Acta Electrotechnica et Informatica, Vol. 14, No. 1, 2014, 9–17, DOI: 10.15546/aeei-2014-0002 9 APPLICATION OF LAMBERT W FUNCTION IN OSCILLATION THEORY Irena JADLOVSKA´ Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Kosice,ˇ Letna´ 9, 042 00 Kosice,ˇ Slovak Republic, e-mail: [email protected] ABSTRACT This paper is primarily concerned with analyzing some special properties of the characteristic equation of the linear delay differ- ential equation of the first and odd order. The basic concepts and results keen to the oscillatory nature of solutions, as well as stability analysis are re-investigated using the properties of Lambert W function and compared with existing ones. A brief survey of the function origin and possible applications is presented together with a short behavior analysis. Keywords: Lambert W function, odd-order delay differential equation, oscillation, asymptotic properties 1. INTRODUCTION 2. LAMBERT W FUNCTION Best known today for proving the irrationality of p We begin here by describing some basic notation and and introducing the concept of hyperbolic functions facts about the mentioned special function which we will in trigonometry, the 18th-century Swiss mathematician, use in later section. physicist, philosopher and astronomer Johann Heinrich The Lambert W function, henceforth also denoted by Lambert W(z), can be defined in analogy with the complex logarithm considered in 1758 the series solution to the sym- w metric case of the trinomial equation xa −xb = (a−b)vxa+b. as the multivalued inverse of the function z = f (w) = we , where w and z are assumed to be complex variables, satis- Six years later, the subsequent work of Euler on a more spe- W(z) cial case, which reduces to the form waw = ax, resulted fying the so-called defining equation z = W(z)e . Thus, in some of the mathematical properties of this equation, the mentioned special function, also known as a Product where Lambert’s initial contribution was explicitly empha- Log function, which clearly indicates an extension of a nat- sized. However, the importance and great applicability of ural logarithm, is an example of multivalued functions f (z), this formulation was not appreciated and obtained no spe- which are characterized, unlike single-valued-ones, by two cial notation until the last decade of the 20th century, when or more distinct values for each value of z. The multival- it was included in the computer algebra software Maple. uedness is represented by infinitely many branches indexed An influential work was done by Corless et al., as by k 2 Z as Wk’s, which restrict the original domain to a a result of a systematic literature search of the simplest smaller subsets of the complex plane, on which our func- nontrivial transcendental function in many isolated contri- tion is already single-valued and continuous. Using the La- butions collected in a compact overview [5], it was shown grange’s Inverse Theorem, the principal branch (i.e., k = 0) that the Lambert W function had been re-discovered at var- of the Lambert W function can be represented by the fol- ious moments in history. Moreover, a new discussion of lowing power series [6] complex branches, symbolic calculus and mainly the accu- ¥ (−n)n−1 W (z) = zn rate implementation of the function has encouraged many 0 ∑ n! researchers. Due to its simplicity, the wide range of appli- n=1 cations associated with the Lambert W function arises in with z 2 C, while the other branches are defined in a series the modeling of diverse phenomena in various fields of sci- form given below: ence and engineering. In physics, it is utilized for the pur- ¥ ¥ m pose of quantum theory, plasma physics and solar physics (ln(lnk(z))) Wk(z) = lnk(z) − ln(lnk(z)) + ∑ ∑ Clm m+l (see e.g. [8, 17, 18]). Applications in the field of theoreti- l=0 m=1 (lnk(z)) cal computer science include the analysis of algorithms and where ln (z) represents the kth logarithm branch and the co- counting search trees and graphs occuring in combinatorial k efficients C can be expressed in terms of nonnegative Stir- applications as well as the iterated exponentiation [13]. lm ling numbers of first kind. Complete and detailed derivation From a strictly mathematical point of view, the most sig- is provided by Corless et al. in [5]. nificant use of the Lambert W function is related to finding The branches of W (z) are defined in detail in the com- zeros of transcendental functions. Hence, without aware- k plex plane (i.e., Re[W (z)] ! Im[W (z)]), however, our in- ness of the Lambert W function, the location of zeros for k k terest is primarily focused on the real and imaginary parts such functions was determined either numerically or aprox- of W (z);k = 0;−1 depending on z, i.e. z ! Re[W(z)] and imately [12]. When applying elementary analytical tech- k z ! Im[W(z)]. niques, an explicit solution to various exponential or loga- Recall the definition of the Lambert W function and de- rithmic polynomials can be formulated in terms of the Lam- note their variables: z = f (w) = wew; w = f −1(z) = W(z), bert W function, especially in the field of time-delay sys- where w = x + iy; z = u + iv. We have tems [1, 14, 19–22]. Moreover, an efficient implementation of this special function is available in various mathematical- z = (x + iy)ex(cosy + isiny): (1) engineering software tools. ISSN 1335-8243 (print) c 2014 FEI TUKE ISSN 1338-3957 (online), www.aei.tuke.sk 10 Application of Lambert W Function in Oscillation Theory 6 6 4 4 2 2 L L z z H 0 H 0 W W -2 -2 -4 -4 -6 -6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 z z Fig. 1 The real (solid) and imaginary (dotted) part of W0(z) Fig. 2 The real and imaginary part of the −1th branch. Separating real and imaginary parts of z, the following Theorem 2.1. For any given z 2 R, the principal branch of equations are obtained: the Lambert W function W0(z) defined by u = Re[z] = xex cosy − yex siny ( x x (2) x; for x ≥ −1; where z ≥ −1=e v = Im[z] = ye cosy + xe siny W0(z) = −ycotgy + iy; for y 2 (0;p); where z < −1=e For further needs, we restrict our interest to a case where v = 0 (it follows from the fact that the arguments has the following properties: of the Lambert W function used in the next section are al- ways real), therefore yex cosy+xex siny = 0, which implies 1.z 2 (−1=e;¥) : W0(z) is real and increasing. x = −ycotgy or y = 0. Both cases will be discussed. 2.z 2 (−p=2;−1=e) : W0(z) is complex valued with de- creasing negative real part. – If y = 0 : W(z) = x is real-valued and z = f (x) = xex. A real-valued function f (x) is strictly decreasing on 3.z = −p=2 : W0(z) is complex with purely imaginary (−¥;−1), strictly increasing on (−1;¥) and has a parts. global minimum point at x = −1 of z = −1=e. A sec- ond derivative indicates an inflection point at x = −2 4.z 2 (−¥;−p=2) : W0(z) is complex valued with de- of z = −2e−2. The function f (x) is not injective, creasing positive real part. hence the inverse function cannot be constructed on the whole interval. For this reason, when restricted The properties are shown in Figure 1. Similarly, the on each of monotone intervals, two real branches of −1th branch, which is real-valued only for z 2 [−1=e;0] is the Lambert W function are defined as follows: illustrated (see Figure 2). The important values are high- lighted by vertical dashed lines passing throught the x-axis – for z 2 [−1=e;0): the principal branch satisfy- at −1=e and −p=2. ing W(z) ≥ −1 and denoted by W0(x) is real and increasing. Similarly, the branch satisfying Lemma 2.1. A solution of the following generalized odd-degree exponential polynomial kax+b = (cx + d)n, W−1 < −1 and denoted by W−1(z) is real and decreasing from −¥ to −1, where it meets with k > 0, a;c 6= 0 can be expressed in terms of the Lambert function as W0(z) at the so-called branch point. – for z 2 [0;¥), the only part on the principal n alnk bc−ad d branch is real, which implies exactly one real x = − W − k nc − root of the defining equation, which is positive alnk cn c except for W(0) = 0. ax+b – with z < −1=e, every Wk(z) is complex-valued. Proof. Taking the nth root, we find k n = cx + d, while separation of the unknown to the right side of the equa- ax 1 b=n − n lnk For our future needs, only the principal branch will tion yields c k = (x + d=c)e . Performing straight- be assumed. forward mathematical calculation, one can easily obtain bc−ad alnk alnk nc alnk − n (x+d=c) − cn e = − n (x + d=c)e . Hence, di- – If y 6= 0, W0(z) = −ycotgy + iy where y 2 (0;p). In rectly from the definition of the Lambert W function, alnk d such a case, Re[W0(z)] = 0 , (y = p=2 _ y = 0). alnk d W(− n (x+ c )) alnk d we have W(− n (x + c ))e = − n (x + c ), Therefore, zeros of the principal branch on the real bc−ad alnk d alnk nc ip ip=2 which is satisfied if − n (x + c ) = W(− cn e ); from line are z = 2 e = −p=2; and z = 0, respectively.