Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 8267951, 15 pages
https://doi.org/10.1155/2019/8267951

Research Article

PSEM Approximations for Both Branches of Lambert Function with Applications

H. Vazquez-Leal,¹ M. A. Sandoval-Hernandez,2,3 J. L. Garcia-Gervacio,¹ A. L. Herrera-May,4,5 and U. A. Filobello-Nino¹

¹Facultad de Instrumentación Electrónica, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz, Mexico
²National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No. 1, Sta. María Tonantzintla, 72840 Puebla, Mexico
³Universidad de Xalapa, Km. 2 Carretera Xalapa-Veracruz, Xalapa 91190, Veracruz, Mexico
⁴Centro de Investigación de Micro y Nanotecnología, Universidad Veracruzana, Calzada Ruíz Cortines 455, Boca del Río 94292, Veracruz, Mexico
⁵Maestría en Ingeniería Aplicada, Facultad de Ingeniería de la Construcción y el Hábitat, Universidad Veracruzana, Calzada Ruíz Cortines 455, Boca del Río 94294, Veracruz, Mexico

Correspondence should be addressed to H. Vazquez-Leal; hvazquez@uv.mx

Received 20 September 2018; Revised 14 January 2019; Accepted 3 February 2019; Published 3 March 2019

Academic Editor: Daniele Fournier-Prunaret

Copyright © 2019 H. Vazquez-Leal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Transcendental functions are a fundamental building block of science and engineering. Among them, a relatively new function denominated as Lambert is highlighted. The importance of such function relies on the fact that it can perform novel isolation of variables. In this work, we propose two accurate piece-wise approximate solutions, one for the lower branch and another one for the upper branch, respectively. The proposed analytic approximations are obtained by using the power series extender method (PSEM) in combination with asymptotic solutions. In addition, we will compare some published approximations with our proposal, highlighting our advantages in terms of significant digits and speed of evaluation. Furthermore, the approximations are validated by the successful simulation of a problem of economy and other acoustic waves of nonlinear ions.

1. Introduction

Transcendental equations are those containing transcendental functions with the variable to be solved for, which have the particularity that can not be expressed in terms of a sequence of algebraic operations [1–4]. This is the case of the Lambert Function [5, 6]. The Lambert W Function as today known, denoted by , is a multivalued function that can be defined as the inverse of the following transcendental function [5–9]:

resulting in

where can be any real or complex number, and is known as Lambert Function.

The Lambert W Function () is a multivalued function (also called multibranched function); branches considering only real numbers of are plotted in Figure 1. The two real branches, and , can be identified. is called the upper branch (also named the principal branch) and satisfies the condition . This branch can be split into two branches, and . is called the lower branch and satisfies the condition . Both branches are jointed at (, ). Then, on the interval , there are two possible values for , the one for and the second for .

Figure 1: The two real branches of the Lambert W Function. is the upper branch and is the lower branch.

The origin of (2) begins in 1758 when Johann Heinrich Lambert solved the next trinomial equation [5, 6] by giving a series for in powers of .

Some years later, in 1783 Leonhard Euler presented (3) into a symmetrical form as follows [5, 6]:

which is developed in series expansion as

and can be expressed as

This series is called the Tree Function and is related to the Lambert W Function (2) because [5, 6]. The Lambert W Function is also called the Omega function or Product Log function [5, 13]. However, the inverse of (1) was first described by Polya and Szego in 1925 [14]. Later, during 1949 up to 1959, Edward Maitland Wright performs significant contribution to the solution of the transcendental equation (1) by means of his effort to solve Delay Differential Equations (DDE) [15, 16]. At the same time as Wright, in 1950, N. D. Hayes [17] collaborated in the solution of (1) with his work on transcendental equations. In 1973, Crowley et al. [18] presented an algorithm (called the 443-algorithm) for the solution of that transcendental equation; an updated version of the 443-algorithm (called the 743-algorithm) was given in 1995 [19]. Different works have been published from 1758 along the years regarding the solution of (1); however, it was until the work of Corless et al. published in 1996 [5] that is given to be known worldwide. Corless et al. [5] made a revision of related works and potential applications and provide approximate representations and numeric algorithms of evaluation to the function. At the same time, this function was included in Maple CAS software [5, 20].

Since then, many works have appeared with reviews and significant contributions to the approximation of the oriented to diverse applications; some relevant works are presented by Corless et al. [21, 22], Brito et al. [6], Abdolhossein et al. [23], Fukushima [24], Marko [25], Fredrik [9], Jeffrey et al. [26], Jeffrey [27], Kheyfits [28], Darko [7], Dence [29], and Mezo et al. [30]. Moreover, a conference was created and celebrated in honour of the Lambert W Function in 2016. Lambert is today used in all fields of science where transcendental functions as (1) are implicit in equations to solve. In general, can be applied in Physics, Mathematics, Chemistry, Engineering, and so on [5, 6, 8]. Some particular applications are biochemical kinetics [25, 31], electromagnetics (capacitor, transmission lines, etc.) [32], semiconductor (diode) circuits [33, 34], photocells [35], acoustics [36], optics [37], particle physics [38], general relativity [39], geophysics [40], risk theory [41], technological systems [42], information theory [43], combinatorial [5, 44], iterative exponentiation [5, 45, 46], graph theory [47], oscillation theory [13], time-delay dynamics [48–51], pedagogical [5], physiology [52, 53], hydrology [54, 55], hydraulic [56], materials and transport research [57–59], electrochemistry [60], microfluidics [61], statistical mechanics [62], etc.

The formal solution of the Lambert W function can be expressed as [56]

This expression clearly requires an infinite iterative calculus, which is not feasible to compute.

From the numerical and analytical point of view, there are several methods to compute transcendental functions like this; for example, Chapeau-Blondeau [10] and Corless et al. [5] proposed to evaluate the Lambert W branches using approximate analytical expressions and increasing accuracy with the Halley numerical method [35, 63–65]. Other proposals to obtain an analytic approximation of the Lambert W function are [11, 66–68]. Fukushima proposed an algorithm [24] to obtain the numerical solution for and by means of an expansion around the zero branch point. Moreover, Chapeau-Blondeau [10], de Brujin [67], and Karamata [68] proposed series expansions to evaluate branch .

In this work, we propose to apply power series extender method (PSEM) [69] to calculate approximate solutions for both branch Lambert . PSEM is a novel tool that is able to produce easy computable, highly accurate approximations without requiring to perform: integrals, solution of differential equations or inclusive without requiring the existence of a perturbative parameter. What is more, the approximate solutions exhibit a large domain of convergence.

Section 2 presents the basic procedure to apply PSEM. The procedure to obtain the approximation of lower and upper branch of Lambert is presented in Section 3. Next, two case studies are presented in Section 5. Later, Section 6 presents a discussion about the main results of this work. Finally, the concluding remarks are presented in Section 7.

2. Basic Procedure of PSEM

In broad sense, a nonlinear differential equation can be expressed as follows:

having as boundary condition,

where and are a linear operator and a nonlinear operator, respectively, is a known analytic function, is a boundary operator, is the boundary of domain , and denotes differentiation along the normal drawn outwards from [70].

Next, we express the solution of (8) as a Taylor expansion:

where are the coefficients of the power series and is the expansion point.

Now, in [69, 71], it is proposed that the solution for (8) can be written as a finite sum of functions as

or

where are constants to be determined by PSEM, are arbitrary trial functions, and and are the orders of approximations (11) and (12), respectively. We will denominate (11) and (12) as a trial function (TF). Next, we calculate the Taylor expansion of (11) or (12), resulting in the power series:

respectively, where Taylor coefficients are expressed in terms of parameters . Finally, we equate/match the coefficients of power series (13) or (14) with the correspondent ones of (10) to obtain the values of and substitute them into (11) or (12) to obtain the PSEM approximation [69].

It is possible to enrich PSEM by means of introducing cancellation points (CP). For every cancellation point several equations depending on the number of desire derivatives to satisfy are constructed. Such new equations are included in PSEM process to complement the system of equations in order to obtain the coefficients (). The cancellation equations are constructed by evaluating the TF and its derivatives at some strategically selected CP to increase the accuracy within a region. It is important to note that every introduced cancellation point and its derivatives would replace the corresponding superior order equations in the regular PSEM procedure.

The TF can be composed by combining different type of functions as long as the Taylor series exist. Note that regrouping terms of the -powers are valid for (13) and (14), because functions were chosen analytic in a well-defined domain in the independent variable , whereby, the correspondent Taylor series are convergent for such values of [72, 73]. By restricting the values of (8) to the mentioned domain of convergence, the sum of the Taylor series of is also convergent [72, 73]. In the same fashion, the quotient of two analytic functions series (see (12)) in is also analytic, whenever the denominator is different from zero at . It is important to notice that PSEM convergence greatly depends on the proper selection of the trial function. Then, it is necessary that the proposed TF can potentially describe the qualitative behaviour of the solution of the nonlinear problem.

3. PSEM Approximation Procedure for Lambert Function

In this section, we will present the approximations for both branches of Lambert function by using the PSEM procedure. Each branch will be divided into some regions in order to reduce the complexity of the approximations increasing in consequence with the speed of evaluation. In addition, all the numerical PSEM approximations of this work are obtained considering at least four significant digits. Readers can find in Appendix A the proposed numerical approximations. It is important to remark that all the power series expansions of this work were obtained using special transformations of and applying the built-in package “MultiSeries” of Maple 18.

3.1. Lower Branch Treatment

The lower branch will be segmented into three pieces. For the first approximation, we propose

where constants and will be determined by the PSEM procedure. In order to obtain an accurate approximation, we propose to obtain build 4 equations for , evaluating them at and other 5 equations for , evaluating them at , and equating with the same derivatives of (15). Then, by solving the resulting equation system for the unknown constants we obtain the approximate solution (A.1) (see Appendix A). The second and third approximations are expressed as

and

where

It is important to note that is obtained by calculating the order two power series expansion at for

and transforming (19) using , we obtain (18).

In summary, the lower branch is approximated (see (A.1)-(A.3) in Appendix A) by

3.2. Upper Branch Treatment

The upper branch is divided into four segments. First, we propose the following transform:

Then, the proposed trial function to apply PSEM to (21) is

with

where is obtained by calculating the order four power series expansion at for (21). Then, the first approximation for the upper branch of Lambert is

Next, the TF for the next two approximations are

and

For the last approximation, we propose the following transform:

Then, we calculate the order 5 power series expansion of (27) and replace ; it results in

In summary, the upper branch is approximated by

Numerical PSEM approximations , , and are (A.4), (A.5), and (A.6), respectively (see Appendix A).

4. Iterative Scheme to Improve Accuracy

In [74] an easy computable iterative algorithm was proposed to increase the accuracy of approximation. The iterative formula is given by

where is number of iterations, is

and

The error term for this iterative scheme is [74, 75]. In general terms, lower and upper branch were approximated by PSEM in order to obtain more than 4 significant digits. Then, after a single pass of 30 we obtain more than 16 digits as depicted in Figure 2. In the same fashion, Figures 3 and 4 show more than 15 digits for the upper branch approximations. Therefore, our scheme is compatible with the double float (64-bit IEEE 754) format of C/C++, which provides 15 significant digits. Furthermore, the same figures presented the significant digits of other works from literature after a singles pass of (30), causing our proposal to be the only one that keeps more than 15 digits for the complete domain of Lambert . If more significant digits are required, user can add more iterations to the scheme. The significant digits were calculated using

The number of employed digits extended to 50 (in Maple CAS software) during the simulation in order to avoid rounding errors during the evaluation of approximations. It is important to notice from Figure 4 that our approximation presents the higher number (more than 23) of significant digits for than all the other approximations.

Figure 2: Significant digits for (20) and the ones reported at [7, 10] after a single pass of iterative scheme of (30).

Figure 3: Significant digits for (29) and other ones reported at [7, 11, 12] after a single pass of iterative scheme of (30) for the interval of .

Figure 4: Significant digits for (29) and other ones reported at [7, 11, 12] after a single pass of iterative scheme of (30) for the interval of .

4.1. Computation Time Measurement

A computation time (CPU time) comparison among our approximation and other ones from literature was performed using Fortran 77/90, using as compiler the gcc 5.4.0 with level 3 of optimization. The computer used for the simulations was an Intel Core Pentium(R) CPU 2117U × 2 running at 1.80GHz under Linux Ubuntu 16.04. In order to circumvent the operative system instabilities, the time measurement was determined by the average time of 10 million of evaluations for each point. It is important to note in Figure 5 that our approximation presents, in general, the best CPU time behaviour in comparison with other reported approximations: [7, 10] for lower branch and [7, 11, 12] for the upper branch. However, for some regions our approximation exhibits a similar or slightly better CPU time. Further work is necessary to obtain faster approximations compatible with double float standard as the one reported [24], which presents a precise and fast computation numerical algorithm to evaluate that exhibits 50 nsec of average time. The key advantage of [24] to obtain such speed is that authors do not use transcendental functions to evaluate Lambert . However, our approximation is simple piece-wise formulas that are straightforward to implement in Fortran (see Appendix A), C/C++, Java, Basic, among many others.

(a) C.T.

(b) C.T. for

Figure 5: Computation time (CT) for and .

5. Case Studies

In this section we present two applications of the Lambert function . The first case of application is to economy and second one is an interesting application to plasma physics.

5.1. Economic Order Quantity with Perishable Inventory

In [76] a problem of Economic Order Quantity (EOQ) with perishable inventory related to the supply chain design of melons was reported. After some assumptions the authors of [76] obtain the optimal transfer batch quantity as

where the batch transfer cost is [$], the batch transfer time is [h], the time in the cold chain is   , the value of the melons at picking is , the deterioration rate at a field temperature at 30°C is   , the deterioration rate in the cold chain is   , and the picking rate is   .

As an example obtained from [76], we assume the values for $7, h, h, $75, days, per day. Allowing the picking rate to vary between 1 and 120 cartons per hour producing an optimal transfer batch quantity, , which is shown in Figure 6, where we have plotted (34) using our proposal and the exact numerical Lambert based solution from Maple 18.

Figure 6: Approximate and exact optimal transfer batch quantity .

5.2. Application on Extremes of the Sagdeev Pseudo-Potential

After several assumptions and manipulations, the electrostatic potential solution [77] is obtained as

and

where the first and second roots correspond, respectively, to the maximum and minimum in the pseudo-potential, in terms of both real branches of Lambert . and are constants of the phenomenon.

The exact ( and ) and approximate ( and ) contour plot for different values of is presented in Figure 7(b). In addition, to find the locus of the conjugation points (see Figure 7(a)), we must insert in (35) and (36).

(a) Contour plot of the roots for and

(b) Locus of conjugation points

Figure 7: Contour plot of the roots and as functions of the parameter .

The conjugation points separate the (upper) region of the root , corresponding to the maximum in the pseudo-potential, from the (lower) region of the root , corresponding to the minimum in the pseudo-potential.

6. Discussion

PSEM procedure produces compact and high accurate approximations as depicted in Figures 2, 3, and 4. Furthermore, as a reference for convergence of PSEM, we use the root mean square error defined as

where means relative error and and represent the valid region of the approximation. The integration procedure will be performed using Simpson’s rule 1/3 taking 100 samples for each valid interval of the approximations (20) and (29). Additionally, in order to calculate (37), we consider the exact value of Lambert function as the numerical values obtained from the built-in command from Maple 18.

Figure 8 depicts the error for each branch approximations, using several -th trial functions (TF) increasing gradually the number of constants accordingly with Tables 1 and 2. Such figure shows how as the number of constants increments the error reduces its value significantly showing the good convergence of the PSEM procedure.

Table 1: Trial (-th) functions (TF) for PSEM for , and .

TF
Approximation

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

1

2

3

4

5

6

♣ .

♠ .

Table 2: Trial (-th) functions (TF) for PSEM for , and .

TF
Approximation

1

2

3

4

5

6

1

2

3

4

5

1

2

3

4

5

6

7

8

9

♣, .

Figure 8: for the different trial functions for PSEM.

As it is expected, the cases study shows how our proposal can be applied to solve practical problems in sciences and engineering. In fact, the simplicity of the piece-wise approximations allows easily implementing the approximations within computer languages as Fortran, C++, among many others. It is also worth mentioning that the computation time of our proposal was compared against other reported approximations from literature presenting the advantage of our proposal. Further work is required to reduce even more the computation time in order to compete with numerical algorithms as the one reported in [24].

7. Concluding Remarks

In this article piece-wise approximations for both branches of Lambert were proposed. The approximations were obtained using PSEM in combination with asymptotic approximations, resulting in handy and highly accurate expressions in comparison with other ones from literature. In fact, we apply the approximations to practical problems in the fields of economic and physics, exhibiting highly accurate results. It is important to highlight that the coupling of PSEM with asymptotic solutions was successful to improve the accuracy in the vicinity of the expansion point and keeping the high accuracy of the original asymptotic approximation for large values of the domain. Additionally, we prove that it is possible to improve the accuracy using an easy computable iterative scheme based on asymptotic methods. The computation time was measured resulting that our proposal is fast in comparison with other approximations from literature. Even more, the significant digits from our proposal are fully compatible with the double float format from C/C++ concluding that our approximations can be applied to practical problems as the study cases of this work.

Appendix

A. Numerical PSEM Approximations

The numerical approximations obtained using PSEM with parameters from Table 3 are for the lower branch:

and

Table 3: Parameters for PSEM procedure.

Equation
Cancellation points
Employed derivatives

Eq. (16)

,

,

,

.

Eq. (17)

.

Eq. (24)

Eq. (25)

,

,

.

Eq. (26)

,

,

,

.

Next, the approximations for the upper branch are

and

B. Fortran Code to Compute the CPU Time

!===================================================================

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!Routine for measure CPU time of the PSEMapproximations for Lambert W

real:: start, finish, tt

integer jend,j

integer8 i,ii

real8 z,x,w,wmin,wmax,dw

real8 V,A, Q,F

real8 k0,k1,k2

real8 k3

ii=10000000

jend=120

wmin=-15.0d0

wmax=15.0d0

dw=(wmax-wmin)/dble(jend)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

write(,"(a20,a20,a25)") "z","W","Tp nS"

do j=0,jend

x=wmin+dwdble(j)

w=x

call cpu_time(start)

x=wexp(w)

do i=0,ii

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Lower branch LambertW!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!! W-1,3(x)

if(-0.1<=x.and.x<0.and. w<-1) then

k0 = (1.01999365162218+(-12.6917365519443-45.1506015092455x)x)x/(1+&

(-22.9809693297808+(-104.692066099727-95.2085341727207x)x)x)

k1 = log(-x)

k2 = k1-log(-k1)+log(-k1)/k1

V = k0+k2

end if

!!!!!!!!!!!!!!!! W-1,2(x)

if(-0.34<=x.and. x<-0.1.and. w<-1) then

V = (-1362.78381643109+(-1386.04132570149+(11892.1649836015+

16904.0507511421x)x)x)x/(1+ &

(251.440197724561+(-1264.99554712435+

(-5687.63429510978-2639.24130979048x)x)x)x)

end if

!

!!!!!!!!!!!!!!!! W-1,1(x)

if(-0.3678794411714423<=x.and.x<-0.34.and. w<-1) then

V= (-7.874564067684664+(-63.11879948166995+(-168.6110850408981-&

150.1089086912451x)x)x)x(x+0.3678794411714423)/(1.0+&

(15.97679839497612+(98.26612857148953+(293.9558944644677+&

(430.4471947824411+247.8576700279611x)x)x)x)x)-1.0

end if

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!

!

!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Upper branch!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!

!!!!!!!!W0,1(x)

if(-0.3678794411714423<=x.and.x<1.and. w>=-1) then

k1 = sqrt(1.+2.7182818284591x)

k2 = 0.33333333333333+.70710678118655/k1-0.058925565098880k1+&

(x+0.36787944117144)(0.050248489761611+(0.11138904851051+

0.040744556245195x)x)/(1.+ &

(2.7090878606183+(1.5510922597820+0.095477712183841x)x)x)

V = -(k2-1)/k2

end if

!

!!!!!!!!!!!!!!!!W0,2(x)

if(1<=x.and.x<40) then

k1 = 1+(5.950065500550155+(13.96586471370701+(10.52192021050505+

(3.065294254265870+0.1204576876518760x)x)x)x)x

V = 0.1600049638651493log (k1)

end if

!

!!!!!!!!!!!!!!!!W0,3(x)

if(40<=x.and.x<20000) then

k1 = 1+(-3.16866642511229e11+(3.420439800038598e10+ &

(-1.501433652432257e9+(3.44887729947585e7+(-4.453783741137856e5+ &

(3257.926478908996+(-10.82545259305382+(0.6898058947898353e-1+ &

0.4703653406071575e-4x)x)x)x)x)x)x)x)x

V = 0.9898045358731312e-1log(k1)

end if

!

!!!!!!!!!!!!!!!!W0,4(x)

if(20000<=x) then

k1 = 1/(1+log(1+x))

k2 = 1/k1

k3 = log(k2)

V = k2-1-k3+(1+k3+(-1/2+(1/2)k32+(-1/6+(-1+(-1/2+

(1/3)k3)k3)k3)k1)k1)k1

end if

!

! Fritsch's formula

A = log(x/V)-V

Q = (2(1+V))(1+V+(2/3)A)

F = A(Q-A)/((1+V)(Q-2A))

z = V(1+F)

  continue

continue

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

enddo

call cpu_time(finish)

tt=((finish-start)/10000000)1000000000

write(,"(0pf25.15,0pf25.15,0PF15.5)")x,z,tt

enddo

end program lambertw_tiempos

Data Availability

All data generated or analysed during this study are included within this research article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Authors wish to thank, M.TN. César Colocia García, Eng. Francisco Martínez Barrios, Roberto Ruiz Gomez and Gerardo César Vélez López for their support. The second author is currently a postdoctoral fellow researcher at National Institute for Astrophysics, Optics and Electronics and expresses his gratitude for the support received. He also acknowledges the financial support from the Secretary of Public Education of México (SEP) through Grant 2703 E476300.0275652.

References

A. Milton and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, DC, USA, 1964.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1996.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions Hardback and CD-ROM, Cambridge University Press, 2010.

M. Kline, Mathematical Thought From Ancient to Modern Times, vol. 3, Oxford University Press, New York, NY, USA, 1972.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, vol. 5, no. 1, pp. 329–359, 1996.

P. B. Brito, F. Fabiao, and A. Staubyn, “Euler, Lambert, and the Lambert W-function today,” The Mathematical Scientist, vol. 33, no. 2, pp. 127–133, 2008.

D. Veberic, “Having fun with Lambert W(x) function,” https://arxiv.org/abs/1003.1628.

D. Veberič, “Lambert W function for applications in physics,” Computer Physics Communications, vol. 183, no. 12, pp. 2622–2628, 2012.

F. Johansson, “Computing the Lambert W function in arbitrary-precision complex interval arithmetic,” https://arxiv.org/abs/1705.03266.

F. Chapeau-Blondeau and A. Monir, “Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2,” IEEE Transactions on Signal Processing, vol. 50, no. 9, pp. 2160–2165, 2002.

D. A. Barry, J.-Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and F. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Mathematics and Computers in Simulation, vol. 53, no. 1-2, pp. 95–103, 2000.

R. Iacono and J. P. Boyd, “New approximations to the principal real-valued branch of the Lambert W-function,” Advances in Computational Mathematics, pp. 1–34, 2017.

I. Jadlovska, “Application of Lambert W function in oscillation theory,” Acta Electrotechnica et Informatica, vol. 14, no. 1, pp. 9–17, 2014.

G. Polya and G. Szego, Problems and Theorems in Analysis, Springer-Verlag, 1972.

E. M. Wright, “Xl the linear difference-differential equation with constant coefficients,” Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 62, no. 4, pp. 387–393, 1949.

E. M. Wright, “Xii solution of the equation ,” Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol. 65, no. 2, pp. 193–203, 1959.

N. D. Hayes, “Roots of the transcendental equation associated with a certain difference-differential equation,” Journal of the London Mathematical Society, vol. 1, no. 3, pp. 226–232, 1950.

F. N. Fritsch, R. E. Shafer, and W. P. Crowley, “Algorithm 443: solution of the transcendental equation [c5],” Communications of the ACM, vol. 16, no. 2, pp. 123-124, 1973.

D. A. Barry, S. J. Barry, and P. J. Culligan-Hensley, “Algorithm 743: Wapr–a fortran routine for calculating real values of the w-function,” ACM Transactions on Mathematical Software (TOMS), vol. 21, no. 2, pp. 172–181, 1995.

R. M. Corless, G. H. Gonnet, D. EG. Hare, D. J. Jeffrey, and D. Knuth, “Lambert’s W function in Maple,” Maple Technical Newsletter, vol. 9, no. 1, pp. 12–22, 1993.

D. J. Jeffrey, D. E. G. Hare, and R. M. Corless, “Unwinding the branches of the Lambert W function,” The Mathematical Scientist, vol. 21, no. 1, pp. 1–7, 1996.

R. M. Corless, D. J. Jeffrey, and D. E. Knuth, “A sequence of series for the Lambert W function,” in Proceedings of the 1997 international symposium on Symbolic and algebraic computation, pp. 197–204, ACM, 1997.

A. Hoorfar and M. Hassani, “Inequalities on the Lambert W function and hyperpower function,” Research Report Collection, vol. 10, no. 2, 2007.

T. Fukushima, “Precise and fast computation of Lambert W-functions without transcendental function evaluations,” Journal of Computational and Applied Mathematics, vol. 244, pp. 77–89, 2013.

M. Goličnik, “On the Lambert W function and its utility in biochemical kinetics,” Biochemical Engineering Journal, vol. 63, pp. 116–123, 2012.

D. J. Jeffrey and J. E. Jankowski, “Branch differences and Lambert W,” in Proceedings of the 16th international symposium on symbolic and numeric algorithms for scientific computing, SYNASC 2014, pp. 61–65, 2014.

D. J. Jeffrey, “Branch structure and implementation of Lambert W,” Mathematics in Computer Science, vol. 11, no. 3-4, pp. 341–350, 2017.

A. I. Kheyfits, “Explicit solutions of transcendental equations and the Lambert W function,” AMS subject classifications. 33B99, 33F05, 65H05, 30D99, 2003.

T. P. Dence, “A Brief Look into the Lambert W Function,” Applied Mathematics, vol. 4, no. 6, pp. 887–892, 2013.

I. Mezö and Á. Baricz, “On the generalization of the Lambert W function,” Transactions of the American Mathematical Society, vol. 369, no. 11, pp. 7917–7934, 2017.

S. Schnell and C. Mendoza, “Closed form solution for time-dependent enzyme kinetics,” Journal of Theoretical Biology, vol. 187, no. 2, pp. 207–212, 1997.

D. C. Jenn, “Applications of the Lambert W function in electromagnetics,” IEEE Antennas and Propagation Magazine, vol. 44, no. 3, pp. 139–142, 2002.

A. Ortiz-Conde, F. J. García Sánchez, and J. Muci, “Exact analytical solutions of the forward non-ideal diode equation with series and shunt parasitic resistances,” Solid-State Electronics, vol. 44, no. 10, pp. 1861–1864, 2000.

J. J. Shynk, Mathematical Foundations for Linear Circuits and Systems in Engineering, John Wiley & Sons, 2016.

M. K. M. Tripathy and P. K. Sadhu, “Photovoltaic system using Lambert W function-based technique,” Solar Energy, vol. 158, pp. 432–439, 2017.

M. A. Casey, “Reduced-rank spectra and minimum entropy priors for generalized sound recognition,” in Proceedings of the workshop on consistent and reliable cues for sound analysis, p. 167, 2001.

S. R. Valluri and R. M. Corless, “Some applications of the Lambert W function to physics,” Canadian Journal of Physics, vol. 78, no. 9, pp. 823–831, 2000.

R. Bartolini, J. I. M. Botman, M. Migliorati, and C. Thomas, “Longitudinal beam distribution, in a storage ring with pure inductance described by the Lambert W function,” in Proceedings of the EPAC, pp. 1565–1567, 2002.

B. Coll, “A universal law of gravitational deformation for general relativity,” in Proceedings of the Spanish Relavistic Meeting. EREs, Salamanca, Spain, 1998.

P. Braun Boni, K. M. Briggs, and P. Boeni, “Analytical solution to matthews and blakeslees critical dislocation formation thickness of epitaxially grown thin films,” Journal of Crystal Growth, vol. 241, no. 1-2, pp. 231–234, 2002.

R. S. Ambagaspitiya and N. Balakrishnan, “On the compound generalized poisson distributions,” ASTIN Bulletin, vol. 24, no. 2, pp. 255–263, 1994.

M. Ditzel and W. A. Serdijn, “Optimal energy assignment for frequency selective fading channels,” in Proceedings of the 12th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC 2001), IEEE, October 2001.

W. Szpankowski, “On asymptotics of certain recurrences arising in universal coding,” Problems of Information Transmission c/C of problemy peredachi, vol. 34, pp. 142–146, 1998.

G. Polya, “Kombinatorische anzahlbestimmungen fiir gruppen, graphen und chemische verbindungen,” Acta Mathematica, vol. 68, no. 1, pp. 145–254, 1937.

I. N. Baker and P. J. Rippon, “A note on complex iteration,” The American Mathematical Monthly, vol. 92, no. 7, pp. 501–504, 1985.

I. N. Galidakis, “On an application of Lambert’s W function to infinite exponentials. complex variables, theory and application,” An International Journal, vol. 49, no. 11, pp. 759–780, 2004.

P. Flajolet, B. Salvy, and G. Schaeffer, “Airy phenomena and analytic combinatorics of connected graphs,” Electronic Journal of Combinatorics, vol. 11, pp. 1–30, 2004.

C. T. H. Baker, C. A. H. Paul, and D. R. Willé, “A bibliography on the numerical solution of delay differential equations,” Numerical Analysis Report M13 9PL, The University of Manchester, Department of Mathematics, Manchester, UK, 1995.

A. Amann, E. Schöll, and W. Just, “Some basic remarks on eigenmode expansions of time-delay dynamics,” Physica A: Statistical Mechanics and its Applications, vol. 373, pp. 191–202, 2007.

A. Sun Y, G. Ulsoy, and W. N. Patrick, “Analysis of systems of linear delay differential equations using the matrix lambert function and the laplace transformation,” Automatica, pp. 1–10, 2006.

W. N. Patrick, Y. Sun, and A. U. Galip, “Time-delay systems: analysis and control using the Lambert W function,” World Scientific, 2010.

R. D. Simitev and V. N. Biktashev, “Asymptotics of conduction velocity restitution in models of electrical excitation in the heart,” Bulletin of Mathematical Biology, vol. 73, no. 1, pp. 72–115, 2011.

A. E. Ehret, M. Böol, and M. Itskov, “A continuum constitutive model for the active behaviour of skeletal muscle,” Journal of the Mechanics and Physics of Solids, vol. 59, no. 3, pp. 625–636, 2011.

F. Heße, F. A. Radu, M. Thullner, and S. Attinger, “Upscaling of the advection-diffusion-reaction equation with Monod reaction,” Advances in Water Resources, vol. 32, no. 8, pp. 1336–1351, 2009.

R. Gong, C. Lu, W.-M. Wu et al., “Estimating reaction rate coefficients within a travel-time modeling framework,” Groundwater, vol. 49, no. 2, pp. 209–218, 2011.

D. Brkic, “Lambert w function in hydraulic problems,” in Proceedings of the MASSEE International Conference on Mathematics MICOM, 2009.

L. Zhang, D. Xing, and J. Sun, “Calculating activation energy of amorphous phase with the Lambert W function,” Journal of Thermal Analysis and Calorimetry, vol. 100, no. 1, pp. 3–10, 2010.

M. Conrath, N. Fries, M. Zhang, and M. E. Dreyer, “Radial capillary transport from an infinite reservoir,” Transport in Porous Media, vol. 84, no. 1, pp. 109–132, 2010.

C. Hadj Belgacem and M. Fnaiech, “Solution for the critical thickness models of dislocation generation in epitaxial thin films using the Lambert W function,” Journal of Materials Science, vol. 46, no. 6, pp. 1913–1915, 2011.

A. Pohjoranta, A. Mendelson, and R. Tenno, “A copper electrolysis cell model including effects of the ohmic potential loss in the cell,” Electrochimica Acta, vol. 55, no. 3, pp. 1001–1012, 2010.

J. Berthier and P. Silberzan, Microfluidics for Biotechnology, Artech House, Norwood, Mass, USA, 2010.

J.-M. Caillol, “Some applications of the Lambert W function to classical statistical mechanics,” Journal of Physics A: Mathematical and General, vol. 36, no. 42, pp. 10431–10442, 2003.

V. K. Kyncheva, V. V. Yotov, and S. I. Ivanov, “Convergence of Newton, Halley and Chebyshev iterative methods as methods for simultaneous determination of multiple polynomial zeros,” Applied Numerical Mathematics, vol. 112, pp. 146–154, 2017.

N. Osada, “Chebyshev-Halley methods for analytic functions,” Journal of Computational and Applied Mathematics, vol. 216, no. 2, pp. 585–599, 2008.

L. Yau and A. Ben-Israel, “The Newton and Halley methods for complex roots,” The American Mathematical Monthly, vol. 105, no. 9, pp. 806–818, 1998.

J. P. Boyd, “Global approximations to the principal real-valued branch of the Lambert W-function,” Applied Mathematics Letters, vol. 11, no. 6, pp. 27–31, 1998.

N. G. de Brujin, Asymptotic Methods in Analysis, North-Holland, 1961.

J. Karamata, “Sur quelques problemes poses par ramanujan,” Journal of the Indian Mathematical Society, vol. 24, no. 3-4, pp. 343–365, 1960.

H. Vazquez-Leal, “The enhanced power series method to find exact or approximate solutions of nonlinear differential equations,” Applied and Computational Mathematics, vol. 14, no. 2, pp. 168–179, 2015.

Y.-G. Wang, W.-H. Lin, and N. Liu, “A homotopy perturbation-based method for large deflection of a cantilever beam under a terminal follower force,” International Journal for Computational Methods in Engineering Science and Mechanics, vol. 13, no. 3, pp. 197–201, 2012.

H. Vazquez-Leal and A. Sarmiento-Reyes, “Power series extender method for the solution of nonlinear differential equations,” Mathematical Problems in Engineering, vol. 2015, no. 1-2, Article ID 717404, 7 pages, 2015.

D. G. Zill, A First Course in Differential Equations with Modeling Applications, Cenage Learning, 10th edition, 2012.

W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, Springer, 1999.

F. N. Fritsch, R. E. Shafer, and W. P. Crowley, “Solution of the transcendental equation wew = x,” Communications of the ACM, vol. 16, no. 2, pp. 123-124, 1973.

D. A. Barry, P. J. Culligan-Hensley, and S. J. Barry, “Real values of the W-function,” ACM Transactions on Mathematical Software, vol. 21, no. 2, pp. 161–171, 1995.

S. M. Disney and R. D. H. Warburton, “On the Lambert W function: Economic Order Quantity applications and pedagogical considerations,” International Journal of Production Economics, vol. 140, no. 2, pp. 756–764, 2012.

I. D. Dubinova, “Application of the Lambert W function in mathematical problems of plasma physics,” Plasma Physics Reports, vol. 30, no. 10, pp. 872–877, 2004.