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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 685485, 21 pages
http://dx.doi.org/10.1155/2011/685485
Research Article

On the Exact Analytical Solutions of Certain Lambert Transcendental Equations

1Department of Electrical Engineering, University of Montenegro, Dobrota 36, 85 330 Kotor, Montenegro
2Department of Electrical Engineering, University of Belgrade, Bul. kralja Aleksandra 73, 11 000 Belgrade, Serbia
3Department of Maritime Sciences, University of Montenegro, Dobrota 36, 85 330 Kotor, Montenegro

Received 13 August 2010; Revised 16 January 2011; Accepted 18 February 2011

Academic Editor: Shi Jian Liao

Copyright © 2011 S. M. Perovich 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

The subject matter of this paper is the Special Trans Functions Theory (STFT) in finding the exact analytical closed form solutions (the new special tran functions) of certain Lambert transcendental equations. Note that these Lambert transcendental equations frequently appear in applied physics and engineering domain. The structure of the theoretical derivations, proofs, and numerical results confirms validity and basic principles of the STFT. Undoubtedly, the proposed exact analytical approach implies the qualitative improvements of the conventional methods and techniques.

1. Introduction

The subject of the theoretical analysis presented here is a broad family of Lambert transcendental equations of the formΨ(𝜁)𝛼(𝜁)Ψ(𝜁)𝛽(𝜁)=𝜆(𝜁)Ψ(𝜁)𝛼(𝜁)+𝛽(𝜁),(1.1) where 𝜆(𝜁)=(𝛼(𝜁)𝛽(𝜁))𝜗(𝜁), Ψ(𝜁)𝑅, 𝛼(𝜁)𝑅+, 𝛽(𝜁)𝑅+, 𝜗(𝜁)𝑅+, and where we will restrict ourselves to the one-dimensional case, when 𝛼, 𝛽, 𝜗, and Ψ are functions of the arbitrary variable 𝜁 in the real domain.

The present paper considers possibilities for finding some exact analytical closed form solutions of the transcendental Lambert equations (1.1), by the Special Trans Functions Theory (S. M. Perovich) [113], since the former methods [14, 15] et al. cannot express solutions in the exact analytical closed form. Equation (1.1) has been firstly proposed by Lambert in 1758 [15] and studied by Euler. In these classical research works Lambert obtained analytical solutions for the special case of (1.1). Namely, (1.1) was reduced to the form ln(Ψ)=𝜗Ψ𝛽, for 𝛼𝛽, and solved by Lambert's W functions [14, 15]. It is to be pointed that this special case of (1.1) was solved exactly in an analytical closed form by the Special Trans Functions Theory [13, 10]. The comparisons between Lambert's W functions and Trans functions are given in some detail in [1, 10]. Though, the transcendental Lambert equation (1.1) has not been solved yet, there is a motivation for solving it analytically, by the STF Theory.

The needs for the exact analytical analysis of transcendental Lambert equations (1.1) arise in variety of different disciplines. In other words, the problem of obtaining the results from an analytical study of certain Lambert transcendental equations (1.1) is the important one, in both the theoretical sense and the practical purpose [1], since there is a broad class of the problems appearing in the applied physics and engineering domain being mathematically described by (1.1).

The simple mathematical analysis of (1.1) implies that it has nontrivial, real solutions: Ψ< (for Ψ<(𝛼/𝛽)1/(𝛼𝛽)), and Ψ> (for Ψ>(𝛼/𝛽)1/(𝛼𝛽))). The condition for solutions existence in real domain takes the form 𝜆<(𝑏1)𝑏1/𝑏𝑏, where 𝑏=𝛼/(𝛼𝛽).

Remark 1.1. The essential for the Special Trans Functions Theory in solving transcendental equations within the real domain is that it always gives the least solution in terms of absolute values. This is general structural characteristic of the STFT, caused, most probably, by convergence dynamics [1, 4, 13].

We remark that the Lambert transcendental equations (1.1) are applicable, for instance, within families of the RC diode nonlinear circuits theoretical analysis [1, 4]. Analogically, the nonlinear system of equations for electrical parameters determination for different unknown materials (fluid and solid) is reduced to (1.1) [1, 16]. Within the previous analysis the observed Lambert transcendental equation has been solved by the application of some interative procedures based on the Special Trans Function Theory. Upon the results obtained in this article, it becomes possible to solve the mentioned problems from the engineering domain-exactly, that is, in the analytically closed form.

2. The Solution of Lambert Transcendental Equation (1.1) when Ψ<(𝛼/𝛽)1/(𝛼𝛽)

In this section, we attempt to find an exact analytical closed form solution of Lambert equation (1.1), Ψ<, when Ψ<(𝛼/𝛽)1/(𝛼𝛽), for arbitrary, real, and positive values of 𝛼, 𝛽, and 𝜗.

Theorem 2.1. Lambert transcendental equation (1.1) for 𝜆<(𝑏1)𝑏1/𝑏𝑏, where 𝑏=𝛼/(𝛼𝛽), and for Ψ<(𝛼/𝛽)1/(𝛼𝛽) has an exact analytical closed form solution Ψ=Ψ<=tran𝐿<(𝛼,𝛽,𝜗), where tran𝐿<(𝛼,𝛽,𝜗) is a new special trans function defined as tran𝐿<(𝛼,𝛽,𝜗)=lim𝑥Ψ𝐿<(𝑥1,𝛼,𝛽,𝜗)𝜎<Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)𝜎<,𝜎<Ψ=𝑜<𝜆,(2.1) where 𝛼=𝛼(𝜁),𝛽=𝛽(𝜁),𝜗=𝜗(𝜁),𝜆=𝜆(𝜁).(2.2)Ψ𝑜< denotes a positive constant. Ψ𝐿<(𝑥,𝛼,𝛽,𝜗) is a function defined as Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=Ψ𝑜<𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥+𝛽(𝛼𝛽)𝑛𝛽𝑘),(2.3) where 𝐻(𝑥+𝛽(𝛼𝛽)𝑛𝛽𝑘) is a Heaviside's unit function. Or, after sumlimit determination, the latter expression for Ψ𝐿<(𝑥,𝛼,𝛽,𝜗) takes the form Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=Ψ𝑜<[(𝑥+𝛽)/𝛼]𝑛=0(1𝜆)𝑛[(𝑥+𝛽)/(𝛼𝛽)]𝑛=[(𝑥+𝛽)/𝛼]+1[(𝑥+𝛽(𝛼𝛽)𝑛)/𝛽]𝑘=0(1)𝑘+1𝜆𝑘𝑛!,(𝑛𝑘)!𝑘!(2.4) where [𝑥] denotes the greatest integer less or equal to 𝑥.

Remark 2.2. The sumlimit determination implies the following reason: from Heaviside's unit function, we have 𝑥+𝛽(𝛼𝛽)𝑛𝛽𝑘>0 or, 𝑘=(𝑥+𝛽)/𝛽((𝛼𝛽)/𝛽)𝑛. Thus, the graphical presentation takes the form (Figure 1).

685485.fig.001
Figure 1: Graphical presentation for sumlimit determination in (2.4).

Proof. The transcendental Lambert equation (1.1) can be identified with an equation for identification (EQID) of type Ψ𝐿<(𝑥𝛼)Ψ𝐿<(𝑥𝛽)𝜆Ψ𝐿<(𝑥𝛼𝛽)=Ψ𝑜<,(2.5) where Ψ𝐿<(𝑥)=Ψ𝐿<(𝑥,𝛼(𝜁),𝛽(𝜁),𝜗(𝜁)) is an arbitrary real function, for 𝑥>0, and Ψ𝐿<(𝑥,𝛼(𝜁),𝛽(𝜁),𝜗(𝜁))=0, for 𝑥<0. Equation (2.5) will be solved in the set of originals of the Laplace Transform. Thus, after taking the Laplace Transform, (2.5) takes the following form 𝑃𝐿<𝑒(𝑠,𝛼,𝛽,𝜗)𝛼𝑠𝑒𝛽𝑠𝜆𝑒(𝛼+𝛽)𝑠=Ψ𝑜<𝑠,(2.6) where 𝑃𝐿<(𝑠,𝛼,𝛽,𝜗)=𝐿[Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)]. Therefore, 𝑃𝐿<(𝑠,𝛼,𝛽,𝜗)𝑒𝛽𝑠Ψ=𝑜<𝑠1𝑒𝑠(𝛼𝛽)1𝜆𝑒𝛽𝑠(2.7) since 𝛼>𝛽. By expanding, we get 𝑃𝐿<(𝑠,𝛼,𝛽,𝜗)𝑒𝛽𝑠Ψ=𝑜<𝑠𝑛=0𝑒𝑛𝑠(𝛼𝛽)1𝜆𝑒𝛽𝑠𝑛.(2.8) The series (2.8) converges for |𝑒𝑠(𝛼𝛽)𝜆𝑒𝛼𝑠|1. Now, we can invert term by term to obtain the original Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=Ψ𝑜<𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥+𝛽(𝛼𝛽)𝑛𝛽𝑘).(2.9) Finally, applying the sumlimit determination, the above equation takes the form Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=Ψ𝑜<[(𝑥+𝛽)/𝛼]𝑛=0(1𝜆)𝑛[(𝑥+𝛽)/(𝛼𝛽)]𝑛=[(𝑥+𝛽)/𝛼]+1[(𝑥+𝛽(𝛼𝛽)𝑛)/𝛽]𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!,(2.10) where Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=Ψ𝐿<(𝑥,𝛼(𝜁),𝛽(𝜁),𝜗(𝜁)). The functional series (2.10) is the unique analytical closed form solution of (2.5) according to Lerch's theorem.

Next, we need the following lemma.

Lemma 2.3. For any 𝑥>𝛼(𝜁), the functional series (2.9) satisfies (2.5).

Proof. When we substitute (2.9) into (2.5), the following is obtained: 𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥𝛼+𝛽(𝛼𝛽)𝑛𝛽𝑘)𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥(𝛼𝛽)𝑛𝛽𝑘)𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘+1𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥𝛼(𝛼𝛽)𝑛𝛽𝑘)=1.(2.11) After simple modification, the above equation takes the form: 𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥(𝛼𝛽)(𝑛+1)𝛽𝑘)𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥(𝛼𝛽)𝑛𝛽𝑘)𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘+1𝑛!(𝑛𝑘)!𝑘!𝐻(𝑥(𝛼𝛽)(𝑛+1)𝛽(1+𝑘))=1,(2.12) or the form 𝑛=𝑛=1𝑛1𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝑛=𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!𝑛=𝑛𝑛=1𝑘=1(1)𝑘+1𝜆𝑘+1(𝑛1)!(𝑛𝑘)!(𝑘1)!𝐻(𝑥(𝛼𝛽)𝑛𝛽𝑘)=1.(2.13) The latter expression can be written in the form 𝑛=𝑛=1𝑛1𝑘=1(1)𝑘+1𝜆𝑘(𝑛1)!1(𝑛𝑘1)!(𝑘1)!𝑘𝑛+1(𝑛𝑘)𝑘(𝑛𝑘)𝐻(𝑥(𝛼𝛽)𝑛𝛽𝑘)+(1)𝐻(𝑥𝛼+𝛽)(1)(𝐻(𝑥))+𝐻(𝑥𝛼)=1(2.14) since (1/𝑘)𝑛/(𝑛𝑘)𝑘+1/(𝑛𝑘)=(𝑛𝑘𝑛+𝑘)/(𝑛𝑘)𝑘=0, and 𝑥>𝛼.
Thus, we have finished our proof.

2.1. On the Particular Solution to (2.5)

Intuitively, we can see that the particular solution of the formΨ𝐿<𝑃Ψ(𝑥,𝛼,𝛽,𝜗)=<(𝜁)𝑥+𝜎<(2.15) satisfies (2.5) under the condition that Ψ<(𝜁) satisfies (1.1). Namely, after substitution of (2.15) into (2.5), we have:Ψ<(𝑥𝛼)+𝜎<Ψ<(𝑥𝛽)𝜎<Ψ𝜆<(𝑥𝛼𝛽)𝜆𝜎<=Ψ𝑜<(2.16) forΨ𝛼<Ψ𝛽<=𝜆Ψ𝛼+𝛽(2.17) and for 𝜎<=Ψ𝑜</𝜆. This means that the particular solution (2.15) satisfies (2.5) under condition that Ψ< (for Ψ<(𝛼/𝛽)1/(𝛼𝛽)) satisfies (1.1). Let us note, that it is beyond the scope of this paper to have indepth theoretical knowledge of the differential form of the particular (or asymptotic) solutions of the EQIDs in the Special Trans Functions Theory [1]. But, it is worth to be mentioned that EQIDs for the broad family of the transcendental Lambert equations of type (1.1) for asymptotic solutions have the functions of the form Ψ<(𝜁)𝑥+𝜎<.

2.2. On the Genesis of the Special Tran Function, tran𝐿<(𝛼,𝛽,𝜗)

Concerning the uniqueness of the inverse Laplace Transform (Lerch's Theorem) (2.10) is an unique analytical closed form solution of (2.5). On the other hand, (2.15) is a particular solution of (2.5). Easily, by (2.10) and (2.15), the following asymptotic relation appearslim𝑥Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)Ψ𝐿<𝑃(𝑥,𝛼,𝛽,𝜗)=1.(2.18) It is based on the functional theory, convergence dynamics and, first of all, on the condition that particular solution (2.15) is an asymptotic solution of EQID. Accordingly, we have the following.

Lemma 2.4. The solution (2.15) is an asymptotic solution of the EQID (2.5).

Proof. For (2.9) as unique analytical closed form solution of EQID (2.5), for (1𝜆)<1, we have lim𝑥Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=lim𝑥Ψ𝑜<𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!=Ψ(𝑛𝑘)!𝑘!𝐻(𝑥+𝛽(𝛼𝛽)𝑛𝛽𝑘)𝑜<𝑛𝑛=0𝑘=0(1)𝑘+1𝜆𝑘𝑛!(𝑛𝑘)!𝑘!=Ψ𝑜<𝑛=0(1𝜆)𝑛=Ψ𝑜<1Ψ1(1𝜆)=𝑜<𝜆.(2.19)
On the other hand, for particular solution (2.15), we havelim𝑥Ψ𝐿𝑃<(𝑥,𝛼,𝛽,𝜗)=lim𝑥Ψ<(𝜁)𝑥+𝜎<=𝜎<Ψ=𝑜<𝜆,lim𝑥Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)=lim𝑥Ψ𝐿𝑃<Ψ(𝑥,𝛼,𝛽,𝜗)=𝑜<𝜆.(2.20) Thus, our proof is completed.

According to the unique solution principle and Lemma 2.4, following appearslim𝑥Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)Ψ𝐿<(𝑥+1,𝛼,𝛽,𝜗)=lim𝑥Ψ𝐿<𝑃(𝑥,𝛼,𝛽,𝜗)Ψ𝐿<𝑃(𝑥+1,𝛼,𝛽,𝜗).(2.21)

It is remarkable that in the STFT, the EQID, on the one hand, has the unique exact analytical closed form solution, and, on the other hand, it has the asymptotic (particular) solution. Thus, asymptotic equalizations (2.18) and (2.21) can be established.

Finally, from (2.15), (2.18) and (2.21), we havelim𝑥Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)𝜎<Ψ𝐿<(𝑥+1,𝛼,𝛽,𝜗)𝜎<=Ψ<(𝜁)𝑥Ψ<(𝜁)(𝑥+1)=Ψ<(𝜁)=Ψ<(2.22) and, Ψ<=tran𝐿<(𝛼,𝛽,𝜗), where tran𝐿<(𝛼,𝛽,𝜗) is a new special tran function defined astran𝐿<(𝛼,𝛽,𝜗)=lim𝑥Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)𝜎<Ψ𝐿<(𝑥+1,𝛼,𝛽,𝜗)𝜎<.(2.23)

More explicitly, after substitution (2.10) in (2.23), we haveΨ<=tran𝐿<(𝛼,𝛽,𝜗)=lim𝑥[(𝑥+𝛽)/𝛼]𝑛=0(1𝜆)𝑛+[(𝑥+𝛽)/(𝛼𝛽)]𝑛=[(𝑥+𝛽)/𝛼]+1[(𝑥+𝛽(𝛼𝛽)𝑛)/𝛽]𝑘=0+(1/𝜆)[(𝑥+1+𝛽)/𝛼]𝑛=0(1𝜆)𝑛+[(𝑥+1+𝛽)/(𝛼𝛽)]𝑛=[(𝑥+1+𝛽)/𝛼]+1[(𝑥+1+𝛽(𝛼𝛽)𝑛)/𝛽]𝑘=0,+(1/𝜆)(2.24) where denotes (((1)𝑘+1𝜆𝑘𝑛!)/(𝑛𝑘)!𝑘!)and Ψ<=Ψ<(𝜁)=tran𝐿<(𝛼(𝜁),𝛽(𝜁),𝜗(𝜁)). The formulae (2.24) is a new special tran function, tran𝐿<(𝛼(𝜁),𝛽(𝜁),𝜗(𝜁)) presented in some detail.

Note that an essential part of the Special Trans Functions Theory is the equalization (2.18) [1]. Due to the previous analysis it becomes clear that by applying unique solution principle [113], we obtain the equalization of type (2.18). Now, we have completed our proof.

3. Concerning a Formula for Practical Applicability of tran𝐿<(𝛼,𝛽,𝜗)

For practical analysis and numerical calculations the formula (2.18) takes the following form:Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)[𝑃<]Ψ𝐿<𝑃(𝑥,𝛼,𝛽,𝜗)[𝑃<]=1,(3.1) where, Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)[𝑃<] denotes the numerical value of the function Ψ𝐿<(𝑥,𝛼,𝛽,𝜗) given with [𝑃<] accurate digits. [𝑃<] is defined as [𝑃<]=[ln(|𝐺<|)], where the error function 𝐺< is defined like 𝐺<=Ψ𝛼<Ψ𝛽<𝜆Ψ<𝛼+𝛽.

So, we get from (2.15), (2.18), (2.21) and(3.1)Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)𝜎<Ψ𝐿<(𝑥+1,𝛼,𝛽,𝜗)𝜎<[𝑃<]=Ψ𝐿<𝑃(𝑥,𝛼,𝛽,𝜗)𝜎<Ψ𝐿<𝑃(𝑥+1,𝛼,𝛽,𝜗)𝜎<[𝑃<],Ψ(3.2)<[𝑃<]=Ψ𝐿<(𝑥,𝛼,𝛽,𝜗)𝜎<Ψ𝐿<(𝑥+1,𝛼,𝛽,𝜗)𝜎<[𝑃<].(3.3)

More explicitly, for fixed variable 𝑥, the Ψ<[𝑃<] takes the formΨ<[𝑃<]=[(𝑥+𝛽)/𝛼]𝑛=0(1𝜆)𝑛+[(𝑥+𝛽)/(𝛼𝛽)]𝑛=[(𝑥+𝛽)/𝛼]+1[(𝑥+𝛽(𝛼𝛽)𝑛)/𝛽]𝑘=0+(1/𝜆)[(𝑥+1+𝛽)/𝛼]𝑛=0(1𝜆)𝑛+[(𝑥+1+𝛽)/(𝛼𝛽)]𝑛=[(𝑥+1+𝛽)/𝛼]+1[(𝑥+1+𝛽(𝛼𝛽)𝑛)/𝛽]𝑘=0+(1/𝜆)[𝑃<].(3.4)

From a theoretical point of view, the solution (3.4) for Ψ<[𝑃<] can be found with an arbitrary order of accuracy by taking an appropriate value of [𝑥]. We remark that for these and other matters related to the numerical accuracy we refer to [1, 3, 10].

Let us note that after some sumlimit modifications, for numerical calculations, formulae (3.4) takes the form:Ψ<[𝑃<]=[𝑥/𝑏]𝑛=0(1𝜆)𝑛+[][𝑥]𝑛=𝑥/𝑏+1[(𝑥𝑛)/(𝑏1)]𝑘=0+(1/𝜆)[(𝑥+1)/𝑏]𝑛=0(1𝜆)𝑛+[][𝑥+1]𝑛=(𝑥+1)/𝑏+1[(𝑥+1𝑛)/(𝑏1)]𝑘=0+(1/𝜆)𝑃1/(𝛼𝛽)<,(3.5) where, 𝑏=𝛼/(𝛼𝛽).

4. The Special Trans Functions Theory for the Transcendental Lambert Equation (1.1) for Ψ>(𝛼/𝛽)1/(𝛼𝛽)

In this section we attempt to find an exact analytical closed form solution of the transcendental Lambert equation (1.1) for Ψ>(𝛼/𝛽)1/(𝛼𝛽), Ψ>.

After simple modification, (1.1) takes the formΨ>𝛽Ψ>𝛼=𝜆,(4.1) or, the form𝑍>𝑍+𝜆=>𝛽/𝛼,(4.2) where 𝑍>=𝑍>(𝜁)=(1/Ψ>(𝜁))𝛼(𝜁), and, finally,Ψ>=𝑍>1/𝛼.(4.3)

Theorem 4.1. The transcendental Lambert equation (4.2) for 𝜆<(𝑏1)𝑏1/𝑏𝑏, where 𝑏=𝛼/(𝛼𝛽), and for Ψ>(𝛼/𝛽)1/(𝛼𝛽) has an analytical closed form solution 𝑍=𝑍>=tran𝑍𝐿>(𝛼,𝛽,𝜆), where tran𝑍𝐿>(𝛼,𝛽,𝜆) is a new special tran function defined as tran𝑍𝐿>(𝛼,𝛽,𝜆)=lim𝑥Ψ𝐿>(𝑥1,𝛼,𝛽,𝜆)𝜎>Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)𝜎>,𝜎>=Ψ𝑜>𝜆,(4.4) where, by Ψ𝑜> we denote a positive constant which may be different in each equality. Ψ𝐿>(𝑥,𝛼,𝛽,𝜆) is a function, defined as Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)=(Ψ𝑜>/𝜆)𝑛=0𝑛𝑘=0((1)𝑘𝑛!/(𝑛𝑘)!𝑘!𝜆𝑛)𝐻(𝑥𝑛(𝛽/𝛼)𝑘(1𝛽/𝛼)), or, it can be rewritten in the form Ψ𝐿>Ψ(𝑥,𝛼,𝛽,𝜆)=𝑜>𝜆[𝑥(𝛼/𝛽)][𝑥]𝑛=+1[(𝑥𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=0(1)𝑘𝑛!(𝑛𝑘)!𝑘!𝜆𝑛.(4.5)
Theorem 4.1 can be proved by similar argument as in Section 3. In fact, we only need to adapt the Special Trans Functions Theory routinely for new case of the transcendental Lambert equation of type (4.2).

Proof. The transcendental Lambert equation (4.2) can be identified with an EQID of type Ψ𝐿>(𝑥1)+𝜆Ψ𝐿>(𝑥)Ψ𝐿>𝛽𝑥𝛼=Ψ𝑜>,(4.6) where Ψ𝐿>(𝑥,𝛼,𝛽,𝜆) is an arbitrary real function for 𝑥>0, and Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)=0 for 𝑥<0. We will solve (4.6) in the set of originals of the Laplace Transform. Thus, after taking the Laplace Transform, (4.6) takes the following form: 𝑃𝐿>𝑒(𝑠,𝛼,𝛽,𝜆)𝑠+𝜆𝑒𝛽𝑠/𝛼=Ψ𝑜>𝑠(4.7) or, 𝑃𝐿>Ψ(𝑠,𝛼,𝛽,𝜆)=𝑜>𝑒𝑠𝜆1𝛽𝑠/𝛼/𝜆1𝑒𝑠(1𝛽/𝛼).(4.8) By expanding, we get 𝑃𝐿>Ψ(𝑠,𝛼,𝛽,𝜆)=𝑜>𝑠𝜆𝑛=0𝑒𝛽𝑠/𝛼𝜆𝑛𝑛𝑘=0𝑛!(1)𝑘𝑒𝑘𝑠(1(𝛽/𝛼)).(𝑛𝑘)!𝑘!(4.9) The series (4.8) converges for |1𝑒𝑠(1(𝛽/𝛼))|1. Now, we can invert term by term to obtain the original Ψ𝐿>Ψ(𝑥,𝛼,𝛽,𝜆)=𝑜>𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝑛𝐻𝛽(𝑛𝑘)!𝑘!𝑥𝑛𝛼𝛽𝑘1𝛼.(4.10)
Finally, by applying the sumlimit determination, the analytical solution to (4.6) can be written in the closed form representationΨ𝐿>Ψ(𝑥,𝛼,𝛽,𝜆)=𝑜>𝜆[𝑥(𝛼/𝛽)]𝑛=[𝑥]+1[(𝑥𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=0(1)𝑘𝑛!(𝑛𝑘)!𝑘!𝜆𝑛.(4.11)
The functional series (4.11) is the unique analytical closed-form solution of (4.6) according to Lerch’s theorem.

Consequently, we have the following lemma.

Lemma 4.2. For any 𝑥>1 the functional series (4.10) satisfies (4.6).

Proof. After substituting (4.10) into (4.6), we obtain 1𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝑛𝐻𝛽(𝑛𝑘)!𝑘!𝑥1𝑛𝛼𝛽𝑘1𝛼1𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝜆𝑛(𝐻𝛽𝑛𝑘)!𝑘!𝑥𝑛𝛼𝛽𝑘1𝛼=1𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝑛𝐻𝛽(𝑛𝑘)!𝑘!𝑥(𝑛+1)𝛼𝛽𝑘1𝛼=1.(4.12)
As a result of a simple modification, the above equation takes the form1𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝑛𝐻𝛽(𝑛𝑘)!𝑘!𝑥(𝑛+1)𝛼𝛽(𝑘+1)1𝛼+1𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝜆𝑛(𝐻𝛽𝑛𝑘)!𝑘!𝑥𝑛𝛼𝛽𝑘1𝛼1𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝑛𝐻𝛽(𝑛𝑘)!𝑘!𝑥(𝑛+1)𝛼𝛽𝑘1𝛼=1(4.13) or the form 1𝜆𝑛𝑛=1𝑘=1(1)𝑘1(𝑛1)!𝜆𝑛+1(𝑛𝑘)!(𝑘1)!𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝜆𝑛1(𝑛𝑘)!𝑘!𝜆𝑛𝑛=1𝑘=0(1)𝑘(𝑛1)!𝜆𝑛𝛽(𝑛1𝑘)!𝑘!𝐻𝑥𝑛𝛼𝛽𝑘1𝛼=1.(4.14)
Namely,1𝜆𝑛𝑛=1𝑘=1(1)𝑘1(𝑛1)!𝜆𝑛+1(𝑛𝑘)!(𝑘1)!𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!𝜆𝜆𝑛1(𝑛𝑘)!𝑘!𝜆𝑛𝑛=1𝑘=0(1)𝑘(𝑛1)!𝜆𝑛𝛽(𝑛1𝑘)!𝑘!𝐻𝑥𝑛𝛼𝛽𝑘1𝛼=1𝜆𝑛𝑛=1𝑘=1(1)𝑘1(𝑛1)!𝜆𝑛1(1𝑛1𝑘)!(𝑘1)!𝑛(𝑛𝑘)+1(𝑛𝑘)𝑘𝑘1+𝐻(𝑥)+𝜆𝐻𝛽𝑥𝛼1𝜆𝐻𝛽𝑥𝛼𝛽1+𝛼=1,(4.15) since 1𝑛(𝑛𝑘)+1(𝑛𝑘)𝑘𝑘=𝑘𝑛+𝑛𝑘(𝑛𝑘)𝑘=0,𝑥>1.(4.16) Thus, the proof is finished.

4.1. On the Particular Solution of (4.6)

Intuitively, we can obtain that particular solution of the formΨ𝐿>𝑃𝑍(𝑥,𝛼,𝛽,𝜆)=>(𝜁)𝑥+𝜎>(4.17) satisfies (4.6), under condition that 𝑍>(𝜁) satisfies (4.2). Namely, after substitution of (4.17) into (4.6), we have𝑍>(𝑥1)+𝜎>𝑍>(𝑥(𝛽/𝛼))𝜎<𝑍+𝜆>𝑥+𝜆𝜎>=Ψ𝑜>(4.18) for𝑍>+𝜆=𝜆𝑍>𝛽/𝛼(4.19) and, for 𝜎>=Ψ𝑜>/𝜆. In other words, the particular solution (4.17) satisfies (4.6), under the condition that 𝑍> satisfies (4.2). The choice of the EQID is determined in a way that after substitution of the particular solution in EQID, we obtain the starting Lambert transcendental equation.

4.2. On the Genesis of the Special Tran Function, tran𝐿>(𝛼,𝛽,𝜗)

Concerning the uniqueness of the inverse Laplace Transform (Lerch's Theorem) (4.11) is an unique analytical closed form solution of (4.6). On the other hand, (4.17) is a particular solution of (4.6). It is easy to verify by (4.11) and (4.17) the existance of the following asymptotic equalizationlim𝑥Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)Ψ𝐿>𝑃(𝑥,𝛼,𝛽,𝜆)=1.(4.20) Note, that the statement (4.20) is based on the simple functional theory.

Lemma 4.3. Solution (4.17) is an asymptotic solution of the EQID (4.6).

Proof. For unique analytical closed form solution of EQID (4.6), for (1𝜆)<1, we have lim𝑥Ψ𝐿>(𝑥,𝛼,𝛽,𝜗)=lim𝑥Ψ𝑜>𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!(𝑛𝑘)!𝑘!𝜆𝑛𝐻𝛽𝑥𝑛𝛼𝛽𝑘1𝛼=Ψ𝑜>𝜆𝑛𝑛=0𝑘=0(1)𝑘𝑛!(𝑛𝑘)!𝑘!𝜆𝑛=Ψ𝑜>𝜆𝑛=01𝜆𝑛(11)𝑛=Ψ𝑜>𝜆.(4.21)
On the other hand, for particular solution (4.17), we havelim𝑥Ψ𝐿𝑃>(𝑥,𝛼,𝛽,𝜗)=lim𝑥Ψ>(𝜁)𝑥+𝜎>=𝜎>=Ψ𝑜>𝜆.(4.22)
Finally, the following appearslim𝑥Ψ𝐿>(𝑥,𝛼,𝛽,𝜗)=lim𝑥Ψ𝐿𝑃>=Ψ(𝑥,𝛼,𝛽,𝜗)𝑜>𝜆(4.23) and, thus the proof is finished.

According to the unique solution principle and Lemma 4.3, we havelim𝑥Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)𝜎>Ψ𝐿>(𝑥+1,𝛼,𝛽,𝜆)𝜎>=lim𝑥Ψ𝐿>𝑃(𝑥,𝛼,𝛽,𝜆)𝜎>Ψ𝐿>𝑃(𝑥+1,𝛼,𝛽,𝜆)𝜎>.(4.24) Now, from (4.17) appearslim𝑥Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)𝜎>Ψ𝐿>(𝑥+1,𝛼,𝛽,𝜆)𝜎>=𝑍>(𝜁)𝑥𝑍>(𝜁)(𝑥+1)=𝑍>(𝜁)=𝑍>(4.25) and 𝑍>=tran𝑍𝐿>(𝛼,𝛽,𝜆), where tran𝑍𝐿>(𝛼,𝛽,𝜆) is a new special tran function defined as𝑍>=tran𝑍𝐿>(𝛼,𝛽,𝜆)=lim𝑥Ψ𝐿>(𝑥,𝛼,𝛽,𝜆)𝜎>Ψ𝐿>(𝑥+1,𝛼,𝛽,𝜆)𝜎>.(4.26)

More explicitly, after substitution (4.11) into (4.26) we have𝑍>=lim𝑥[𝑥][𝑥(𝛼/𝛽)]𝑛=+1[(𝑥𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=0[][(𝑥+1)(𝛼/𝛽)]𝑛=𝑥+1+1[((𝑥+1)𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=0,(4.27) where 𝑍>=𝑍>(𝜁)=tran𝑍𝐿>(𝛼(𝜁),𝛽(𝜁),𝜆(𝜁)). From (4.3), followsΨ>=tran𝐿>(𝛼,𝛽,𝜆)=lim𝑥[𝑥][𝑥(𝛼/𝛽)]𝑛=+1[(𝑥𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=0[][(𝑥+1)(𝛼/𝛽)]𝑛=𝑥+1+1[((𝑥+1)𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=01/𝛼,(4.28) where tran𝐿>(𝛼,𝛽,𝜆) is another special tran function.

Now, the proof is completed.

Analogically, from (4.3) and (4.28), we obtain formulae for the practical applicationΨ>[𝑃>]=[𝑥][𝑥(𝛼/𝛽)]𝑛=+1[(𝑥𝑛(𝛽/𝛼))/(1(𝛽/𝛼))]𝑘=0[][(𝑥+1)(𝛼/𝛽)]𝑛=𝑥+1+1[((𝑥+1)𝑛(𝛽/𝛼))/(1𝛽/𝛼)]𝑘=01/𝛼[𝑃>],(4.29) where Ψ>[𝑃>] denotes the numerical value of Ψ> given with [𝑃>] accurate digits. [𝑃>] is defined as [𝑃>]=[ln(|𝐺>|)], where the error function 𝐺> is defined as 𝐺>=Ψ𝛼>Ψ𝛽>𝜆Ψ>𝛼+𝛽.

Remark 4.4. If there is more than two solutions of the transcendental Lambert equation (1.1), in real domain, than the appropriate modifications are necessary [13].

5. The Numerical Results Analysis

It is not difficult to see that analytical solutions (3.4) and (4.29) obtained by the Special Trans Functions Theory, gives highly accurate numerical results by MATHEMATICA program, suggesting that the STFT numerically works (Tables 1, 2, and 3).

tab1
Table 1
tab2
Table 2
tab3
Table 3

Concerning the numerical results through the number of accurate digits 𝑃< and 𝑃>, we deal with a sumlimit [𝑥]. It may be convenient to consider the 𝑃< and 𝑃> as a functions of [𝑥]. In fact, we have interest of knowing the functional dependence 𝑃<=𝑃<([𝑥]) (or, 𝑃>=𝑃>([𝑥])), which is sometimes the ordinary linear function. Of course, the linear functional form of 𝑃<=𝑃<([𝑥]) (or, 𝑃>=𝑃>([𝑥])) is frequently used when the problems in applied physics, or in engineering domain are in matter. Thus, the number of accurate digits in the numerical structure of Ψ< (or Ψ>) depends on the [𝑥]. An important study concerning the numbers of accurate digits in the numerical structure of the result has been formulated in [13, 6]. On the other hand, the number of accurate digits in the practical applications of Ψ< (or Ψ>) is in accordance with physical requirements of exactness.

Finally, we will illustrate the STFT application by some simple examples of the transcendental Lambert equations (1.1). For convenience we restrict ourselves to the one dimensional case of (1.1).

5.1. The Numerical Results for Some Examples of the Transcendental Lambert Equation (1.1) for Ψ<

The subjects of the numerical analysis presented here are some solutions of two simple examples of (1.1), for Ψ<(𝛼/𝛽)1/(𝛼𝛽).

5.1.1. Example 1 for Ψ<

Equation (1.1) for the following values: 𝛼=4, 𝛽=2, and 𝜆=(𝛼𝛽)𝜗=1/10, takes the formΨ<4Ψ<2=1Ψ10<6.(5.1)

According to (3.4), for above example, we obtain the solutionΨ<[𝑃<]=[(𝑥+2)/4]𝑛=0(9/10)𝑛+[][(𝑥+2)/2]𝑛=(𝑥+2)/4+1[(𝑥+22𝑛)/2]𝑘=0𝒴+10[(𝑥+3)/4]𝑛=0(9/10)𝑛+[][(𝑥+3)/2]𝑛=(𝑥+3)/4+1[(𝑥+32𝑛)/2]𝑘=0𝒴+10[𝑃<],(5.2) where 𝒴 denotes ((1)𝑘+1(1/10)𝑘𝑛!/(𝑛𝑘)!𝑘!) and 𝑃<=[abs(𝐺<)], and 𝐺<=(Ψ<)4(Ψ<)2(1/10)(Ψ<)6. In Table 1, using the Special Trans Functions Theory (5.2) we obtained the following numerical results.

Remark 5.1. From (5.1), by simple quadratic equation approach we have solution of the form Ψ<=5151.06161040584226712581595349425165683(5.3) if we restrict on 36 accurate digits.

Let us note that the result in the last row in Table 1 (for [𝑥]=37), obtained by STFT, is impressive, and, consequently, it proves that STFT numerically works!

5.1.2. Example 2 for Ψ<

Equation (1.1) for a following values: 𝛼=12, 𝛽=9, and 𝜆=(𝛼𝛽)𝜗=1/16, takes the formΨ<12Ψ<9=1Ψ16<21.(5.4)

According to (3.4), for above example, we obtain the solutionΨ<[𝑃<]=[(𝑥+9)/12]𝑛=0(15/16)𝑛+[][(𝑥+9)/3]𝑛=(𝑥+9)/12+1[(𝑥+93𝑛)/3]𝑘=0𝒲+16[(𝑥+10)/12]𝑛=0(15/16)𝑛+[][(𝑥+10)/3]𝑛=(𝑥+10)/12+1[(𝑥+103𝑛)/3]𝑘=0𝒲+16[𝑃<],(5.5) where 𝒲 denotes ((1)𝑘+1(1/16)𝑘𝑛!/(𝑛𝑘)!𝑘!)and 𝑃<=[abs(𝐺<)], and 𝐺<=(Ψ<)12(Ψ<)9(1/16)(Ψ<)21. In Table 2, using the Special Trans Functions Theory (5.5) we obtain the following numerical results.

Remark 5.2. From (5.5), by classical approach, we have solution of the form Ψ<=233333+173333171=1.02831660970359090498285316522474301(5.6) if we restrict on 35 accurate digits.

Let us note that the result in the last row in Table 2 (for [𝑥]=125), obtained by STFT, is correct, and consequently, we have a confirmation that the STFT numerically works.

5.2. The Numerical Results for Some Examples of (1.1) for Ψ>

The subject of the numerical analysis presented here are some solutions of two simple examples of (1.1) for Ψ>(𝛼/𝛽)1/(𝛼𝛽).

5.2.1. Example 1 for Ψ>

Equation (1.1) for the following values: 𝛼=4, 𝛽=2, and 𝜆=(𝛼𝛽)𝜗=1/10, takes the formΨ>4Ψ>2=1Ψ10>6.(5.7)

According to (4.29), for above example, we obtain the solutionΨ>[𝑃>]=[𝑥][2𝑥]𝑛=+1[2𝑥𝑛]𝑘=0(1)𝑘𝑛!/(𝑛𝑘)!𝑘!(1/10)𝑛[][2(𝑥+1)]𝑛=𝑥+1+1[2𝑥+2𝑛]𝑘=0(1)𝑘𝑛!/(𝑛𝑘)!𝑘!(1/10)𝑛1/4[𝑃>],(5.8) where 𝑃>=[abs(𝐺>)], and 𝐺>=(Ψ>)4(Ψ>)2(1/10)(Ψ>)6. In Table 3, using the Special Trans Functions Theory (5.8) we have the following numerical results.

Remark 5.3. Deriving from (5.8), by conventional approach, we get the solution of the form Ψ<=5+152.97875533506990414004149468203763347(5.9) if we restrict on 35 accurate digits.

Let us note that the result in the last row of the Table 3 (for [𝑥]=20) is correct, and we must declare that STFT, for this case, also works!

5.2.2. Example 2 for Ψ>

The transcendental Lambert equation (1.1) for following values: 𝛼=12, 𝛽=9, and 𝜆=(𝛼𝛽)𝜗=1/16, takes the formΨ>12Ψ>9=1Ψ16>21.(5.10) According to (4.2), for above example, we have𝑍>+116=𝑍>3/4,(5.11) or, the form16𝑍>+1=16𝑍>3/4.(5.12) After simple modification, the above equation takes the following form𝑌+1=2𝑌3/4,(5.13) where𝑌=16𝑍>.(5.14) Thus, by application of the STFT we have that EQID for (5.13) that takes the form𝜑>(𝑥1)+𝜑>(𝑥)2𝜑>3𝑥4=𝜑0>.(5.15) Due to (2.15), the particular solution of (5.15) takes the form 𝜑>𝑃(𝑥)=𝑌𝑥+𝜎>.(5.16) From (5.13), (5.15) and (5.16) we have𝜎>=𝜑0>22.(5.17) Consequently, from (2.23), analogically, we have for 𝑌𝑌=lim𝑥𝜑>(𝑥1)𝜎>𝜑>(𝑥)𝜎>1(5.18) since 𝜎>.  From (5.14) we have 𝑍=1/16, and Ψ>=(𝑍)1/𝛼=32. From (5.13) we can observed that 𝑌=1 is its trivial solution (1+1=213/4).

Remark 5.4. An example when the number of nontrivial solutions is equal three (𝜃3𝜃=𝜆𝜃4), has been presented in some detail in [13].

6. Conclusions

From the theoretical point of the analysis presented in this paper, the Special Trans Functions Theory is a consistent general approach for solving a broad class of Lambert transcendental equations (1.1), exactly, analytically, independently of particular case. This means that in the some manner we can obtain the Special Trans Functions for variety of different Lambert transcendental equations.

Accordingly, in this paper the family of the transcendental Lambert equations (1.1), is solved analytically in the closed form, by the Special Trans Functions Theory, in the real domain.

Formulae (2.24) (or (3.4)) and (4.27) (or (4.29)) derived in this paper, by using the STFT, are valid in the mathematical sense, and, are correct in the numerical sense (see Tables 1, 2, and 3), as well. Namely, the obtained analytical solutions (2.24) and (4.27), apart from the theoretical values have practical applications (3.4) and (4.29).

Let us note that the Special Trans Functions Theory is recently developed theory, and any numerical confirmation of the STFT validity is very important, because it is relevant argument for our statement concerning the STFT application validity independent on the comprehensiveness of the author`s explanations of it.

Of course, the theoretical confirmations are comprised in: the initial idea (born in neutron slowing down analysis [13]), consistent theoretical structure of the Special Trans Functions genesis, correct mathematical derivations, analysis of convergence dynamics between analytical unique and asymptotic solutions of the EQID, and so forth.

Let us note that one of the advantages of the Special Trans Functions Theory, as a mathematical method, is its applicability to a broad class of transcendental Lambert functional forms.

The theoretical accuracy of the numerical structures in (3.4) and (4.29) is unlimited and extremely precise [1, 3].

The forms of the EQID are routinely applied, according to the intuitive assumptions.

Finally, by STFT application, we have the possibility to obtain different gradient parameters. It is not difficult to see that the latter statement implies the rigorous analytical analysis for any nonlinear problem solved by STFT. Thus, definition of some new analytically sensitive parameters directly follows in the forms𝜕Ψ<,𝜕𝛼𝜕Ψ<,𝜕𝛽𝜕Ψ<,𝜕𝜗𝜕Ψ<,𝜕𝜆𝜕Ψ>,𝜕𝛼𝜕Ψ>,𝜕𝛽𝜕Ψ>,𝜕𝜗𝜕Ψ>.𝜕𝜆(6.1)

Or, from (2.24) and (4.28), we have, respectively,𝜕tran𝐿<,𝜕𝛼𝜕tran𝐿<,𝜕𝛽𝜕tran𝐿<,𝜕𝜗𝜕tran𝐿<,𝜕𝜆𝜕tran𝐿>,𝜕𝛼𝜕tran𝐿>,𝜕𝛽𝜕tran𝐿>,𝜕𝜗𝜕tran𝐿>.𝜕𝜆(6.2)

For example, from (3.3), we have:𝜕Ψ<𝜕𝛼[𝑃<]=𝜕Ψ𝐿<Ψ(𝑥)/𝜕𝛼𝐿<(𝑥+1)𝜎<𝜕Ψ𝐿<Ψ(𝑥+1)/𝜕𝛼𝐿<(𝑥)𝜎<Ψ𝐿<(𝑥+1)𝜎<2[𝑃<],(6.3) or we have𝜕Ψ<𝜕𝛼[𝑃<]=𝜕Ψ𝐿<(𝑥)/𝜕𝛼𝜕Ψ𝐿<Ψ(𝑥+1)/𝜕𝛼<[𝐺]Ψ𝐿<(𝑥+1)𝜎<[𝑃<],(6.4) and analogically𝜕Ψ>𝜕𝛼[𝑃>]=𝜕Ψ𝐿>(𝑥)/𝜕𝛼𝜕Ψ𝐿>Ψ(𝑥+1)/𝜕𝛼>[𝐺]Ψ𝐿>(𝑥+1)𝜎>[𝑃>].(6.5) The latter expressions are a significant contribution of the Special Trans Function Theory in applied physics and engineering domain for analytical approach to theoretical processes analysis.

According to the authors' knowledge, this is the first direct application of the STFT to the mathematical genesis of the analytical closed form solution to the broad family of the transcendental Lambert equations (1.1), independently of any physical process or phenomena.

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