\displaystyle \frac{-\frac{\partial p\left[k-1,w\right]}{\partial w}}{\frac{\partial p\left[k-1,w\right]}{\partial w}-(k-1)p\left[k-1,w\right]}}$$

(A25)

The ratio ${\displaystyle \frac{\frac{\partial p\left[k,w\right]}{\partial w}}{p[k,w]}}·{\displaystyle \frac{1}{k}}$ is largely independent of $k,$ i.e., ${\displaystyle \frac{\frac{\partial p\left[k,w\right]}{\partial w}}{p[k,w]}}·{\displaystyle \frac{1}{k}}\approx {\displaystyle \frac{\frac{\partial p\left[k-1,w\right]}{\partial w}}{p[k-1,w]}}·{\displaystyle \frac{1}{k-1}}$ when $\left|{\displaystyle \frac{\partial p\left[k-1,w\right]}{\partial w}}\right|\ll (k-1)p\left[k-1,w\right]$. The validity of this inequality, for the case of $k,w\gg 1$, is evident in the results shown in Figure A1.