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On the structure of the solution set of a generalized Euler–Lambert equation
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On the structure of the solution set of a generalized Euler–Lambert equation István Mező 1 Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing, PR China
a r t i c l e
i n f o
Article history: Received 13 January 2017 Available online xxxx Submitted by B.C. Berndt Keywords: Lambert W function r-Lambert function Branch structure Complex functions Transcendental equations
a b s t r a c t The transcendental equation xex = z and its solutions, described by the Lambert W function, often occur in physics and mathematics. In the last decades it turned out that the study of similar but more general equations is necessary in molecular physics, in the theory of general relativity and also in the description of Bose– Fermi mixtures as well as in some combinatorial problems. In this paper we offer a full description for the solution set of a one parameter generalization of the above mentioned equation. © 2017 Elsevier Inc. All rights reserved.
1. Introduction The equation xex = z
(1)
has been studied by a large number of authors since centuries, including Euler, Lambert, Wright, and more recently by R.M. Corless, D.E. Knuth, and many others. The solutions of this transcendental equation are given by the Lambert W function W (z), that is, by definition, W (z)eW (z) = z. If z = 0 there are infinitely many solutions so the Lambert function has infinitely many branches, indexed by the integers: Wk (z) (k ∈ Z). These branches and a number of properties of W were described by R.M. Corless et al. [15]. In [15] the reader can find many applications as well as in [28,44]. For additional applications E-mail address: [email protected]. The research of István Mező was supported by the Startup Foundation for Introducing Talent of NUIST (Project no.: S8113062001), and the National Natural Science Foundation of China (Grant no. 11501299). 1
http://dx.doi.org/10.1016/j.jmaa.2017.05.061 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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of W in the study of projectile motion see [2,3,7], for material science one can read [4–6,22], for molecular physics [25], electromagnetics [29,32], quantum statistics [44,43], plasma physics [19,23,24,26], solar physics [18], differential equations [1,10,12,20,21], classical mechanics [9], signal processing [11]. For an occurrence in the theory of zeta functions, see [45] and an appearance in biology (vision) see [33]. Mathematical properties of W can be found, beside the already mentioned [15], in [16,17,30,31], inequalities with respect to W were studied by Hoorfar [27] and Stewart [42]. In a number of recent studies it turned out that the equation (1) sometimes is insufficient in applications and therefore generalizations are necessary [34,37–41]. T.C. Scott et al. defined the generalized W function as the solution(s) of the equation ex
(x − t1 )(x − t2 ) · · · (x − tn ) = z. (x − s1 )(x − s2 ) · · · (x − sm )
(2)
In [36] we introduced the notation W
t1 t2 . . . tn ;z s1 s2 . . . sm
for the solution function of (2). In particular, W
1 ; z = −W − ; z = log(z), W ; z = W (z), W 0 z s e t ;z = s − W − W . ; z = t + W (ze−t ), W s z
0
Apart from the above literature we mention [35] for simple applications of the generalized Lambert function in the study of electromagnetic materials and water wave physics. In [35] it was also pointed out that the one upper and one lower parameter generalized Lambert function W st ; z is actually equivalent to the solution function of the equation xex + rx = z,
(3)
which was extensively studied in [36]: Taylor-expansion, asymptotics, differential and integral formulas were deduced. This equation also appears in combinatorial studies, see [13,14]. The solution function of (3) is denoted by Wr (z) and is called as the r-Lambert function in [36]. The equivalence with W st ; z is established by the relation [35] W
t ;a s
= t + W−ae−t ae−t (t − s) .
As (3) is analytically easier to handle we will consider the r-Lambert function in this article in place of its equivalent W st ; a . Our main goal is to determine the structure of the solution set of (3) when z is an arbitrary complex number. This program is equivalent to describing the branch structure of the r-Lambert function as a complex multi-valued function. 2. The branch structure of the r-Lambert function – general considerations Letting z be an arbitrary complex number, equation (3) has infinitely many solutions (even when z = 0 if r = 0, in contrary with the classical Lambert function). This is equivalent to saying that the w = Wr (z) r-Lambert function that satisfies the transcendental equation
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Wr (z)eWr (z) + rWr (z) = z
(4)
has infinitely many branches partitioning the w-plane. To determine the boundary of these regions first we set z = x + iy in the domain of Wr , and fix w = ξ + iη, so, by (4), x = rξ + eξ (ξ cos η − η sin η),
(5)
y = rη + eξ (η cos η + ξ sin η).
(6)
We want to partition the w-plane such that each component corresponds bijectively to the z-plane in the domain of Wr (z). In order to reach this aim we determine the critical points of the mapping (5)–(6) through the Jacobian of the function fr (ξ, η) = (rξ + eξ (ξ cos η − η sin η), rη + eξ (η cos η + ξ sin η)) =: (fr1 (ξ, η), fr2 (ξ, η)). Then ∂ 1 ∂ξ fr (ξ, η) J (fr (ξ, η)) = ∂ f 2 (ξ, η) ∂ξ r
As
∂ 1 ∂ξ fr (ξ, η)
=
∂ 2 ∂η fr (ξ, η)
and
∂ 1 ∂η fr (ξ, η)
=
∂ 2 ∂η fr (ξ, η)
∂ 1 ∂η fr (ξ, η)
∂ 1 ∂ ∂ 1 ∂ fr (ξ, η) fr2 (ξ, η) − fr (ξ, η) fr2 (ξ, η). ∂ξ ∂η ∂η ∂ξ
∂ 2 = − ∂ξ fr (ξ, η), we have
J (fr (ξ, η)) =
2
∂ 2 f (ξ, η) ∂η r
+
2
∂ 2 f (ξ, η) ∂ξ r
.
The critical points (ξ ∗ , η ∗ ) are those for which this Jacobian is zero. A short algebra yields that (ξ ∗ , η ∗ ) must satisfy the equations ξ ∗ = −η cot η − 1,
(7)
η ∗ = re sin η ∗ exp(η ∗ cot η ∗ ).
(8)
On the other hand, the derivative of the r-Lambert function is [36] Wr (x) =
1 , eWr (x) (1 + Wr (x)) + r
and this is singular when Wr (x) = W (−re) − 1.
(9)
To see this, considering the denominator in the derivative, one needs to solve the equation et (1 + t) + r, from where it comes that the solution is (9), indeed. Note that (9) encodes infinitely many different numbers, one solution for each branch of the Lambert function. Thus, denoting the individual solutions of (7)–(8) by (ξk∗ , ηk∗ ) where k ∈ Z, we see that ξk∗ = (Wk (−re) − 1), ηk∗ = (Wk (−re) − 1).
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The r-dependence will not be written out in order to keep the formulas simpler. We remark now that the branch indices that we will use later will somewhat differ from this indexing of ξk∗ and ηk∗ when r > 1/e2 . That there must be infinitely many solutions can easily be seen also from the oscillatory term (8). Note also that ∗ η−k = −ηk∗ ,
∗ and ξ−k = ξk∗
for all non-negative integer k. This is very helpful in order to make the r-Lambert function near conjugate symmetric: Wr,k (z) = Wr,−k (z) where Wr,k denotes the kth branch. This will be discussed later on. We will also study the asymptotics of ξk∗ and ηk∗ , see Section 6. The points wk∗ = ξk∗ + iηk∗ in the range of Wr (z) are the branch points belong to more than one branch. The points in the domain of Wr that maps to these wk∗ ’s will be denoted by zk∗ = ak + bk . Hence ak = [(Wk (−re) − 1)eWk (−re)−1 + r(Wk (−re) − 1)],
(10)
bk = [(Wk (−re) − 1)eWk (−re)−1 + r(Wk (−re) − 1)].
(11)
Note that if r = 0 then there is only one pair (ξ ∗ , η ∗ ) = (−1, 0) belonging to the unique branch point of the classical Lambert function. For this point z ∗ = −1/e, and W−1 (−1/e) = W0 (−1/e) = −1.
(12)
That here are infinitely many wk∗ when r = 0 shows the complexity of the branch structure of Wr comparing to the one of W . These numbers are also useful to determine the branch cut curves in the w-plane as for all branches there is a given wk∗ . The branch cut curves in the domain (the z-plane) of Wr will be a part of z = {x + ibk | x ∈ R, k ∈ Z\0 } with the above defined bk . (That why only a part of it will be discussed in the subsequent sections.) These correspond to the implicitly defined curves rη + eξ (η cos η + ξ sin η) = b∗k
(13)
in the range (the w-plane or (ξ, η)-plane). The discussion of the details are divided into three sections as the structure of the branches is radically different when r > 1/e2 , 0 < r ≤ 1/e2 and r < 0. The case r = 0 can be found in the fundamental paper [15]. 3. The branch structure of the r-Lambert function when r > 1/e2 3.1. The partition of the w-plane If r > 1/e2 the curves in (13) determine one branch that extends to minus infinity. This branch will be denoted by Wr,0 and we will call it the principal branch.2 Moreover, (13) determines infinitely many branches bounded from the left. These will be denoted by Wr,k where k ∈ Z\0 := Z \ {0}. 2 We try to keep as many analogies as possible with the classical Lambert function. However, here the principal branch has no analogue in the r = 0 case as we will see. When r tends to zero this branch will disappear and another branch emerges (already
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Here we chose the indices of ξk∗ and ηk∗ such that for k > 0 ∗ wk+1 = Wk (−re) − 1,
and for k < 0 wk∗ = Wk+1 (−re) − 1. We keep w0∗ undefined until we will need it, namely, when r < 0. The general situation is depicted above where all the curves come from (13). The solid lines constitute the branch border lines. The dashed line segments on the left of the points wk∗ (when ξ < ξk∗ ) belong to the image set of Wr,0 (z) where z = {x + ibk | x ∈ R, x < ak }, while the dashed line segments on the right of wk∗ (for ξ > ξk∗ ) belong to Wr,k (z). The small circles symbolize the points wk∗ as it is written on the figure. To make the formulas shorter we apply and rewrite (10)–(11) as ∗
ak = rξk∗ + eξk (ξk∗ cos ηk∗ − ηk∗ sin ηk∗ ), ∗ ξk
bk = rηk∗ + e (ηk∗ cos ηk∗ + ξk∗ sin ηk∗ ),
(14) (15)
for k ∈ Z\0 keeping in mind that these constants depend on r and also that ξk∗ and ηk∗ are simple expressions of the Lambert W function. Now, when z = x + iy such that x = ak ,
y = bk
for some non-zero k, then the point z belongs to two branches, to Wr,0 and to Wr,k . Such z’s are the pre-images of the wk∗ points. As we mentioned earlier this happens in only one point with respect to the Lambert function, namely, when z = −1/e (with η ∗ = 0 and ξ ∗ = −1), see (12). The principal branch Wr,0 . This branch of the r-Lambert function contains the whole real line in its image (the real line is mapped onto itself by this branch), and this branch is unbounded from the left and from the right on the complex plane. Horizontal lines x + iy (x ∈ R, y fixed) maps by Wr,0 to curves from minus infinity to plus infinity such that if 0 < y < b1 , then the curve runs above the ξ axis but below the first dashed line in the upper half plane (see on Fig. 2). As x grows the curve {Wr,0 (x + iy) | x ∈ R} runs in the gap between Wr,−1 and Wr,1 . Based on Fig. 1 one might think that this gap is finite but, in fact, Wr,−1 and Wr,1 never meet in the finite plane.3 The curves separating them are very close to each other indeed, but they meet only when ξ → ∞. The same is true for all the other branches, so Wr,0 is actually not bounded from the right. when r = 1/e2 ) that we will call principal branch in that case. It is still natural to call Wr,0 principal because it contains the real line in its image, and also, its structure and shape is unique among all the branches. 3 The numerical difference between the boundary points of Wr,k and Wr,k+1 is so small that at the applied level of magnification they seem to be overlapping.
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Fig. 1. The branch structure of the r-Lambert function when r = 2 in the (ξ, η)-plane.
If, in turn, y is negative such that 0 > y > b−1 , then {Wr,0 (x + iy) | x ∈ R} runs below the ξ-axis, and it penetrates into the area between the ξ-axis and the branch Wr,−1 . Moreover, if y is a fixed real number in the interval bk < y < bk+1 , with a given k > 0, then the curve {Wr,0 (x + iy) | x ∈ R} runs between two consecutive dashed lines in the upper half plane. As x grows this curve penetrates into the gap between Wr,k and Wr,k+1 . If for some k < 0 bk > y > bk−1 ,
(16)
then the image of x + iy by the principal branch runs between two consecutive dashed lines in the lower half plane such that it runs towards the area between Wr,−k and Wr,−k−1 . A typical situation for y > 0 is shown in Fig. 2, where we have chosen y = y± to be b1 ± 1 when r = 2. The lower dotted curve in the image of the principal branch corresponds to y− while the higher dotted curve in Wr,0 belongs to y+ . The another two dotted curves are the images of the same set x + iy by Wr,1 . If, finally, y equals to some bk then the curve {Wr,0 (x + iy) | x ∈ R} runs on the corresponding dashed line until reaching the point wk∗ . That from here it runs downwards or upwards depends on the branch closure we choose. By reasons made clear below we choose {Wr,0 (x + iy) | x ∈ R} to run downwards for every k regardless the sign. The non-principal branches Wr,k when k ∈ Z\0 . These branches are bounded from the left; each branch has its leftmost point at (ξk∗ , ηk∗ ). They extend to the right unbounded and their boundaries are asymptotic to the horizontal multiples of π. (This fact comes from (13): η is bounded so as ξ grows the factor η cos η+ξ sin η must tend to zero. This is possible only if η → 0 + kπ for some k ∈ Z.)
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Fig. 2. Images of horizontal lines. The meaning of the dotted lines is described in the text after (16).
These branches are divided into two parts by the dashed lines in Fig. 1 we were discussing in Section 3.1. Regardless4 the sign of k, the upper half of the branch Wr,k (z) contains the image of those points z = x + iy for which y < bk , and the lower half contains the image of z = x + iy for which y is greater than this quantity (and x spans R). The horizontal line z = x + ibk partially maps onto the dashed line in the range of Wr,k till the branch point wk∗ . Here Wr,k shares its value with the principal branch, then the curve turns upwards forming the upper limiting curve of the set of the range of Wr,k . This will further be discussed in the following subsection. 3.2. The branch closures After having described the boundary curves of the images of the different branches we also need to clarify where the points on the boundaries (solid lines on Fig. 2) belong to. The closure will be chosen such that in the limit r → 0 we get the closure structure of the branches of the classical Lambert function. This will be more apparent in the following section when we study the case 0 < r < 1/e2 . Thus we fix the following boundaries. For k ∈ Z\0 the boundary curve above of the point wk∗ and wk∗ itself belongs to the image set of Wr,k , while the curve below wk∗ belongs to Wr,0 . See Fig. 3 for a visual guide. For example, a part of the curve Wr,0− (x + ib2 ) (x ∈ R) is on the dashed line on the left of w2∗ . It runs into w2∗ from the left and here turns downwards on the dotted line of Fig. 3 and goes forward on this line to the right between Wr,2 and Wr,1 . This part of the curve belongs to Wr,0− hence Wr,2 is open here. In turn, the same horizontal line {x + ib2 } maps by Wr,2 onto the dashed line in the image set of Wr,2 , it reaches w2∗ then it turns upwards and closes the branch by going on the solid line above w2∗ . 4 Let y = bk − ε for some positive small ε such that y − ε does not go below bk−1 . If now z = x + iy then z = −bk + ε = b−k + ε > b−k and this point, as we wrote, maps to the lower half of the branch of Wr,−k .
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Fig. 3. Branch closures in the case r > 1/e2 .
According to the previous description, in Fig. 3 the dotted lines belong to the principal branch, while the solid lines belong to the branches below them. Note that the dotted curves go to the right to plus infinity but on the figure they seem to be merged to the solid lines. This is only virtual as the numerical difference between these points is tiny. Thus, for example, the real line maps onto the real line by Wr,0 (not presented in Fig. 3) and this set in the range of the principal branch lies between Wr,1 and Wr,−1 so these two branches do not meet in the finite plane. 4. The branch structure of the r-Lambert function when 0 < r ≤ 1/e2 4.1. The partition of the w-plane when r = 1/e2 When r → 1/e2 + an interesting phenomenon happens: (ξ1∗ , η1∗ ) → (−2, 0), ∗ ∗ (ξ−1 , η−1 ) → (−2, 0), ∗ so w1∗ = w−1 = −2. Geometrically, the branch curves that previously formed the boundary of Wr,1 and Wr,−1 now share the point −2 and the area that was lying between these two sets and formerly belonged to the principal branch disappears. The point −2 in the range of the 1/e2 -Lambert function is the image of z = −4/e2 . The special role of this point can be explained by (9). In the particular case when r = 1/e2 we have that
W1/e2 (x) = W (−1/e) − 1 = −2, i.e., by (4), −2e−2 + e−2 (−2) = x, so x = −4/e2 . In this point the real W1/e2 function is not differentiable. As we will see, in this point three branches meet.
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Fig. 4. Branch structure at r = 1/e2 without closure information.
The lower half part of Wr,1 and the upper half part of Wr,−1 form a new branch containing the set of real numbers > −2. This branch will be denoted by Wr,0+ while the old principal branch will be renamed to Wr,0− . As Wr,0+ will become the principal W0 branch of the Lambert function when r → 0+ this naming convention seems to be appropriate. The branch formed by the upper part of Wr,1 and by the lower part of Wr,−1 will be denoted by Wr,±1 . See Fig. 4. When r = 1/e2 , by the definitions (14)–(15), we have, as we have said already, a1 = a−1 = −4/e2 ,
b1 = b−1 = 0,
hence there will be three branches containing real numbers: W1/e2 ,0− contains the set ]−∞, −2], the branch W1/e2 ,±1 contains the single real point −2, while W1/e2 ,0+ contains the real interval [−2, +∞[ in its range. In Fig. 4 the dashed lines are the images of x + iy where y = bk , as before. The dotted lines depict the set
W1/e2 ,0− (x + iy), W1/e2 ,±1 (x + iy), W1/e2 ,0+ (x + iy) | x ∈ R, y = 1/2 ,
from where it seems that W1/e2 ,0− maps the upper half plane to the upper half plane, W1/e2 ,0+ maps this plane to its upper half part, and W1/e2 ,±1 maps it into the lower half of its image set. The dash–dotted lines belong to the set
W1/e2 ,0− (x + iy), W1/e2 ,±1 (x + iy), W1/e2 ,0+ (x + iy) | x ∈ R, y = −1/2 .
For |k| ≥ 2 the upper half plane {x + iy | y > bk } maps into the lower half of W1/e2 ,k , while the lower half plane {x + iy | y < bk } is mapped into the upper half of these branches. ∗ = The small circle in Fig. 4 in the intersection of W1/e2 ,0− , W1/e2 ,0+ , and W1/e2 ,±1 is the point w1∗ = w−1 2 −2. As we have said, the branch point z = −4/e maps to the common point −2 by these three branches: W1/e2 ,0− (−4/e2 ) = W1/e2 ,0+ (−4/e2 ) = W1/e2 ,±1 (−4/e2 ) = −2.
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Fig. 5. Branch closure at r = 1/e2 .
4.2. The branch closures when r = 1/e2 The branch closure is taken as follows: we keep the closure for the branches W1/e2,k when |k| ≥ 2 as they were for r > 1/e2 . The upper part of W1/e2 ,±1 till the point −2 belongs to W1/e2 ,±1 , while the part below −2 limited by the dotted curve in Fig. 5 belongs to W1/e2 ,0− . Moreover, the upper limiting curve of W1/e2 ,0+ belongs to W1/e2 ,0+ while the lower limiting curve belongs to W1/e2 ,±1 instead. All of these are necessary if we would like to enforce agreement with the branch closures of the Lambert function when r → 0+. The horizontal lines {x + iy | y = bk } are mapped by the W1/e2 ,0− branch such that their images are curves from minus infinity till wk∗ , including this point, then they turn downwards and go to plus infinity on the corresponding dotted line. In particular, {W1/e2 ,0− (x) | x ∈ R} = ]−∞, −2] ∪ {the limiting dotted curve of the lower part of W1/e2 ,±1 }. Moreover, {W1/e2 ,0+ (x) | x ∈ R} = [−2, +∞[ ∪ {the solid line running upwards from −2 in W1/e2 ,0+ }. 4.3. The partition of the w-plane when 0 < r < 1/e2 and the closures ∗ When r gets smaller than 1/e2 but still positive, the point w1∗ and w−1 will separate again but both stay real, forming a real interval ∗ I = [w−1 , w1∗ ] = [W−1 (−re) − 1, W0 (−re) − 1].
(17)
This interval belongs to the image set of Wr,±1 such that the left endpoint is shared with Wr,0− while the right endpoint is shared with Wr,0+ , see Fig. 6. The branches are the same as for r = 1/e2 as well as the closures, the only difference is that the point −2 enlarges into an interval. See the below graph.
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Fig. 6. Branch structure and closure at r = 1/e4 .
4.4. The branch structure as r → 0+ When r → 0+ equation (3) turns to be the defining equation (1) of the Lambert function and the branch structure should be the same as the one of W . In this limit the real part ξk∗ of the branch points tends to minus infinity (see Theorem 1 below) and when |k| ≥ 2 the Wr,k branches will be stretched to minus infinity. These correspond to the Wk branches of the Lambert function described in [15]. The behavior of the interval I is as follows. The left endpoint of I, W−1 (−re) − 1, tends to −∞ and so the Wr,±1 branch also runs out to minus infinity. This together with the run out of the other branches Wr,k (|k| ≥ 2) result that the Wr,0− branch disappears. Moreover, the upper and lower parts of Wr,±1 separate such that these parts form, when r = 0, two separate branches that correspond to W1 and W−1 of the Lambert function. This is why we chose this notation for Wr,±1 before. These branches are separated by the half line ]−∞, −1] in the w-plane. The other endpoint of I, W0 (−re) − 1 turns to be W0 (0) − 1 = −1 so that it forms the only branch point of W . The Wr,0+ branch turns to be W0 when r = 0, becoming the principal branch of W . Now one can also see why we have chosen the branch closures as we did. The Wk branches in the limit r = 0 [15] are closed exactly as we see on Fig. 5 when we stretch Wr,k (|k| ≥ 2) to minus infinity: closed from above and open from below. The upper part of Wr,±1 that becomes the new W1 branch is open on the curve separating it from W0 . The ]−∞, −1] half line belongs to W−1 . The lower half of W0 , below the real line, is separated from W−1 such that the separating curve belongs to W−1 , while the curve separating W0 from W1 belongs to the principal branch W0 . 5. The branch structure of the r-Lambert function when r < 0 When r < 0 and k = 0 the Wr,k branches are bounded from the left again by the quantities wk∗ as they were for r > 1/e2 . Now we go back to the non-shifted indices again: wk∗ = ξk∗ + iηk∗ = Wk (−re) − 1. The importance of this will be clear soon.
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Fig. 7. Branch structure and closure when r < 0.
The branch structure thus formally looks like on Fig. 1. There are some important differences that we are going to describe now. If r < 0 the number η0∗ = (W0 (−re) − 1) is always 0 (W0 (z) is real for real non-negative z), so there will be a real branch point at w0∗ = ξ0∗ = W0 (−re) − 1. The image of the real line thus belongs to two branches that we denote by Wr,0− and Wr,0+ . A part of the real line maps to two half lines: the interval [fr (w0∗ ), +∞[ maps to [−∞, w0∗ ] by Wr,0− , while the same interval [fr (w0∗ ), +∞[ maps to [w0∗ , +∞] by Wr,0+ . Here fr (z) = zez + rz. The union of these two image sets forms the horizontal dashed line in the below figure. The image of the real line by Wr,0− , as we have just described, runs from −∞ till w0∗ . Here, to keep consistence with the branch closure of the other cases, we choose the continuation of this curve such that in the point w0∗ it turns downwards. Hence the boundary of Wr,0+ belongs to Wr,0− from the point w0∗ and below. The boundary of Wr,0+ from w0∗ and above belongs to Wr,0+ . This is also depicted on Fig. 6. We note that for r < 0 the sign of bk is negative when k > 0 (and b−k = −bk ). The Wr,0− branch maps the lower half plane into the upper half plane and vice versa. This is in accordance with the remarks on the sign of bk . Thus, for example, the set Wr,0− ({x + iy | x ∈ R}) will be a curve below the horizontal dashed line and it goes to plus infinity in the gap between Wr,0+ and Wr,1 for b1 < y < 0. If y = b1 the set Wr,0− ({x + iy | x ∈ R}) is the dashed line going through w1∗ and continuing on the dotted line, forming the lower boundary of Wr,1 . The other branches, including Wr,0+ , map the upper half plane into the upper half of their range while the lower half plane is mapped to their lower half. The two halves are separated by the corresponding dashed lines on Fig. 7. The horizontal lines {x + ibk | x ∈ R} are mapped partially onto the corresponding dashed lines in the branch Wr,k until wk∗ and at this point they turn upwards forming the closure of the given branch.
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6. More on the points wk∗ In this section we are going to examine the asymptotic behavior of the points wk∗ as k → ±∞. As these determine the extremal points of the branches they have special importance. Recall that (if we do not shift the indices) ξk∗ = (Wk (−re) − 1),
(18)
ηk∗ = (Wk (−re) − 1),
(19)
and ∗ η−k = −ηk∗ ,
∗ and ξ−k = ξk∗
The following statement is true. Theorem 1. Let r be a non-zero real number. Then, for large k π ξk∗ ∼ log |r| − log 2kπ + sgn(r) , 2 π ηk∗ ∼ 2kπ + sgn(r) . 2
(20) (21)
Here sgn is the signum function, and ak ∼ bk means that ak /bk → 1 as k → ∞. Proof. The asymptotic behavior of the Lambert function was discussed by de Bruijn [8], and was completed by Corless et al. [15]. For all the branches it holds true that Wk (z) = log z + 2kπi − log(log z + 2kπi) + O
log log z log z
,
when z → ∞ as well as for z → 0. In this expansion the principal branch of the logarithm is taken at each occurrence. We make use of this asymptotic expansion when the branch index k is big and z simply equals to −re. We then have Wk (−re) − 1 ≈ −1 + log |re| + (0, π)i + 2kπi − log(log |re| + (0, π)i + 2kπi). The big “O” term will be neglected, and the symbol (0, π) is defined to be 0 if −re > 0 (i.e., if r < 0) and it is defined to be π if −re < 0 (if r > 0). Thus ηk∗ = (Wk (−re) − 1) ≈ 2kπ + (0, π) − (log(log |re| + (0, π)i + 2kπi)). If k is big, log |re| and (0, π)i can be neglected in the outer logarithm’s argument, log |re| + (0, π)i + 2kπi) is a complex number with angle close to π/2. Hence (log(log |re| + (0, π)i + 2kπi)) ≈ (log(2kπi)) = From here we get that ηk∗ ≈ 2kπ + (0, π) −
π . 2
This, by the definition of the symbol (0, π), is equivalent to (21). The statement for ξk∗ can be treated similarly. 2
π . 2
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We know from the former sections that the Wr,k branches are bounded from the left if r = 0 and k = 0. This theorem says more: it states that the left extremal points of the individual branches (not being equal to Wr,0− , Wr,0+ , Wr,±1 ) go to left as k grows, and the ordinate logarithmically decreases to minus infinity. 6.1. Tables of wk∗ We compare the estimations of the above theorem to the exact expressions for wk∗ = ξk∗ + iηk∗ given by (18)–(19). First we choose r = 2. The exact value of ξ50 is ∗ ξ50 = −5.06178,
while (20) results in the estimation ∗ ξ50,estim = −5.06174. ∗ For η50 we have ∗ η50 = 315.717, ∗ η50,estim = 315.73.
For r = 1/e4 the corresponding values for k = 50 are ∗ ξ50 = −8.16961, ∗ ξ50,estim = −8.16515,
and ∗ η50 = 315.702, ∗ η50,estim = 315.73.
While, for example, taking r = −3 ∗ ξ50 = −4.64631, ∗ ξ50,estim = −4.64628,
and ∗ η50 = 312.577, ∗ η50,estim = 312.588.
For larger k the performance of these estimations is even better. Moreover, (20) and (21) are extremely useful when one wants to find a good starting point for numerical root finding methods to solve (3). This way one does not need to use the Lambert W function in order to determine wk∗ to have an initial guess where the solutions on the different branches could be. As W is not given in many software packages, this is of practical importance.
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7. Conclusion In this paper we have studied the structure of the solution set of the transcendental equation xex + rx = z where z is a fixed complex parameter and r is an arbitrary but fixed real number. The solutions are represented by the r-Lambert function. We pointed out that the solutions fall into infinitely many components of the complex plane and we determined the shape of these components for all possible values of r. A full description of the boundaries of these components is also given together with the consistent determination of the closure of these components. References [1] B.V. Alekseev, A.E. Dubinov, I.D. Dubinova, Analytical and numerical solutions of generalized dispersion equations for one-dimensional damped plasma oscillation, High Temp. 43 (4) (2005) 479–485, translated from Teplofiz. Vys. Temp. 43 (4) (2005) 485–491. [2] C.H. Belgacem, Range and flight time of quadratic resisted projectile motion using the Lambert W function, Eur. J. Phys. 35 (2014) 055025, http://dx.doi.org/10.1088/0143-0807/35/5/055025. [3] C.H. Belgacem, Analysis of projectile motion with quadratic air resistance from a nonzero height using the Lambert W function, J. Taibah Univ. Sci. 11 (2) (2017) 328–331, http://dx.doi.org/10.1016/j.jtusci.2016.02.009. [4] C.H. Belgacem, Explicit solution for critical thickness of semicircular misfit dislocation loops in strained semiconductors heterostructures, Silicon 8 (2016) 397–399. [5] C.H. Belgacem, M. Fnaiceh, Exact analytical solution for the critical layer thickness of a lattice-mismatched heteroepitaxial layer, J. Electron. Mater. 39 (10) (2010) 2248–2250. [6] C.H. Belgacem, M. Fnaiceh, Solution for the critical thickness models of dislocation generation in epitaxial thin films using the Lambert W function, J. Mater. Sci. 46 (2011) 1913–1915. [7] R.C. Bernardo, J.P. Esguerra, J.D. Vallejos, J.J. Canda, Wind-influenced projectile motion, Eur. J. Phys. 36 (2015) 025016, http://dx.doi.org/10.1088/0143-0807/36/2/025016. [8] N.G. de Bruijn, Asymptotic Methods in Analysis, North-Holland, 1961. [9] S.A. Campbell, Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops, Dyn. Contin. Discrete Impuls. Syst. 5 (1999) 225–235. [10] Ll.G. Chambers, Solutions of the neutral differential-difference equation αx (t) + βx (t − r) + γx(t) + δx(t − r) = f (t), Int. J. Math. Math. Sci. 15 (4) (1989) 773–780. [11] François Chapeau-Blondeau, Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2, IEEE Trans. Signal Process. 50 (9) (2002) 2160–2165. [12] B. Cogan, A.M. de Paor, Analytic root locus and Lambert W function in control of a process with time delay, J. Electr. Eng. 62 (6) (2011) 327–334. [13] C.B. Corcino, An asymptotic formula for the r-Bell numbers, Matimyás Mat. 24 (2001) 9–18. [14] C.B. Corcino, R.B. Corcino, An asymptotic formula for r-Bell numbers with real arguments, ISRN Discrete Math. 2013 (2013) 274697. [15] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) 329–359. [16] R.M. Corless, D.J. Jeffrey, The unwinding number, SIGSAM Bull. 30 (2) (1996) 28–35. [17] R.M. Corless, D.J. Jeffrey, D.E. Knuth, A sequence of series for the Lambert W function, in: ISSAC ’97 Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, 1997, pp. 197–204. [18] S.R. Cranmer, New views of the solar wind with the Lambert W function, Am. J. Phys. 72 (11) (2004) 1397–1403. [19] I.D. Dubinova, Application of the Lambert W function in mathematical problems of plasma physics, Plasma Phys. Rep. 30 (10) (2004) 872–877, translated from Fiz. Plazmy 30 (10) (2004) 937–943. [20] I.D. Dubinova, Exact closed-form solutions of some nonlinear differential equations, Differ. Equ. 40 (8) (2004) 1195–1196, translated from Differ. Uravn. 40 (8) (2004) 1129–1130. [21] A.E. Dubinov, I.D. Dubinova, S.K. Saikov, Characteristic roots and stability domains of one dynamic delay system, Autom. Remote Control 66 (8) (2005) 1212–1213, translated from Avtomat. i Telemekh. 8 (2005) 22–23. [22] A.E. Dubinov, Dynamics of virtual-cathode formation in a viscous-friction medium, Dokl. Phys. 49 (12) (2004) 697–700, translated from Dokl. Akad. Nauk 399 (4) (2004) 468–471. [23] A.E. Dubinov, I.D. Dubinova, How can one solve exactly some problems in plasma theory, J. Plasma Phys. 71 (5) (2005) 715–728. [24] A.E. Dubinov, I.D. Dubinova, Exact solution of the Landau dispersion equation for electron plasma oscillations, Tech. Phys. Lett. 32 (1) (2006) 36–37. [25] A.A. Frost, Delta-function model. I. Electronic energies of hydrogen-like atoms and diatomic molecules, J. Chem. Phys. 25 (1956) 1150. [26] V.A. Gordienko, I.D. Dubinova, A.E. Dubinov, Nonlinear theory of large-amplitude stationary solitary waves in symmetric − + unmagnetized e− e+ and C60 C60 plasmas, Plasma Phys. Rep. 32 (11) (2006) 910–915. [27] A. Hoorfar, M. Hassani, Inequalities on the Lambert W function and hyperpower function, J. Inequal. Pure Appl. Math. 9 (2) (2008) 51, available at http://www.emis.de/journals/JIPAM/article983.html. [28] A. Houari, Additional applications of the Lambert W function in physics, Eur. J. Phys. 34 (3) (2013) 695–702.
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[29] D. Jaisson, Simple formula for the wave number of the Goubau line, Electromagnetics 34 (2014) 85–91. [30] D.J. Jeffrey, D.E.G. Hare, R.M. Corless, Unwinding the branches of the Lambert W function, Math. Sci. 21 (1996) 1–7. [31] D.J. Jeffrey, J.E. Jankowski, Branch differences and Lambert W , in: Proceedings of the 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2014, available online at http://www.apmaths. uwo.ca/~djeffrey/Offprints/JeffreyJankowskiSYNASC2014.pdf. [32] D.C. Jenn, Applications of the Lambert W function in electromagnetics, IEEE Antennas Propag. Mag. 44 (3) (2002) 139–142. [33] T. Lamb, E. Pugh, Dark adaptation and the retinoid cycle of vision, Prog. Retin. Eye Res. 23 (3) (2004) 307–380. [34] R.B. Mann, T. Ohta, Exact solution for the metric and the motion of two bodies in (1 + 1)-dimensional gravity, Phys. Rev. D 55 (1997) 4723–4747. [35] I. Mező, G. Keady, Some physical applications of generalized Lambert functions, Eur. J. Phys. 37 (2016) 065802, http:// dx.doi.org/10.1088/0143-0807/37/6/065802. [36] I. Mező, Á. Baricz, On the generalization of the Lambert W function, Trans. Amer. Math. Soc. (2017), in press, available online (under a different title) at http://arxiv.org/abs/1408.3999. [37] T.C. Scott, M. Aubert-Frécon, J. Grotendorst, New approach for the electronic energies of the hydrogen molecular ion, Chem. Phys. 324 (2–3) (2006) 323–338. [38] T.C. Scott, J.F. Babb, A. Dalgarno, J.D. Morgan III, J. Chem. Phys. 99 (1993) 2841–2854. [39] T.C. Scott, G. Fee, J. Grotendorst, Asymptotic series of generalized Lambert W function, SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation) 47 (185) (2013) 75–83. [40] T.C. Scott, G. Fee, J. Grotendorst, W.Z. Zhang, Numerics of the generalized Lambert W function, SIGSAM 48 (188) (2014) 42–56. [41] T.C. Scott, R. Mann, R.E. Martinez, General relativity and quantum mechanics: towards a generalization of the Lambert W function, Appl. Algebra Engrg. Comm. Comput. 17 (1) (2006) 41–47. [42] S.M. Stewart, On certain inequalities involving the Lambert W function, J. Inequal. Pure Appl. Math. 10 (4) (2009) 96, available at http://www.emis.de/journals/JIPAM/article1152.html. [43] S.R. Valluri, M. Gil, D.J. Jeffrey, S. Basu, The Lambert W function and quantum statistics, J. Math. Phys. 50 (2009), 11 pages. [44] S.R. Valluri, D.J. Jeffrey, R.M. Corless, Some applications of the Lambert W function to physics, Canad. J. Phys. 78 (9) (2000) 823–831. [45] M. Visser, Primes and the Lambert W function, arXiv preprint, available online at http://arxiv.org/pdf/1311.2324v1.pdf.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On the structure of the solution set of a generalized Euler–Lambert equation István Mező 1 Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing, PR China
a r t i c l e
i n f o
Article history: Received 13 January 2017 Available online xxxx Submitted by B.C. Berndt Keywords: Lambert W function r-Lambert function Branch structure Complex functions Transcendental equations
a b s t r a c t The transcendental equation xex = z and its solutions, described by the Lambert W function, often occur in physics and mathematics. In the last decades it turned out that the study of similar but more general equations is necessary in molecular physics, in the theory of general relativity and also in the description of Bose– Fermi mixtures as well as in some combinatorial problems. In this paper we offer a full description for the solution set of a one parameter generalization of the above mentioned equation. © 2017 Elsevier Inc. All rights reserved.
1. Introduction The equation xex = z
(1)
has been studied by a large number of authors since centuries, including Euler, Lambert, Wright, and more recently by R.M. Corless, D.E. Knuth, and many others. The solutions of this transcendental equation are given by the Lambert W function W (z), that is, by definition, W (z)eW (z) = z. If z = 0 there are infinitely many solutions so the Lambert function has infinitely many branches, indexed by the integers: Wk (z) (k ∈ Z). These branches and a number of properties of W were described by R.M. Corless et al. [15]. In [15] the reader can find many applications as well as in [28,44]. For additional applications E-mail address: [email protected]. The research of István Mező was supported by the Startup Foundation for Introducing Talent of NUIST (Project no.: S8113062001), and the National Natural Science Foundation of China (Grant no. 11501299). 1
http://dx.doi.org/10.1016/j.jmaa.2017.05.061 0022-247X/© 2017 Elsevier Inc. All rights reserved.
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of W in the study of projectile motion see [2,3,7], for material science one can read [4–6,22], for molecular physics [25], electromagnetics [29,32], quantum statistics [44,43], plasma physics [19,23,24,26], solar physics [18], differential equations [1,10,12,20,21], classical mechanics [9], signal processing [11]. For an occurrence in the theory of zeta functions, see [45] and an appearance in biology (vision) see [33]. Mathematical properties of W can be found, beside the already mentioned [15], in [16,17,30,31], inequalities with respect to W were studied by Hoorfar [27] and Stewart [42]. In a number of recent studies it turned out that the equation (1) sometimes is insufficient in applications and therefore generalizations are necessary [34,37–41]. T.C. Scott et al. defined the generalized W function as the solution(s) of the equation ex
(x − t1 )(x − t2 ) · · · (x − tn ) = z. (x − s1 )(x − s2 ) · · · (x − sm )
(2)
In [36] we introduced the notation W
t1 t2 . . . tn ;z s1 s2 . . . sm
for the solution function of (2). In particular, W
1 ; z = −W − ; z = log(z), W ; z = W (z), W 0 z s e t ;z = s − W − W . ; z = t + W (ze−t ), W s z
0
Apart from the above literature we mention [35] for simple applications of the generalized Lambert function in the study of electromagnetic materials and water wave physics. In [35] it was also pointed out that the one upper and one lower parameter generalized Lambert function W st ; z is actually equivalent to the solution function of the equation xex + rx = z,
(3)
which was extensively studied in [36]: Taylor-expansion, asymptotics, differential and integral formulas were deduced. This equation also appears in combinatorial studies, see [13,14]. The solution function of (3) is denoted by Wr (z) and is called as the r-Lambert function in [36]. The equivalence with W st ; z is established by the relation [35] W
t ;a s
= t + W−ae−t ae−t (t − s) .
As (3) is analytically easier to handle we will consider the r-Lambert function in this article in place of its equivalent W st ; a . Our main goal is to determine the structure of the solution set of (3) when z is an arbitrary complex number. This program is equivalent to describing the branch structure of the r-Lambert function as a complex multi-valued function. 2. The branch structure of the r-Lambert function – general considerations Letting z be an arbitrary complex number, equation (3) has infinitely many solutions (even when z = 0 if r = 0, in contrary with the classical Lambert function). This is equivalent to saying that the w = Wr (z) r-Lambert function that satisfies the transcendental equation
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Wr (z)eWr (z) + rWr (z) = z
(4)
has infinitely many branches partitioning the w-plane. To determine the boundary of these regions first we set z = x + iy in the domain of Wr , and fix w = ξ + iη, so, by (4), x = rξ + eξ (ξ cos η − η sin η),
(5)
y = rη + eξ (η cos η + ξ sin η).
(6)
We want to partition the w-plane such that each component corresponds bijectively to the z-plane in the domain of Wr (z). In order to reach this aim we determine the critical points of the mapping (5)–(6) through the Jacobian of the function fr (ξ, η) = (rξ + eξ (ξ cos η − η sin η), rη + eξ (η cos η + ξ sin η)) =: (fr1 (ξ, η), fr2 (ξ, η)). Then ∂ 1 ∂ξ fr (ξ, η) J (fr (ξ, η)) = ∂ f 2 (ξ, η) ∂ξ r
As
∂ 1 ∂ξ fr (ξ, η)
=
∂ 2 ∂η fr (ξ, η)
and
∂ 1 ∂η fr (ξ, η)
=
∂ 2 ∂η fr (ξ, η)
∂ 1 ∂η fr (ξ, η)
∂ 1 ∂ ∂ 1 ∂ fr (ξ, η) fr2 (ξ, η) − fr (ξ, η) fr2 (ξ, η). ∂ξ ∂η ∂η ∂ξ
∂ 2 = − ∂ξ fr (ξ, η), we have
J (fr (ξ, η)) =
2
∂ 2 f (ξ, η) ∂η r
+
2
∂ 2 f (ξ, η) ∂ξ r
.
The critical points (ξ ∗ , η ∗ ) are those for which this Jacobian is zero. A short algebra yields that (ξ ∗ , η ∗ ) must satisfy the equations ξ ∗ = −η cot η − 1,
(7)
η ∗ = re sin η ∗ exp(η ∗ cot η ∗ ).
(8)
On the other hand, the derivative of the r-Lambert function is [36] Wr (x) =
1 , eWr (x) (1 + Wr (x)) + r
and this is singular when Wr (x) = W (−re) − 1.
(9)
To see this, considering the denominator in the derivative, one needs to solve the equation et (1 + t) + r, from where it comes that the solution is (9), indeed. Note that (9) encodes infinitely many different numbers, one solution for each branch of the Lambert function. Thus, denoting the individual solutions of (7)–(8) by (ξk∗ , ηk∗ ) where k ∈ Z, we see that ξk∗ = (Wk (−re) − 1), ηk∗ = (Wk (−re) − 1).
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The r-dependence will not be written out in order to keep the formulas simpler. We remark now that the branch indices that we will use later will somewhat differ from this indexing of ξk∗ and ηk∗ when r > 1/e2 . That there must be infinitely many solutions can easily be seen also from the oscillatory term (8). Note also that ∗ η−k = −ηk∗ ,
∗ and ξ−k = ξk∗
for all non-negative integer k. This is very helpful in order to make the r-Lambert function near conjugate symmetric: Wr,k (z) = Wr,−k (z) where Wr,k denotes the kth branch. This will be discussed later on. We will also study the asymptotics of ξk∗ and ηk∗ , see Section 6. The points wk∗ = ξk∗ + iηk∗ in the range of Wr (z) are the branch points belong to more than one branch. The points in the domain of Wr that maps to these wk∗ ’s will be denoted by zk∗ = ak + bk . Hence ak = [(Wk (−re) − 1)eWk (−re)−1 + r(Wk (−re) − 1)],
(10)
bk = [(Wk (−re) − 1)eWk (−re)−1 + r(Wk (−re) − 1)].
(11)
Note that if r = 0 then there is only one pair (ξ ∗ , η ∗ ) = (−1, 0) belonging to the unique branch point of the classical Lambert function. For this point z ∗ = −1/e, and W−1 (−1/e) = W0 (−1/e) = −1.
(12)
That here are infinitely many wk∗ when r = 0 shows the complexity of the branch structure of Wr comparing to the one of W . These numbers are also useful to determine the branch cut curves in the w-plane as for all branches there is a given wk∗ . The branch cut curves in the domain (the z-plane) of Wr will be a part of z = {x + ibk | x ∈ R, k ∈ Z\0 } with the above defined bk . (That why only a part of it will be discussed in the subsequent sections.) These correspond to the implicitly defined curves rη + eξ (η cos η + ξ sin η) = b∗k
(13)
in the range (the w-plane or (ξ, η)-plane). The discussion of the details are divided into three sections as the structure of the branches is radically different when r > 1/e2 , 0 < r ≤ 1/e2 and r < 0. The case r = 0 can be found in the fundamental paper [15]. 3. The branch structure of the r-Lambert function when r > 1/e2 3.1. The partition of the w-plane If r > 1/e2 the curves in (13) determine one branch that extends to minus infinity. This branch will be denoted by Wr,0 and we will call it the principal branch.2 Moreover, (13) determines infinitely many branches bounded from the left. These will be denoted by Wr,k where k ∈ Z\0 := Z \ {0}. 2 We try to keep as many analogies as possible with the classical Lambert function. However, here the principal branch has no analogue in the r = 0 case as we will see. When r tends to zero this branch will disappear and another branch emerges (already
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Here we chose the indices of ξk∗ and ηk∗ such that for k > 0 ∗ wk+1 = Wk (−re) − 1,
and for k < 0 wk∗ = Wk+1 (−re) − 1. We keep w0∗ undefined until we will need it, namely, when r < 0. The general situation is depicted above where all the curves come from (13). The solid lines constitute the branch border lines. The dashed line segments on the left of the points wk∗ (when ξ < ξk∗ ) belong to the image set of Wr,0 (z) where z = {x + ibk | x ∈ R, x < ak }, while the dashed line segments on the right of wk∗ (for ξ > ξk∗ ) belong to Wr,k (z). The small circles symbolize the points wk∗ as it is written on the figure. To make the formulas shorter we apply and rewrite (10)–(11) as ∗
ak = rξk∗ + eξk (ξk∗ cos ηk∗ − ηk∗ sin ηk∗ ), ∗ ξk
bk = rηk∗ + e (ηk∗ cos ηk∗ + ξk∗ sin ηk∗ ),
(14) (15)
for k ∈ Z\0 keeping in mind that these constants depend on r and also that ξk∗ and ηk∗ are simple expressions of the Lambert W function. Now, when z = x + iy such that x = ak ,
y = bk
for some non-zero k, then the point z belongs to two branches, to Wr,0 and to Wr,k . Such z’s are the pre-images of the wk∗ points. As we mentioned earlier this happens in only one point with respect to the Lambert function, namely, when z = −1/e (with η ∗ = 0 and ξ ∗ = −1), see (12). The principal branch Wr,0 . This branch of the r-Lambert function contains the whole real line in its image (the real line is mapped onto itself by this branch), and this branch is unbounded from the left and from the right on the complex plane. Horizontal lines x + iy (x ∈ R, y fixed) maps by Wr,0 to curves from minus infinity to plus infinity such that if 0 < y < b1 , then the curve runs above the ξ axis but below the first dashed line in the upper half plane (see on Fig. 2). As x grows the curve {Wr,0 (x + iy) | x ∈ R} runs in the gap between Wr,−1 and Wr,1 . Based on Fig. 1 one might think that this gap is finite but, in fact, Wr,−1 and Wr,1 never meet in the finite plane.3 The curves separating them are very close to each other indeed, but they meet only when ξ → ∞. The same is true for all the other branches, so Wr,0 is actually not bounded from the right. when r = 1/e2 ) that we will call principal branch in that case. It is still natural to call Wr,0 principal because it contains the real line in its image, and also, its structure and shape is unique among all the branches. 3 The numerical difference between the boundary points of Wr,k and Wr,k+1 is so small that at the applied level of magnification they seem to be overlapping.
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Fig. 1. The branch structure of the r-Lambert function when r = 2 in the (ξ, η)-plane.
If, in turn, y is negative such that 0 > y > b−1 , then {Wr,0 (x + iy) | x ∈ R} runs below the ξ-axis, and it penetrates into the area between the ξ-axis and the branch Wr,−1 . Moreover, if y is a fixed real number in the interval bk < y < bk+1 , with a given k > 0, then the curve {Wr,0 (x + iy) | x ∈ R} runs between two consecutive dashed lines in the upper half plane. As x grows this curve penetrates into the gap between Wr,k and Wr,k+1 . If for some k < 0 bk > y > bk−1 ,
(16)
then the image of x + iy by the principal branch runs between two consecutive dashed lines in the lower half plane such that it runs towards the area between Wr,−k and Wr,−k−1 . A typical situation for y > 0 is shown in Fig. 2, where we have chosen y = y± to be b1 ± 1 when r = 2. The lower dotted curve in the image of the principal branch corresponds to y− while the higher dotted curve in Wr,0 belongs to y+ . The another two dotted curves are the images of the same set x + iy by Wr,1 . If, finally, y equals to some bk then the curve {Wr,0 (x + iy) | x ∈ R} runs on the corresponding dashed line until reaching the point wk∗ . That from here it runs downwards or upwards depends on the branch closure we choose. By reasons made clear below we choose {Wr,0 (x + iy) | x ∈ R} to run downwards for every k regardless the sign. The non-principal branches Wr,k when k ∈ Z\0 . These branches are bounded from the left; each branch has its leftmost point at (ξk∗ , ηk∗ ). They extend to the right unbounded and their boundaries are asymptotic to the horizontal multiples of π. (This fact comes from (13): η is bounded so as ξ grows the factor η cos η+ξ sin η must tend to zero. This is possible only if η → 0 + kπ for some k ∈ Z.)
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Fig. 2. Images of horizontal lines. The meaning of the dotted lines is described in the text after (16).
These branches are divided into two parts by the dashed lines in Fig. 1 we were discussing in Section 3.1. Regardless4 the sign of k, the upper half of the branch Wr,k (z) contains the image of those points z = x + iy for which y < bk , and the lower half contains the image of z = x + iy for which y is greater than this quantity (and x spans R). The horizontal line z = x + ibk partially maps onto the dashed line in the range of Wr,k till the branch point wk∗ . Here Wr,k shares its value with the principal branch, then the curve turns upwards forming the upper limiting curve of the set of the range of Wr,k . This will further be discussed in the following subsection. 3.2. The branch closures After having described the boundary curves of the images of the different branches we also need to clarify where the points on the boundaries (solid lines on Fig. 2) belong to. The closure will be chosen such that in the limit r → 0 we get the closure structure of the branches of the classical Lambert function. This will be more apparent in the following section when we study the case 0 < r < 1/e2 . Thus we fix the following boundaries. For k ∈ Z\0 the boundary curve above of the point wk∗ and wk∗ itself belongs to the image set of Wr,k , while the curve below wk∗ belongs to Wr,0 . See Fig. 3 for a visual guide. For example, a part of the curve Wr,0− (x + ib2 ) (x ∈ R) is on the dashed line on the left of w2∗ . It runs into w2∗ from the left and here turns downwards on the dotted line of Fig. 3 and goes forward on this line to the right between Wr,2 and Wr,1 . This part of the curve belongs to Wr,0− hence Wr,2 is open here. In turn, the same horizontal line {x + ib2 } maps by Wr,2 onto the dashed line in the image set of Wr,2 , it reaches w2∗ then it turns upwards and closes the branch by going on the solid line above w2∗ . 4 Let y = bk − ε for some positive small ε such that y − ε does not go below bk−1 . If now z = x + iy then z = −bk + ε = b−k + ε > b−k and this point, as we wrote, maps to the lower half of the branch of Wr,−k .
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Fig. 3. Branch closures in the case r > 1/e2 .
According to the previous description, in Fig. 3 the dotted lines belong to the principal branch, while the solid lines belong to the branches below them. Note that the dotted curves go to the right to plus infinity but on the figure they seem to be merged to the solid lines. This is only virtual as the numerical difference between these points is tiny. Thus, for example, the real line maps onto the real line by Wr,0 (not presented in Fig. 3) and this set in the range of the principal branch lies between Wr,1 and Wr,−1 so these two branches do not meet in the finite plane. 4. The branch structure of the r-Lambert function when 0 < r ≤ 1/e2 4.1. The partition of the w-plane when r = 1/e2 When r → 1/e2 + an interesting phenomenon happens: (ξ1∗ , η1∗ ) → (−2, 0), ∗ ∗ (ξ−1 , η−1 ) → (−2, 0), ∗ so w1∗ = w−1 = −2. Geometrically, the branch curves that previously formed the boundary of Wr,1 and Wr,−1 now share the point −2 and the area that was lying between these two sets and formerly belonged to the principal branch disappears. The point −2 in the range of the 1/e2 -Lambert function is the image of z = −4/e2 . The special role of this point can be explained by (9). In the particular case when r = 1/e2 we have that
W1/e2 (x) = W (−1/e) − 1 = −2, i.e., by (4), −2e−2 + e−2 (−2) = x, so x = −4/e2 . In this point the real W1/e2 function is not differentiable. As we will see, in this point three branches meet.
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Fig. 4. Branch structure at r = 1/e2 without closure information.
The lower half part of Wr,1 and the upper half part of Wr,−1 form a new branch containing the set of real numbers > −2. This branch will be denoted by Wr,0+ while the old principal branch will be renamed to Wr,0− . As Wr,0+ will become the principal W0 branch of the Lambert function when r → 0+ this naming convention seems to be appropriate. The branch formed by the upper part of Wr,1 and by the lower part of Wr,−1 will be denoted by Wr,±1 . See Fig. 4. When r = 1/e2 , by the definitions (14)–(15), we have, as we have said already, a1 = a−1 = −4/e2 ,
b1 = b−1 = 0,
hence there will be three branches containing real numbers: W1/e2 ,0− contains the set ]−∞, −2], the branch W1/e2 ,±1 contains the single real point −2, while W1/e2 ,0+ contains the real interval [−2, +∞[ in its range. In Fig. 4 the dashed lines are the images of x + iy where y = bk , as before. The dotted lines depict the set
W1/e2 ,0− (x + iy), W1/e2 ,±1 (x + iy), W1/e2 ,0+ (x + iy) | x ∈ R, y = 1/2 ,
from where it seems that W1/e2 ,0− maps the upper half plane to the upper half plane, W1/e2 ,0+ maps this plane to its upper half part, and W1/e2 ,±1 maps it into the lower half of its image set. The dash–dotted lines belong to the set
W1/e2 ,0− (x + iy), W1/e2 ,±1 (x + iy), W1/e2 ,0+ (x + iy) | x ∈ R, y = −1/2 .
For |k| ≥ 2 the upper half plane {x + iy | y > bk } maps into the lower half of W1/e2 ,k , while the lower half plane {x + iy | y < bk } is mapped into the upper half of these branches. ∗ = The small circle in Fig. 4 in the intersection of W1/e2 ,0− , W1/e2 ,0+ , and W1/e2 ,±1 is the point w1∗ = w−1 2 −2. As we have said, the branch point z = −4/e maps to the common point −2 by these three branches: W1/e2 ,0− (−4/e2 ) = W1/e2 ,0+ (−4/e2 ) = W1/e2 ,±1 (−4/e2 ) = −2.
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Fig. 5. Branch closure at r = 1/e2 .
4.2. The branch closures when r = 1/e2 The branch closure is taken as follows: we keep the closure for the branches W1/e2,k when |k| ≥ 2 as they were for r > 1/e2 . The upper part of W1/e2 ,±1 till the point −2 belongs to W1/e2 ,±1 , while the part below −2 limited by the dotted curve in Fig. 5 belongs to W1/e2 ,0− . Moreover, the upper limiting curve of W1/e2 ,0+ belongs to W1/e2 ,0+ while the lower limiting curve belongs to W1/e2 ,±1 instead. All of these are necessary if we would like to enforce agreement with the branch closures of the Lambert function when r → 0+. The horizontal lines {x + iy | y = bk } are mapped by the W1/e2 ,0− branch such that their images are curves from minus infinity till wk∗ , including this point, then they turn downwards and go to plus infinity on the corresponding dotted line. In particular, {W1/e2 ,0− (x) | x ∈ R} = ]−∞, −2] ∪ {the limiting dotted curve of the lower part of W1/e2 ,±1 }. Moreover, {W1/e2 ,0+ (x) | x ∈ R} = [−2, +∞[ ∪ {the solid line running upwards from −2 in W1/e2 ,0+ }. 4.3. The partition of the w-plane when 0 < r < 1/e2 and the closures ∗ When r gets smaller than 1/e2 but still positive, the point w1∗ and w−1 will separate again but both stay real, forming a real interval ∗ I = [w−1 , w1∗ ] = [W−1 (−re) − 1, W0 (−re) − 1].
(17)
This interval belongs to the image set of Wr,±1 such that the left endpoint is shared with Wr,0− while the right endpoint is shared with Wr,0+ , see Fig. 6. The branches are the same as for r = 1/e2 as well as the closures, the only difference is that the point −2 enlarges into an interval. See the below graph.
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Fig. 6. Branch structure and closure at r = 1/e4 .
4.4. The branch structure as r → 0+ When r → 0+ equation (3) turns to be the defining equation (1) of the Lambert function and the branch structure should be the same as the one of W . In this limit the real part ξk∗ of the branch points tends to minus infinity (see Theorem 1 below) and when |k| ≥ 2 the Wr,k branches will be stretched to minus infinity. These correspond to the Wk branches of the Lambert function described in [15]. The behavior of the interval I is as follows. The left endpoint of I, W−1 (−re) − 1, tends to −∞ and so the Wr,±1 branch also runs out to minus infinity. This together with the run out of the other branches Wr,k (|k| ≥ 2) result that the Wr,0− branch disappears. Moreover, the upper and lower parts of Wr,±1 separate such that these parts form, when r = 0, two separate branches that correspond to W1 and W−1 of the Lambert function. This is why we chose this notation for Wr,±1 before. These branches are separated by the half line ]−∞, −1] in the w-plane. The other endpoint of I, W0 (−re) − 1 turns to be W0 (0) − 1 = −1 so that it forms the only branch point of W . The Wr,0+ branch turns to be W0 when r = 0, becoming the principal branch of W . Now one can also see why we have chosen the branch closures as we did. The Wk branches in the limit r = 0 [15] are closed exactly as we see on Fig. 5 when we stretch Wr,k (|k| ≥ 2) to minus infinity: closed from above and open from below. The upper part of Wr,±1 that becomes the new W1 branch is open on the curve separating it from W0 . The ]−∞, −1] half line belongs to W−1 . The lower half of W0 , below the real line, is separated from W−1 such that the separating curve belongs to W−1 , while the curve separating W0 from W1 belongs to the principal branch W0 . 5. The branch structure of the r-Lambert function when r < 0 When r < 0 and k = 0 the Wr,k branches are bounded from the left again by the quantities wk∗ as they were for r > 1/e2 . Now we go back to the non-shifted indices again: wk∗ = ξk∗ + iηk∗ = Wk (−re) − 1. The importance of this will be clear soon.
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Fig. 7. Branch structure and closure when r < 0.
The branch structure thus formally looks like on Fig. 1. There are some important differences that we are going to describe now. If r < 0 the number η0∗ = (W0 (−re) − 1) is always 0 (W0 (z) is real for real non-negative z), so there will be a real branch point at w0∗ = ξ0∗ = W0 (−re) − 1. The image of the real line thus belongs to two branches that we denote by Wr,0− and Wr,0+ . A part of the real line maps to two half lines: the interval [fr (w0∗ ), +∞[ maps to [−∞, w0∗ ] by Wr,0− , while the same interval [fr (w0∗ ), +∞[ maps to [w0∗ , +∞] by Wr,0+ . Here fr (z) = zez + rz. The union of these two image sets forms the horizontal dashed line in the below figure. The image of the real line by Wr,0− , as we have just described, runs from −∞ till w0∗ . Here, to keep consistence with the branch closure of the other cases, we choose the continuation of this curve such that in the point w0∗ it turns downwards. Hence the boundary of Wr,0+ belongs to Wr,0− from the point w0∗ and below. The boundary of Wr,0+ from w0∗ and above belongs to Wr,0+ . This is also depicted on Fig. 6. We note that for r < 0 the sign of bk is negative when k > 0 (and b−k = −bk ). The Wr,0− branch maps the lower half plane into the upper half plane and vice versa. This is in accordance with the remarks on the sign of bk . Thus, for example, the set Wr,0− ({x + iy | x ∈ R}) will be a curve below the horizontal dashed line and it goes to plus infinity in the gap between Wr,0+ and Wr,1 for b1 < y < 0. If y = b1 the set Wr,0− ({x + iy | x ∈ R}) is the dashed line going through w1∗ and continuing on the dotted line, forming the lower boundary of Wr,1 . The other branches, including Wr,0+ , map the upper half plane into the upper half of their range while the lower half plane is mapped to their lower half. The two halves are separated by the corresponding dashed lines on Fig. 7. The horizontal lines {x + ibk | x ∈ R} are mapped partially onto the corresponding dashed lines in the branch Wr,k until wk∗ and at this point they turn upwards forming the closure of the given branch.
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6. More on the points wk∗ In this section we are going to examine the asymptotic behavior of the points wk∗ as k → ±∞. As these determine the extremal points of the branches they have special importance. Recall that (if we do not shift the indices) ξk∗ = (Wk (−re) − 1),
(18)
ηk∗ = (Wk (−re) − 1),
(19)
and ∗ η−k = −ηk∗ ,
∗ and ξ−k = ξk∗
The following statement is true. Theorem 1. Let r be a non-zero real number. Then, for large k π ξk∗ ∼ log |r| − log 2kπ + sgn(r) , 2 π ηk∗ ∼ 2kπ + sgn(r) . 2
(20) (21)
Here sgn is the signum function, and ak ∼ bk means that ak /bk → 1 as k → ∞. Proof. The asymptotic behavior of the Lambert function was discussed by de Bruijn [8], and was completed by Corless et al. [15]. For all the branches it holds true that Wk (z) = log z + 2kπi − log(log z + 2kπi) + O
log log z log z
,
when z → ∞ as well as for z → 0. In this expansion the principal branch of the logarithm is taken at each occurrence. We make use of this asymptotic expansion when the branch index k is big and z simply equals to −re. We then have Wk (−re) − 1 ≈ −1 + log |re| + (0, π)i + 2kπi − log(log |re| + (0, π)i + 2kπi). The big “O” term will be neglected, and the symbol (0, π) is defined to be 0 if −re > 0 (i.e., if r < 0) and it is defined to be π if −re < 0 (if r > 0). Thus ηk∗ = (Wk (−re) − 1) ≈ 2kπ + (0, π) − (log(log |re| + (0, π)i + 2kπi)). If k is big, log |re| and (0, π)i can be neglected in the outer logarithm’s argument, log |re| + (0, π)i + 2kπi) is a complex number with angle close to π/2. Hence (log(log |re| + (0, π)i + 2kπi)) ≈ (log(2kπi)) = From here we get that ηk∗ ≈ 2kπ + (0, π) −
π . 2
This, by the definition of the symbol (0, π), is equivalent to (21). The statement for ξk∗ can be treated similarly. 2
π . 2
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We know from the former sections that the Wr,k branches are bounded from the left if r = 0 and k = 0. This theorem says more: it states that the left extremal points of the individual branches (not being equal to Wr,0− , Wr,0+ , Wr,±1 ) go to left as k grows, and the ordinate logarithmically decreases to minus infinity. 6.1. Tables of wk∗ We compare the estimations of the above theorem to the exact expressions for wk∗ = ξk∗ + iηk∗ given by (18)–(19). First we choose r = 2. The exact value of ξ50 is ∗ ξ50 = −5.06178,
while (20) results in the estimation ∗ ξ50,estim = −5.06174. ∗ For η50 we have ∗ η50 = 315.717, ∗ η50,estim = 315.73.
For r = 1/e4 the corresponding values for k = 50 are ∗ ξ50 = −8.16961, ∗ ξ50,estim = −8.16515,
and ∗ η50 = 315.702, ∗ η50,estim = 315.73.
While, for example, taking r = −3 ∗ ξ50 = −4.64631, ∗ ξ50,estim = −4.64628,
and ∗ η50 = 312.577, ∗ η50,estim = 312.588.
For larger k the performance of these estimations is even better. Moreover, (20) and (21) are extremely useful when one wants to find a good starting point for numerical root finding methods to solve (3). This way one does not need to use the Lambert W function in order to determine wk∗ to have an initial guess where the solutions on the different branches could be. As W is not given in many software packages, this is of practical importance.
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7. Conclusion In this paper we have studied the structure of the solution set of the transcendental equation xex + rx = z where z is a fixed complex parameter and r is an arbitrary but fixed real number. The solutions are represented by the r-Lambert function. We pointed out that the solutions fall into infinitely many components of the complex plane and we determined the shape of these components for all possible values of r. A full description of the boundaries of these components is also given together with the consistent determination of the closure of these components. References [1] B.V. Alekseev, A.E. Dubinov, I.D. Dubinova, Analytical and numerical solutions of generalized dispersion equations for one-dimensional damped plasma oscillation, High Temp. 43 (4) (2005) 479–485, translated from Teplofiz. Vys. Temp. 43 (4) (2005) 485–491. [2] C.H. Belgacem, Range and flight time of quadratic resisted projectile motion using the Lambert W function, Eur. J. Phys. 35 (2014) 055025, http://dx.doi.org/10.1088/0143-0807/35/5/055025. [3] C.H. Belgacem, Analysis of projectile motion with quadratic air resistance from a nonzero height using the Lambert W function, J. Taibah Univ. Sci. 11 (2) (2017) 328–331, http://dx.doi.org/10.1016/j.jtusci.2016.02.009. [4] C.H. Belgacem, Explicit solution for critical thickness of semicircular misfit dislocation loops in strained semiconductors heterostructures, Silicon 8 (2016) 397–399. [5] C.H. Belgacem, M. Fnaiceh, Exact analytical solution for the critical layer thickness of a lattice-mismatched heteroepitaxial layer, J. Electron. Mater. 39 (10) (2010) 2248–2250. [6] C.H. Belgacem, M. Fnaiceh, Solution for the critical thickness models of dislocation generation in epitaxial thin films using the Lambert W function, J. Mater. Sci. 46 (2011) 1913–1915. [7] R.C. Bernardo, J.P. Esguerra, J.D. Vallejos, J.J. Canda, Wind-influenced projectile motion, Eur. J. 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