Resolution of the Plane-symmetric Einstein-Maxwell fields with a generalization of the Lambert W function

Here, the Einstein-Maxwell field equation with plane-symmetry is resolved and a solution involving a generalized Lambert W function is obtained. This generalized Lambert W function is studied and its derivative, Taylor series, Mellin transform and approximation formula are derived and its real branches are determined. Implications of these analytic properties to the solution function are also presented.


Introduction
The Einstein's field equations are ten equations, contained in the tensor (see [1]) where R μv is the Ricci curvature tensor, R is the scalar curvature, g μv is the metric tensor, Λ is the Cosmological constant, G is Newton's gravitational constant, c is speed of light and T μv is the Stress energy tensor. These equations describe the gravity as a result of spacetime being curved by mass and energy.
The basic characteristic of Einstein's theory is the Lorentz transformation which relates scientific measurements in one frame of reference to another. Maxwell's equations together with the transformations of the special theory of relativity, are referred to as the Einstein-Maxwell equations [2].
A plane-symmetric Einstein-Maxwell field with (L = 0) is given by (see [3]) For ¹ e 0, both e and m are constant, and the metric is either static, or spatially homogeneous. By a transformation of the r-coordinate, the line element (2) can be transformed into the form [3-53-5] (5) is a special case of (7). However, we cannot say that the general form (7) is solvable with a generalized Lambert function because we have not yet explored such general case. Although the proposed generalization (5) can be cosidered a special case of the more general form (7), this generalization (5) in particular, is not yet considered in the literature. The generalized forms - x t x t e a ; 9 with parameters t, s, t 1 , t 2 were studied in [8]. Some applications of the generalized Lambert functions defined as the solutions to the equations of the form (9) were presented in [9]. Motivated by the above mentioned resolution of the plane symmetric Einstein-Maxwell field equation we proceed to study the quadratic Lambert W function. Formal notation of the function is given in section 2 and its analytic properties such as derivative, Taylor series and Mellin transform are derived. The real branches are discussed in section 3 and an asymptotic approximation formula is proved in section 4. A discussion on implications of the results from sections 2-4 is given in section 5. Finally, conclusion is given in section 6.

The quadratic Lambert function and some basic properties
Denote by W a x ; ( ) the quadratic Lambert function which is the solution to (5). The proposition that follows shows that (4) can be transformed to (5) and a solution is expressed in terms of the quadratic Lambert function.
Proof. Substituting E=Y+A to (4) and performing simple algebraic manipulation yield

the last equation transforms into
The left hand side is already of the form (5). This means that With the result in proposition 1, knowing the behavior of the quadratic Lambert W(a; x) will enable us to describe the behavior of the solution Y(z) of (4). Thus, we proceed to establish some analytic properties of the quadratic Lambert function.
As W(a; x) is an inverse function, its derivative can be found readily.
Theorem 2. The derivatives of the quadratic Lambert W function are is used to derive (10). To prove (11), take the derivative of the defining equation (5) with respect to a. Without writing out the arguments: Here the prime refers to the derivative with respect to a. The exponential terms cancel out, and a rearrangement gives (11). , . Hence, up to the first order term, The Taylor series around x=0 can be explicitly determined as the following theorem shows.
Theorem 3. The Taylor series of the quadratic Lambert W function around x=0 is given as The binomial theorem gives that is the falling factorial with = x 1 0 .) Taking  x 0, the terms that survive are only those for which + =i m k 1. Since im, A simplification and rearrangement gives the desired formula. , Few terms of the Taylor series are given below, Integral transformations of functions are often important in applications. The Mellin transform in particular, has applications in geophysics [10] and time series models [11]. The Mellin transform of the quadratic Lambert is derived in the next theorem.
Proof. By the definition of Mellin transform, W a x s t W a t t ; ; d 1 4 Using this expression for I s a b , , with Î b 0, 1, 2 { }and some algebra will give the formula in the theorem. The signs of s and a must be opposite in order to have convergent integrals. Moreover, as W(a; t) behaves around t=0 as t, so the integrand in (14) behaves as t s around zero. This explains why s>−1 is necessary. ,

The real branches of the quadratic Lambert function
It is important to know how many real solutions (5) has, and how this number depends on the parameter a. This analysis is done below.
We  . Depending on the the value of the parameter a, the quadratic Lambert W(a; x) has one, two or three real branches. These branches are described in more detail in the following theorem. ] is a strictly increasing function. These three functions are differentiable on the interior of their domains.
) is undefined only at = x 0 but because = W a; 0 0 ( ) , this singularity is removable. In this case we only have one branch given by  W a x ; : ( ) R Ris a strictly increasing function which is differentiable for all x.
Proof. In each case, the domains are determined by looking at the singularities of ¢ W a x ; ( ). We will prove Case II only. The other cases may be done similarly. Let u=W(a; x). Since x=0 is a removable singularity, we have only three possibilities a  u a , b a   u a a , and b  u a . These constitute the branches of the function. The domains can then be readily determined to be those specified in the theorem. To see whether the function is increasing or decreasing on the branch, the first derivative test is used. We