Lambertian W function

The graph of W ( x ) for W > −4 and x <6. The upper branch W ≥ −1 is the function W ₀ (principal branch), the lower branch with W ≤ −1 is the function W ₋₁ .

In mathematics , the Lambertian W function (or Lambert W function ), also known as the Omega function or product logarithm , named after Johann Heinrich Lambert , is the inverse function of

${\displaystyle f\colon x\mapsto xe^{x},}$

where is the exponential function . The Lambertian W function is usually referred to as. It applies ${\displaystyle e^{x}}$ ${\displaystyle W(x)}$

${\displaystyle z=W(z)e^{W(z)},z\in \mathbb {C} .}$

properties

In real life

The two functional branches and ${\displaystyle W_{0}}$ ${\displaystyle W_{-1}}$

Since the function is not injective on the interval , the Lambertian W-function has two function branches and on the interval . With but is usually referred to the top of the branches. ${\displaystyle f}$ ${\displaystyle \left(-\infty ,0\right]}$ ${\displaystyle \left[-{\tfrac {1}{e}},0\right)}$ ${\displaystyle W_{0}(x)}$ ${\displaystyle W_{-1}(x)}$ ${\displaystyle W(x)}$

The W function cannot be expressed as an elementary function .

It is mostly used in combinatorics , for example to evaluate trees or to determine the Bell numbers asymptotically .

The derivative function of a branch of the W-function can be found with the help of the inverse rule of differential calculus ( the derivative does not exist at this point, its amount increases with sufficient approximation to this point in each branch over all bounds): ${\displaystyle -1/e}$

${\displaystyle W'(x)={\frac {W(x)}{x(1+W(x))}}{\text{ für }}x>-{\frac {1}{e}},x\neq 0}$

as well as for the upper branch (the lower branch is not defined for at all). ${\displaystyle W'_{0}(0)=1}$ ${\displaystyle x\geq 0}$

The higher order derivatives have the form

${\displaystyle {\frac {\mathrm {d} ^{n}W(x)}{\mathrm {d} x^{n}}}={\frac {(-1)^{n+1}W^{n}(x)}{x^{n}(1+W(x))^{2n-1}}}\cdot P_{n}(W(x)),}$

where are the polynomials that can be calculated from the following recursion formula: ${\displaystyle P_{n}}$

${\displaystyle P_{n+1}(t)=(nt+3n-1)\cdot P_{n}(t)-(t+1)\cdot P_{n}'(t),\quad n\geq 1}$

Based on this, the next three derivatives result: ${\displaystyle P_{1}(t)=1}$

${\displaystyle {\begin{aligned}W''(x)&=-{\frac {W^{2}(x)}{x^{2}(1+W(x))^{3}}}\cdot (W(x)+2)\\W^{(3)}(x)&=+{\frac {W^{3}(x)}{x^{3}(1+W(x))^{5}}}\cdot (2W^{2}(x)+8W(x)+9)\\W^{(4)}(x)&=-{\frac {W^{4}(x)}{x^{4}(1+W(x))^{7}}}\cdot (6W^{3}(x)+36W^{2}(x)+79W(x)+64)\end{aligned}}}$

An antiderivative is obtained by substituting the whole integrand:

${\displaystyle \int W(x)\,\mathrm {d} x=x\left(W(x)-1+{\frac {1}{W(x)}}\right)+C}$

By implicit differentiation one can show that the following differential equation is sufficient: ${\displaystyle W}$

${\displaystyle z(1+W){\frac {\mathrm {d} W}{\mathrm {d} z}}=W\quad {\text{mit }}z\neq -{\frac {1}{e}}}$

The Taylor series of um is given by ${\displaystyle W}$ ${\displaystyle x_{0}=0}$

${\displaystyle W(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}\ x^{n}=x-x^{2}+{\frac {3}{2}}x^{3}-{\frac {8}{3}}x^{4}+{\frac {125}{24}}x^{5}-\dotsb .}$

The radius of convergence is . ${\displaystyle {\tfrac {1}{e}}}$

In the complex

The main branch of the W-function on the complex number level. Note the fraction along the negative real semiaxis . The coordinates of a point describe the real and imaginary part of the argument, the brightness describes the amount and the hue the phase of the result. ${\displaystyle -e^{-1}}$

Radius of the main branch of the W function as height, hue the phase

For each there is a branch of the W-function, where and represent the real branches mentioned above. The main branch is special in that it is defined on the entire complex number plane; all other branches (secondary branches) have a definition gap at . Specifically, ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle k=0}$ ${\displaystyle k=-1}$ ${\displaystyle W_{0}}$ ${\displaystyle z=0}$

${\displaystyle W_{0}(0)=0}$ and

${\displaystyle \lim _{z\to 0}W_{k}(z)=-\infty }$for everyone . ${\displaystyle k\neq 0}$

This behavior can be seen in the diagram above for the real cases as an example.

The branch point for the main branch is at , which extends over the remainder of the negative semi-axis in the direction . This branch separates the main branch from the secondary branches and . On the secondary branches, the branch already begins at and continues in the direction of the main branch . ${\displaystyle z=-{\tfrac {1}{e}}}$ ${\displaystyle -\infty }$ ${\displaystyle W_{-1}}$ ${\displaystyle W_{+1}}$ ${\displaystyle z=0}$ ${\displaystyle -\infty }$

All branches are injective and their ranges of values ​​are disjoint . Considered as a function with two parameters and the W function has the entire complex number level as a range of values. The image of the real axis is the union of the real axis with the quadratic rix of Hippias , the parametric curve defined for , whereby the limit value is understood, whereby the point is continuously continued. ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle t\in \mathbb {R} \setminus \{k\pi \mid k\in \mathbb {Z} \setminus \{0\}\}}$ ${\displaystyle w(t)=-t\cot t+it}$ ${\displaystyle w(0)}$ ${\displaystyle \lim _{t\to 0}w(t)=-1}$ ${\displaystyle w}$ ${\displaystyle t=0}$

Special values

${\displaystyle W\left(-{\frac {\pi }{2}}\right)={\frac {\mathrm {i} \pi }{2}}}$

${\displaystyle W\left(-{\frac {1}{e}}\right)=-1}$

${\displaystyle W\left(-{\frac {\ln 2}{2}}\right)=-\ln 2}$

${\displaystyle W\left(0\right)=0}$

${\displaystyle W\left(1\right)=0{,}5671432904\dots =:\Omega }$   (the omega constant )

${\displaystyle W\left(e\right)=1}$

properties

• ${\displaystyle \int _{0}^{+\infty }W\left({\frac {1}{x^{2}}}\right)\,\mathrm {d} x={\sqrt {2\pi }}}$
• ${\displaystyle \int _{0}^{+\infty }{\frac {W(x)}{x{\sqrt {x}}}}\,\mathrm {d} x={\sqrt {8\pi }}}$
• ${\displaystyle \int _{0}^{\pi }W\left(2\cot ^{2}(x)\right)\sec ^{2}(x)\,\mathrm {d} x={\sqrt {16\pi }}}$

Use outside of the combinatorics

The Lambertian W-function can be used to find equations of the type

${\displaystyle a(x)e^{a(x)}=y}$

to solve ( is any expression that depends on). ${\displaystyle a(x)}$ ${\displaystyle x}$

The equation too

${\displaystyle x^{x}=z}$

can be solved with the help of Lambert's W function. The solution is

${\displaystyle x={\frac {\ln z}{W(\ln z)}}=\exp \left(W(\ln z)\right).}$

The infinite power tower

${\displaystyle x\uparrow \uparrow \infty :=x^{x^{x^{\cdot ^{\cdot ^{\cdot }}}}}}$

can be brought into closed form at the convergent points with the W function:

${\displaystyle x\uparrow \uparrow \infty ={\frac {W(\ln {\frac {1}{x}})}{\ln {\frac {1}{x}}}}.}$

Generalizations

With the help of the normal Lambertian W function, the exact solutions of "transcendent algebraic" equations (in x ) can be expressed in the  following form:

${\displaystyle e^{-cx}=a_{0}(x-r)~~\quad \qquad \qquad \qquad \quad (1)}$

with real constants and . The solution is . Generalizations of the Lambertian W-function include: ${\displaystyle a_{0},c}$ ${\displaystyle r}$ ${\displaystyle x=r+{\frac {1}{c}}W\left({\frac {ce^{-cr}}{a_{0}}}\right)}$

• For an application in the field of general relativity and quantum mechanics ( quantum gravity ) in lower dimensions that demonstrated a previously unknown link between the two fields, see Journal of Classical and Quantum Gravity, with the right-hand side of equation (1) now a quadratic polynomial in is: ${\displaystyle x}$

${\displaystyle e^{-cx}=a_{0}(x-r_{1})(x-r_{2})~~\qquad \qquad (2)}$

Here and are different real constants, the roots of the quadratic polynomial. The solution is a function of the argument alone , but and are parameters of this function. In this respect, this generalization is similar to the hypergeometric function and Meijer's G-function , but it belongs to a different “class” of functions. If so, both sides of (2) can be factored and reduced to (1) so that the solution is reduced to the normal Lambertian W-function. Equation (2) corresponds to the equation for the “Dilaton” field, of which the metric of the “linear” two-body gravitation problem in 1 + 1 dimensions (one spatial and one temporal dimension) for the case of unequal (rest) masses is derived, as well as the problem of the eigenvalue calculation for the quantum mechanical double minimum Dirac delta function model in one dimension and with "unequal" charges. ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle x}$ ${\displaystyle r_{i}}$ ${\displaystyle a_{0}}$ ${\displaystyle r_{1}=r_{2}}$

• Analytical solutions of the energy eigenvalues ​​for a special case of the quantum mechanical analogue of Euler's three-body problem, namely the (three-dimensional) hydrogen molecule ion . Here now the right hand side of (1) (or (2)) is the ratio of two polynomials of infinite order in : ${\displaystyle x}$

${\displaystyle e^{-cx}=a_{0}{\frac {\prod _{i=1}^{\infty }(x-r_{i})}{\prod _{i=1}^{\infty }(x-s_{i})}}\qquad \qquad \qquad (3)}$

with pairwise different real constants and as well as a function of the energy eigenvalue and the core-core distance . Equation (3), with the special cases (1) and (2), is related to a large class of retarded differential equations . Using Hardy's notion of "false derivative", exact multiple roots were found for special cases of equation (3). The applications of the Lambertian W-function to fundamental physical problems are by no means exhausted even for the normal Lambertian W-function, see (1). This is shown by recent examples from the field of atomic, molecular and optical physics. ${\displaystyle r_{i}}$ ${\displaystyle s_{i}}$ ${\displaystyle x}$ ${\displaystyle r}$

Numerical calculation

A sequence of approximations to the W function can be recursive using the relationship

${\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}(w_{j}+1)-{\frac {(w_{j}+2)(w_{j}e^{w_{j}}-z)}{2w_{j}+2}}}}}$

be calculated. Alternatively, Newton's method can be used to solve the equation : ${\displaystyle we^{w}-z=0}$

${\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}+e^{w_{j}}w_{j}}}}$.

Table of real function values

${\displaystyle W_{0},}$ upper branch:

${\displaystyle {\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}x&-0{,}3679&-0{,}34&-0{,}2&0&0{,}3&0{,}7&1{,}2&2&3&4&6&10&20&40&+\infty \\\hline y&-1&-0{,}6537&-0{,}2592&0&0{,}2368&0{,}4475&0{,}6356&0{,}8526&1{,}0499&1{,}2022&1{,}4324&1{,}7455&2{,}205&2{,}6968&+\infty \\\end{array}}}$

${\displaystyle W_{-1},}$ lower branch:

${\displaystyle {\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}x&-0{,}3679&-0{,}365&-0{,}355&-0{,}31&-0{,}25&-0{,}18&-0{,}1&-0{,}05&-0{,}025&-0{,}01&-0{,}005&-0{,}001&-0{,}0001&0\\\hline y&-1&-1{,}1307&-1{,}2912&-1{,}7044&-2{,}1533&-2{,}7128&-3{,}5772&-4{,}4998&-5{,}3696&-6{,}4728&-7{,}284&-9{,}118&-11{,}6671&-\infty \\\end{array}}}$

Other values ​​can easily be calculated via . ${\displaystyle x=y\,e^{y}}$

An approximation of for large is ${\displaystyle W_{0}(x)}$ ${\displaystyle x}$

${\displaystyle W_{0}(x)\approx \ln(x)-\ln(\ln(x))+\ln(\ln(x))/\ln(x).}$

Individual evidence

1. ^ TC Scott, RB Mann: General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function. In: AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 No. 1, April 2006. pp. 41-47. acm.org ; Arxiv article.
2. ^ TC Scott, G. Fee, J. Grotendorst: Asymptotic series of Generalized Lambert W Function . In: SIGSAM (ACM Special Interest Group in Symbolic and Algebraic Manipulation) . 47, No. 185, 2013, pp. 75-83.  ( Page no longer available , search in web archives )
3. ^ TC Scott, G. Fee, J. Grotendorst, WZ Zhang: Numerics of the Generalized Lambert W Function . In: SIGSAM . 48, No. 188, 2014, pp. 42–56.  ( Page no longer available , search in web archives )
4. ^ PS Farrugia, RB Mann, TC Scott: N-body Gravity and the Schrödinger Equation. In: Class. Quantum Grav. 24, 2007, pp. 4647-4659. doi: 10.1088 / 0264-9381 / 24/18/006 ; Arxiv article.
5. ^ TC Scott, M. Aubert-Frécon, J. Grotendorst: New Approach for the Electronic Energies of the Hydrogen Molecular Ion. In: Chem. Phys. 324: 2006. pp. 323-338. doi: 10.1016 / j.chemphys.2005.10.031 ; Arxiv article.
6. ^ Aude Maignan, TC Scott: Fleshing out the Generalized Lambert W Function . In: SIGSAM . 50, No. 2, 2016, pp. 45-60. doi : 10.1145 / 2992274.2992275 .
7. ^ TC Scott, A. Lüchow, D. Bressanini, JD Morgan III: The Nodal Surfaces of Helium Atom Eigenfunctions. In: Phys. Rev. A. 75: 060101, 2007. scitation.aip.org. ( Memento from July 17, 2012 in the web archive archive.today ).
8. ^ RM Corless et al: On the Lambert W function. ( Memento of December 14, 2010 in the Internet Archive ). (PDF; 304 kB). In: Adv. Computational Maths. 5, 1996, pp. 329-359.
9. ↑ Eric W. Weisstein : Lambert W-Function . In: MathWorld (English).