Movatterモバイル変換

[0]ホーム

Jump to content

LambertW function

Edit links

From Wikipedia, the free encyclopedia

Multivalued function in mathematics

The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i — The product logarithm Lambert W function plotted in the complex plane from−2 − 2i to2 + 2i

The graph of $y=W(x)$ for real $x<6$ and $y>-4$ . The upper branch (blue) with $y\geq -1$ is the graph of the function $W_{0}$ (principal branch), the lower branch (magenta) with $y\leq -1$ is the graph of the function $W_{-1}$ . The minimum value of $x {\displaystyle x}$ is at $\left\{-1/e,-1\right\}$ .

{\displaystyle y=W(x)} — The graph of $y=W(x)$ for real $x<6$ and $y>-4$ . The upper branch (blue) with $y\geq -1$ is the graph of the function $W_{0}$ (principal branch), the lower branch (magenta) with $y\leq -1$ is the graph of the function $W_{-1}$ . The minimum value of $x {\displaystyle x}$ is at $\left\{-1/e,-1\right\}$ .

Inmathematics, theLambert W function, also called theomega function orproduct logarithm,^[1] is amultivalued function, namely thebranches of theconverse relation of the function $f(w)=we^{w}$ , wherew is anycomplex number and $e^{w}$ is theexponential function. The function is named afterJohann Lambert, who considered a related problem in 1758. Building on Lambert's work,Leonhard Euler described the W function per se in 1783.^[2] Despite its early origins and wide use, its properties were not widely recognized until the 1990s thanks primarily to the work of Corless.

For each integer $k {\displaystyle k}$ there is one branch, denoted by $W_{k}\left(z\right)$ , which is a complex-valued function of one complex argument. $W_{0}$ is known as theprincipal branch. These functions have the following property: if $z {\displaystyle z}$ and $w {\displaystyle w}$ are any complex numbers, then

we^{w}=z

holds if and only if

w=W_{k}(z)\ \ {\text{ for some integer }}k.

When dealing with real numbers only, the two branches $W_{0}$ and $W_{-1}$ suffice: for real numbers $x {\displaystyle x}$ and $y {\displaystyle y}$ the equation

ye^{y}=x

can be solved for $y {\displaystyle y}$ only if ${\textstyle x\geq {\frac {-1}{e}}}$ ; yields $y=W_{0}\left(x\right)$ if $x\geq 0$ and the two values $y=W_{0}\left(x\right)$ and $y=W_{-1}\left(x\right)$ if ${\textstyle {\frac {-1}{e}}\leq x<0}$ .

The Lambert W function's branches cannot be expressed in terms ofelementary functions.^[3] It is useful incombinatorics, for instance, in the enumeration oftrees. It can be used to solve various equations involving exponentials (e.g. the maxima of thePlanck,Bose–Einstein, andFermi–Dirac distributions) and also occurs in the solution ofdelay differential equations, such as $y'\left(t\right)=a\ y\left(t-1\right)$ . Inbiochemistry, and in particularenzyme kinetics, an opened-form solution for the time-course kinetics analysis ofMichaelis–Menten kinetics is described in terms of the Lambert W function.

Main branch of the Lambert W function in the complex plane, plotted withdomain coloring. Note thebranch cut along the negative real axis, ending at ${\textstyle -{\frac {1}{e}}}$ .

Main branch of the Lambert W function in the complex plane, plotted withdomain coloring. Note thebranch cut along the negative real axis, ending at ${\textstyle -{\frac {1}{e}}}$ .

The modulus of the principal branch of the Lambert W function, colored according to $\arg W\left(z\right)$

{\displaystyle \arg W\left(z\right)} — The modulus of the principal branch of the Lambert W function, colored according to $\arg W\left(z\right)$

Terminology

[edit]

The notation convention chosen here (with $W_{0}$ and $W_{-1}$ ) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey andKnuth.^[4]

The name "product logarithm" can be understood as follows: since theinverse function of $f\left(w\right)=e^{w}$ is termed thelogarithm, it makes sense to call the inverse "function" of theproduct $we^{w}$ the "product logarithm". (Technical note: like thecomplex logarithm, it is multivalued and thus W is described as aconverse relation rather than inverse function.) It is related to theomega constant, which is equal to $W_{0}\left(1\right)$ .

History

[edit]

Lambert first considered the relatedLambert's Transcendental Equation in 1758,^[5] which led to an article byLeonhard Euler in 1783^[6] that discussed the special case of $we^{w}$ .

The equation Lambert considered was

x=x^{m}+q.

Euler transformed this equation into the form

x^{a}-x^{b}=(a-b)cx^{a+b}.

Both authors derived a series solution for their equations.

Once Euler had solved this equation, he considered the case⁠ $a=b$ ⁠. Taking limits, he derived the equation

\ln x=cx^{a}.

He then put⁠ $a=1$ ⁠ and obtained a convergent series solution for the resulting equation, expressing⁠ $x {\displaystyle x}$ ⁠ in terms of ⁠ $c {\displaystyle c}$ ⁠.

After taking derivatives with respect to⁠ $x {\displaystyle x}$ ⁠ and some manipulation, the standard form of the Lambert function is obtained.

In 1993, it was reported that the Lambert⁠ $W {\displaystyle W}$ ⁠ function provides an exact solution to the quantum-mechanicaldouble-well Dirac delta function model for equal charges^[7]—a fundamental problem in physics. Prompted by this, Rob Corless and developers of theMaple computer algebra system realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."^[4]^[8]

Another example where this function is found is inMichaelis–Menten kinetics.^[9]

Although it was widely believed that the Lambert⁠ $W {\displaystyle W}$ ⁠ function cannot be expressed in terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008.^[10]

Elementary properties, branches and range

[edit]

The range of theW function, showing all branches. The black curves (including the real axis) form the image of the real axis, the orange curves are the image of the imaginary axis. The purple curve and circle are the image of a small circle around the pointz = 0; the red curves are the image of a small circle around the pointz = −1/e.

Plot of the imaginary part ofW_n(x +iy) for branchesn = −2, −1, 0, 1, 2. The plot is similar to that of the multivaluedcomplex logarithm function except that the spacing between sheets is not constant and the connection of the principal sheet is different

There are countably many branches of theW function, denoted byW_k(z), for integerk;W₀(z) being the main (or principal) branch.W₀(z) is defined for all complex numbersz whileW_k(z) withk ≠ 0 is defined for all non-zeroz, withW₀(0) = 0 and $\lim \limits _{z\to 0}W_{k}(z)=\;-\infty ,$ for allk ≠ 0.

The branch point for the principal branch is at $z=-e^{-1}$ , with the standard branch cut extending along the negative real axis to−∞+0i. This branch cut separates the principal branch from the two branchesW₋₁ andW₁. In all branchesW_k withk ≠ 0, there is a branch point atz = 0 and a branch cut is conventionally taken along the entire negative real axis.

The functionsW_k(z),k ∈Z are allinjective and their ranges are disjoint. The range of the entire multivalued functionW is the complex plane. The image of the real axis is the union of the real axis and thequadratrix of Hippias, the parametric curvew = −t cott +it.

Inverse

[edit]

Regions of the complex plane for whichW(n,ze^z) =z, wherez =x +iy. The darker boundaries of a particular region are included in the lighter region of the same color. The point at{−1, 0} is included in both then = −1 (blue) region and then = 0 (gray) region. Horizontal grid lines are in multiples ofπ.

The range plot above also delineates the regions in the complex plane where the simple inverse relationship⁠ $W(n,ze^{z})=z$ ⁠ is true.⁠ $f=ze^{z}$ ⁠ implies that there exists an⁠ $n {\displaystyle n}$ ⁠ such that⁠ $z=W(n,f)=W(n,ze^{z})$ ⁠, where⁠ $n {\displaystyle n}$ ⁠ depends upon the value of⁠ $z {\displaystyle z}$ ⁠. The value of the integer⁠ $n {\displaystyle n}$ ⁠ changes abruptly when⁠ $ze^{z}$ ⁠ is at the branch cut of⁠ $W(n,ze^{z})$ ⁠, which means that⁠ $ze^{z}$ ⁠ ≤ 0, except for⁠ $n=0$ ⁠ where it is⁠ $ze^{z}$ ⁠ ≤ −1/⁠ $e {\displaystyle e}$ ⁠.

Defining⁠ $z=x+iy$ ⁠, where⁠ $x {\displaystyle x}$ ⁠ and⁠ $y {\displaystyle y}$ ⁠ are real, and expressing⁠ $e^{z}$ ⁠ in polar coordinates, it is seen that

{\begin{aligned}ze^{z}&=(x+iy)e^{x}(\cos y+i\sin y)\\&=e^{x}(x\cos y-y\sin y)+ie^{x}(x\sin y+y\cos y)\\\end{aligned}}

For $n\neq 0$ , the branch cut for⁠ $W(n,ze^{z})$ ⁠ is the non-positive real axis, so that

x\sin y+y\cos y=0\Rightarrow x=-y/\tan(y),

and

(x\cos y-y\sin y)e^{x}\leq 0.

For $n=0$ , the branch cut for⁠ $W[n,ze^{z}]$ ⁠ is the real axis with $-\infty <z\leq -1/e$ , so that the inequality becomes

(x\cos y-y\sin y)e^{x}\leq -1/e.

Inside the regions bounded by the above, there are no discontinuous changes in⁠ $W(n,ze^{z})$ ⁠, and those regions specify where the⁠ $W {\displaystyle W}$ ⁠ function is simply invertible, i.e.⁠ $W(n,ze^{z})=z$ ⁠.

Transcendence

[edit]

For each algebraic number $z\neq 0$ , the numbers $W_{k}(z)$ are transcendental. This can be proved as follows. Suppose that $W_{k}(z)$ is algebraic. Then by theLindemann–Weierstrass theorem we have $e^{W_{k}(z)}$ is transcendental, but $e^{W_{k}(z)}={\frac {z}{W_{k}(z)}}$ which is algebraic, giving a contradiction.

Calculus

[edit]

Derivative

[edit]

Byimplicit differentiation, one can show that all branches ofW satisfy thedifferential equation

z(1+W){\frac {dW}{dz}}=W\quad {\text{for }}z\neq -{\frac {1}{e}}.

(W is notdifferentiable forz = −⁠1/e⁠.) As a consequence, that gets the following formula for the derivative ofW:

{\frac {dW}{dz}}={\frac {W(z)}{z(1+W(z))}}\quad {\text{for }}z\not \in \left\{0,-{\frac {1}{e}}\right\}.

Using the identitye^W(z) =⁠z/W(z)⁠, gives the following equivalent formula:

{\frac {dW}{dz}}={\frac {1}{z+e^{W(z)}}}\quad {\text{for }}z\neq -{\frac {1}{e}}.

At the origin we have

W'_{0}(0)=1.

The n-th derivative ofW is of the form:

{\frac {d^{n}W}{dz^{n}}}={\frac {P_{n}(W(z))}{(z+e^{W(z)})^{n}(W(z)+1)^{n-1}}}\quad {\text{for }}n>0,\,z\neq -{\frac {1}{e}}.

WhereP_n is a polynomial function with coefficients defined inA042977. If and only ifz is a root ofP_n thenze^z is a root of the n-th derivative ofW.

Taking the derivative of the n-th derivative ofW yields:

{\frac {d^{n+1}W}{dz^{n+1}}}={\frac {(W(z)+1)P_{n}'(W(z))+(1-3n-nW(z))P_{n}(W(z))}{(n+e^{W(z)})^{n+1}(W(z)+1)^{n}}}\quad {\text{for }}n>0,\,z\neq -{\frac {1}{e}}.

Inductively proving the n-th derivative equation.

Integral

[edit]

The functionW(x), and many other expressions involvingW(x), can beintegrated using thesubstitutionw =W(x), i.e.x =we^w:

{\begin{aligned}\int W(x)\,dx&=xW(x)-x+e^{W(x)}+C\\&=x\left(W(x)-1+{\frac {1}{W(x)}}\right)+C.\end{aligned}}

(The last equation is more common in the literature but is undefined atx = 0). One consequence of this (using the fact thatW₀(e) = 1) is the identity

\int _{0}^{e}W_{0}(x)\,dx=e-1.

Asymptotic expansions

[edit]

By theLagrange inversion theorem, theTaylor series of the principal branch $W_{0}(x)$ around $x=0$ is:

W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n}=x-x^{2}+{\tfrac {3}{2}}x^{3}-{\tfrac {16}{6}}x^{4}+{\tfrac {125}{24}}x^{5}-\cdots .

Theradius of convergence is $1/e$ by theratio test, and the function defined by the series can be extended to aholomorphic function defined on all complex numbers except abranch cut along theinterval $(-\infty ,1/e]$ .

For large values $x\to \infty$ , the real function $W_{0}(x)$ is asymptotic to

{\begin{aligned}W_{0}(x)&=L_{1}-L_{2}+{\frac {L_{2}}{L_{1}}}+{\frac {L_{2}\left(-2+L_{2}\right)}{2L_{1}^{2}}}+{\frac {L_{2}\left(6-9L_{2}+2L_{2}^{2}\right)}{6L_{1}^{3}}}+{\frac {L_{2}\left(-12+36L_{2}-22L_{2}^{2}+3L_{2}^{3}\right)}{12L_{1}^{4}}}+\cdots \\[5pt]&=L_{1}-L_{2}+\sum _{\ell =1}^{\infty }\sum _{m=1}^{\ell }{\frac {(-1)^{\ell }\left[{\begin{smallmatrix}\ell \\\ell -m+1\end{smallmatrix}}\right]}{m!}}{\frac {L_{2}^{m}}{L_{1}^{\ell }}},\end{aligned}}

whereL₁ = lnx,L₂ = ln lnx, and[ⁿ
_k] is a non-negativeStirling number of the first kind.^[4] Keeping only the first two terms of the expansion,

W_{0}(x)=\ln x-\ln \ln x+{\mathcal {o}}(1).

The other real branch,W₋₁, defined in the interval[−⁠1/e⁠, 0), has an approximation of the same form asx approaches zero, in this case withL₁ = ln(−x) andL₂ = ln(−ln(−x)).^[4]

Integer and complex powers

[edit]

Integer powers ofW₀ also admit simpleTaylor (orLaurent) series expansions at zero:

W_{0}(x)^{2}=\sum _{n=2}^{\infty }{\frac {-2\left(-n\right)^{n-3}}{(n-2)!}}x^{n}=x^{2}-2x^{3}+4x^{4}-{\tfrac {25}{3}}x^{5}+18x^{6}-\cdots .

More generally, forr ∈Z, theLagrange inversion formula gives

W_{0}(x)^{r}=\sum _{n=r}^{\infty }{\frac {-r\left(-n\right)^{n-r-1}}{(n-r)!}}x^{n},

which is, in general, a Laurent series of orderr. Equivalently, the latter can be written in the form of a Taylor expansion of powers ofW₀(x) /x:

\left({\frac {W_{0}(x)}{x}}\right)^{r}=e^{-rW_{0}(x)}=\sum _{n=0}^{\infty }{\frac {r\left(n+r\right)^{n-1}}{n!}}\left(-x\right)^{n},

which holds for anyr ∈C and|x| <⁠1/e⁠.

Bounds and inequalities

[edit]

A number of non-asymptotic bounds are known for the Lambert function.

Principal branch

[edit]

Hoorfar and Hassani^[11] showed that the following bound holds forx ≥e:

\ln x-\ln \ln x+{\frac {\ln \ln x}{2\ln x}}\leq W_{0}(x)\leq \ln x-\ln \ln x+{\frac {e}{e-1}}{\frac {\ln \ln x}{\ln x}}.

Roberto Iacono and John P. Boyd^[12] enhanced the bounds forx ≥e as follows:

\ln \left({\frac {x}{\ln x}}\right)-{\frac {\ln \left({\frac {x}{\ln x}}\right)}{1+\ln \left({\frac {x}{\ln x}}\right)}}\ln \left(1-{\frac {\ln \ln x}{\ln x}}\right)\leq W_{0}(x)\leq \ln \left({\frac {x}{\ln x}}\right)-\ln \left(\left(1-{\frac {\ln \ln x}{\ln x}}\right)\left(1-{\frac {\ln \left(1-{\frac {\ln \ln x}{\ln x}}\right)}{1+\ln \left({\frac {x}{\ln x}}\right)}}\right)\right).

Hoorfar and Hassani^[11] also showed the general bound

W_{0}(x)\leq \ln \left({\frac {x+y}{1+\ln(y)}}\right),

for every $y>1/e$ and $x\geq -1/e$ , with equality only for $x=y\ln(y)$ .The bound allows many other bounds to be derived, such as taking $y=x+1$ which gives the bound

W_{0}(x)\leq \ln \left({\frac {2x+1}{1+\ln(x+1)}}\right).

Bounds for the function $W_{0}(-xe^{-x})$ for $x\geq 1$ are obtained by Stewart.^[13]

Secondary branch

[edit]

The branchW₋₁ can be bounded as follows:^[14]

-1-{\sqrt {2u}}-u<W_{-1}\left(-e^{-u-1}\right)<-1-{\sqrt {2u}}-{\tfrac {2}{3}}u\quad {\text{for }}u>0.

Identities

[edit]

A plot ofW_j(xe^x) where blue is forj = 0 and red is forj = −1. The diagonal line represents the intervals whereW_j(xe^x) =x.

The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i — The product logarithm LambertW functionW₂(z) plotted in the complex plane from−2 − 2i to2 + 2i

A few identities follow from the definition:

{\begin{aligned}W_{0}(xe^{x})&=x&{\text{for }}x&\geq -1,\\W_{-1}(xe^{x})&=x&{\text{for }}x&\leq -1.\end{aligned}}

Sincef(x) =xe^x is notinjective, it does not always hold thatW(f(x)) =x, much like with theinverse trigonometric functions. For fixedx < 0 andx ≠ −1, the equationxe^x =ye^y has two real solutions iny, one of which is of coursey =x. Then, fori = 0 andx < −1, as well as fori = −1 andx ∈ (−1, 0),y =W_i(xe^x) is the other solution.

Some other identities:^[15]

{\begin{aligned}&W(x)e^{W(x)}=x,\quad {\text{therefore:}}\\[5pt]&e^{W(x)}={\frac {x}{W(x)}},\qquad e^{-W(x)}={\frac {W(x)}{x}},\qquad e^{nW(x)}=\left({\frac {x}{W(x)}}\right)^{n}.\end{aligned}}

\ln W_{0}(x)=\ln x-W_{0}(x)\quad {\text{for }}x>0.

^[16]

W_{0}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{0}\left(x\ln x\right)}=x\quad {\text{for }}{\frac {1}{e}}\leq x.

W_{-1}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{-1}\left(x\ln x\right)}=x\quad {\text{for }}0<x\leq {\frac {1}{e}}.

{\begin{aligned}&W(x)=\ln {\frac {x}{W(x)}}&&{\text{for }}x\geq -{\frac {1}{e}},\\[5pt]&W\left({\frac {nx^{n}}{W\left(x\right)^{n-1}}}\right)=nW(x)&&{\text{for }}n,x>0\end{aligned}}

(which can be extended to othern andx if the correct branch is chosen).

W(x)+W(y)=W\left(xy\left({\frac {1}{W(x)}}+{\frac {1}{W(y)}}\right)\right)\quad {\text{for }}x,y>0.

Substituting−lnx in the definition:^[17]

{\begin{aligned}W_{0}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}0&<x\leq e,\\[5pt]W_{-1}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}x&>e.\end{aligned}}

With Euler's iterated exponentialh(x):

{\begin{aligned}h(x)&=e^{-W(-\ln x)}\\&={\frac {W(-\ln x)}{-\ln x}}\quad {\text{for }}x\neq 1.\end{aligned}}

$\forall c\in \left[-{\frac {1}{e}},0\right),{\text{let }}t={\frac {W_{-1}(c)}{W_{0}(c)}}\geq 1\implies W_{0}(c)={\frac {\ln t}{1-t}},W_{-1}(c)={\frac {t\ln t}{1-t}}$

Special values

[edit]

The following^[18]^[19]^[20] are special values of the principal branch: $W_{0}\left(-{\frac {\pi }{2}}\right)={\frac {i\pi }{2}}$ $W_{0}\left(-{\frac {1}{e}}\right)=-1$ $W_{0}\left(2\ln 2\right)=\ln 2$ $W_{0}\left(x\ln x\right)=\ln x\quad \left(x\geqslant {\tfrac {1}{e}}\approx 0.36788\right)$ $W_{0}\left(x^{x+1}\ln x\right)=x\ln x\quad \left(x>0\right)$ $W_{0}(0)=0$

W_{0}(1)=\Omega \approx 0.56714329\quad

(theomega constant)

$W_{0}(1)=e^{-W_{0}(1)}=\ln {\frac {1}{W_{0}(1)}}=-\ln W_{0}(1)$ $W_{0}(e)=1$ $W_{0}\left(e^{1+e}\right)=e$ $W_{0}\left({\frac {\sqrt {e}}{2}}\right)={\frac {1}{2}}$ $W_{0}\left({\frac {\sqrt[{n}]{e}}{n}}\right)={\frac {1}{n}}$ $W_{0}(-1)\approx -0.31813+1.33723i$

Special values of the branchW₋₁: $W_{-1}\left(-{\frac {\ln 2}{2}}\right)=-\ln 4$

Representations

[edit]

The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:^[21]

-{\frac {\pi }{2}}W_{0}(-x)=\int _{0}^{\pi }{\frac {\sin \left({\tfrac {3}{2}}t\right)-xe^{\cos t}\sin \left({\tfrac {5}{2}}t-\sin t\right)}{1-2xe^{\cos t}\cos(t-\sin t)+x^{2}e^{2\cos t}}}\sin \left({\tfrac {1}{2}}t\right)\,dt\quad {\text{for }}|x|<{\frac {1}{e}}.

Another representation of the principal branch was found by Kalugin–Jeffrey–Corless:^[22]

W_{0}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\ln \left(1+x{\frac {\sin t}{t}}e^{t\cot t}\right)dt.

The followingcontinued fraction representation also holds for the principal branch:^[23]

W_{0}(x)={\cfrac {x}{1+{\cfrac {x}{1+{\cfrac {x}{2+{\cfrac {5x}{3+{\cfrac {17x}{10+{\cfrac {133x}{17+{\cfrac {1927x}{190+{\cfrac {13582711x}{94423+\ddots }}}}}}}}}}}}}}}}.

Also, if|W₀(x)| < 1:^[24]

W_{0}(x)={\cfrac {x}{\exp {\cfrac {x}{\exp {\cfrac {x}{\ddots }}}}}}.

In turn, if|W₀(x)| > 1, then

W_{0}(x)=\ln {\cfrac {x}{\ln {\cfrac {x}{\ln {\cfrac {x}{\ddots }}}}}}.

Other formulas

[edit]

Definite integrals

[edit]

There are several useful definite integral formulas involving the principal branch of theW function, including the following:

{\begin{aligned}&\int _{0}^{\pi }W_{0}\left(2\cot ^{2}x\right)\sec ^{2}x\,dx=4{\sqrt {\pi }},\\[5pt]&\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx=2{\sqrt {2\pi }},\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{2}}}\right)\,dx={\sqrt {2\pi }},{\text{ and more generally}}\\[5pt]&\int _{0}^{\infty }W_{0}\left({\frac {1}{x^{N}}}\right)\,dx=N^{1-{\frac {1}{N}}}\Gamma \left(1-{\frac {1}{N}}\right)\qquad {\text{for }}N>1\end{aligned}}

where $\Gamma$ denotes thegamma function.

The first identity can be found by writing theGaussian integral inpolar coordinates.

The second identity can be derived by making the substitutionu =W₀(x), which gives

{\begin{aligned}x&=ue^{u},\\[5pt]{\frac {dx}{du}}&=(u+1)e^{u}.\end{aligned}}

Thus

{\begin{aligned}\int _{0}^{\infty }{\frac {W_{0}(x)}{x{\sqrt {x}}}}\,dx&=\int _{0}^{\infty }{\frac {u}{ue^{u}{\sqrt {ue^{u}}}}}(u+1)e^{u}\,du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {ue^{u}}}}du\\[5pt]&=\int _{0}^{\infty }{\frac {u+1}{\sqrt {u}}}{\frac {1}{\sqrt {e^{u}}}}du\\[5pt]&=\int _{0}^{\infty }u^{\tfrac {1}{2}}e^{-{\frac {u}{2}}}du+\int _{0}^{\infty }u^{-{\tfrac {1}{2}}}e^{-{\frac {u}{2}}}du\\[5pt]&=2\int _{0}^{\infty }(2w)^{\tfrac {1}{2}}e^{-w}\,dw+2\int _{0}^{\infty }(2w)^{-{\tfrac {1}{2}}}e^{-w}\,dw&&\quad (u=2w)\\[5pt]&=2{\sqrt {2}}\int _{0}^{\infty }w^{\tfrac {1}{2}}e^{-w}\,dw+{\sqrt {2}}\int _{0}^{\infty }w^{-{\tfrac {1}{2}}}e^{-w}\,dw\\[5pt]&=2{\sqrt {2}}\cdot \Gamma \left({\tfrac {3}{2}}\right)+{\sqrt {2}}\cdot \Gamma \left({\tfrac {1}{2}}\right)\\[5pt]&=2{\sqrt {2}}\left({\tfrac {1}{2}}{\sqrt {\pi }}\right)+{\sqrt {2}}\left({\sqrt {\pi }}\right)\\[5pt]&=2{\sqrt {2\pi }}.\end{aligned}}

The third identity may be derived from the second by making the substitutionu =x⁻² and the first can also be derived from the third by the substitutionz =⁠1/√2⁠ tanx. Deriving its generalization, the fourth identity, is only slightly more involved and can be done by substituting, in turn, $u={\frac {1}{x^{N}}}$ , $t=W_{0}(u)$ , and $z={\frac {t}{N}}$ , observing that one obtains two integrals matching the definition of the gamma function, and finally using the properties of the gamma function to collect terms and simplify.

Except forz along the branch cut(−∞, −⁠1/e⁠] (where the integral does not converge), the principal branch of the LambertW function can be computed by the following integral:^[25]

{\begin{aligned}W_{0}(z)&={\frac {z}{2\pi }}\int _{-\pi }^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \left(\nu \right)e^{-\nu \cot \nu }}}\,d\nu \\[5pt]&={\frac {z}{\pi }}\int _{0}^{\pi }{\frac {\left(1-\nu \cot \nu \right)^{2}+\nu ^{2}}{z+\nu \csc \left(\nu \right)e^{-\nu \cot \nu }}}\,d\nu ,\end{aligned}}

where the two integral expressions are equivalent due to the symmetry of the integrand.

Indefinite integrals

[edit]

$\int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C$

1st proof

Introduce substitution variable $u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{du}}ue^{u}=(u+1)e^{u}$

\int {\frac {W(x)}{x}}\,dx\;=\;\int {\frac {u}{ue^{u}}}(u+1)e^{u}\,du

\int {\frac {W(x)}{x}}\,dx\;=\;\int {\frac {\cancel {\color {OliveGreen}{u}}}{{\cancel {\color {OliveGreen}{u}}}{\cancel {\color {BrickRed}{e^{u}}}}}}\left(u+1\right){\cancel {\color {BrickRed}{e^{u}}}}\,du

\int {\frac {W(x)}{x}}\,dx\;=\;\int (u+1)\,du

\int {\frac {W(x)}{x}}\,dx\;=\;{\frac {u^{2}}{2}}+u+C

u=W(x)

\int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C

2nd proof

$W(x)e^{W(x)}=x\rightarrow {\frac {W(x)}{x}}=e^{-W(x)}$

$\int {\frac {W(x)}{x}}\,dx\;=\;\int e^{-W(x)}\,dx$

u=W(x)\rightarrow ue^{u}=x\;\;\;\;{\frac {d}{\,du}}ue^{u}=\left(u+1\right)e^{u}

$\int {\frac {W(x)}{x}}\,dx\;=\;\int e^{-u}(u+1)e^{u}\,du$

$\int {\frac {W(x)}{x}}\,dx\;=\;\int {\cancel {\color {OliveGreen}{e^{-u}}}}\left(u+1\right){\cancel {\color {OliveGreen}{e^{u}}}}\,du$

$\int {\frac {W(x)}{x}}\,dx\;=\;\int (u+1)\,du$

$\int {\frac {W(x)}{x}}\,dx\;=\;{\frac {u^{2}}{2}}+u+C$

u=W(x)

$\int {\frac {W(x)}{x}}\,dx\;=\;{\frac {W(x)^{2}}{2}}+W(x)+C$

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{Bx}\right)^{2}}{2B}}+{\frac {W\left(Ae^{Bx}\right)}{B}}+C$

Proof

$\int W\left(Ae^{Bx}\right)\,dx\;=\;\int W\left(Ae^{Bx}\right)\,dx$

u=Bx\rightarrow {\frac {u}{B}}=x\;\;\;\;{\frac {d}{du}}{\frac {u}{B}}={\frac {1}{B}}

$\int W\left(Ae^{Bx}\right)\,dx\;=\;\int W\left(Ae^{u}\right){\frac {1}{B}}du$

v=e^{u}\rightarrow \ln \left(v\right)=u\;\;\;\;{\frac {d}{dv}}\ln \left(v\right)={\frac {1}{v}}

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {W\left(Av\right)}{v}}dv$

w=Av\rightarrow {\frac {w}{A}}=v\;\;\;\;{\frac {d}{dw}}{\frac {w}{A}}={\frac {1}{A}}

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {{\cancel {\color {OliveGreen}{A}}}W(w)}{w}}{\cancel {\color {OliveGreen}{\frac {1}{A}}}}dw$

t=W\left(w\right)\rightarrow te^{t}=w\;\;\;\;{\frac {d}{dt}}te^{t}=\left(t+1\right)e^{t}

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {t}{te^{t}}}\left(t+1\right)e^{t}dt$

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int {\frac {\cancel {\color {OliveGreen}{t}}}{{\cancel {\color {OliveGreen}{t}}}{\cancel {\color {BrickRed}{e^{t}}}}}}\left(t+1\right){\cancel {\color {BrickRed}{e^{t}}}}dt$

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {1}{B}}\int (t+1)dt$

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {t^{2}}{2B}}+{\frac {t}{B}}+C$

t=W\left(w\right)

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(w\right)^{2}}{2B}}+{\frac {W\left(w\right)}{B}}+C$

w=Av

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Av\right)^{2}}{2B}}+{\frac {W\left(Av\right)}{B}}+C$

v=e^{u}

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{u}\right)^{2}}{2B}}+{\frac {W\left(Ae^{u}\right)}{B}}+C$

u=Bx

$\int W\left(Ae^{Bx}\right)\,dx\;=\;{\frac {W\left(Ae^{Bx}\right)^{2}}{2B}}+{\frac {W\left(Ae^{Bx}\right)}{B}}+C$

$\int {\frac {W(x)}{x^{2}}}\,dx\;=\;\operatorname {Ei} \left(-W(x)\right)-e^{-W(x)}+C$

Proof

Introduce substitution variable $u=W(x)$ , which gives us $ue^{u}=x$ and ${\frac {d}{du}}ue^{u}=\left(u+1\right)e^{u}$

${\begin{aligned}\int {\frac {W(x)}{x^{2}}}\,dx\;&=\;\int {\frac {u}{\left(ue^{u}\right)^{2}}}\left(u+1\right)e^{u}du\\&=\;\int {\frac {u+1}{ue^{u}}}du\\&=\;\int {\frac {u}{ue^{u}}}du\;+\;\int {\frac {1}{ue^{u}}}du\\&=\;\int e^{-u}du\;+\;\int {\frac {e^{-u}}{u}}du\end{aligned}}$

v=-u\rightarrow -v=u\;\;\;\;{\frac {d}{dv}}-v=-1

$\int {\frac {W(x)}{x^{2}}}\,dx\;=\;\int e^{v}\left(-1\right)dv\;+\;\int {\frac {e^{-u}}{u}}du$

$\int {\frac {W(x)}{x^{2}}}\,dx\;=\;-e^{v}+\operatorname {Ei} \left(-u\right)+C$

v=-u

$\int {\frac {W(x)}{x^{2}}}\,dx\;=\;-e^{-u}+\operatorname {Ei} \left(-u\right)+C$

u=W(x)

${\begin{aligned}\int {\frac {W(x)}{x^{2}}}\,dx\;&=\;-e^{-W(x)}+\operatorname {Ei} \left(-W(x)\right)+C\\&=\;\operatorname {Ei} \left(-W(x)\right)-e^{-W(x)}+C\end{aligned}}$

Applications

[edit]

Solving equations

[edit]

General case

[edit]

The LambertW function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm. The strategy is to convert such an equation into one of the formze^z =w and then to solve forz using theW function.

For example, the equation

3^{x}=2x+2

(wherex is an unknownreal number) can be solved by rewriting it as

{\begin{aligned}&(x+1)\ 3^{-x}={\frac {1}{2}}&({\mbox{multiply by }}3^{-x}/2)\\\Leftrightarrow \ &(-x-1)\ 3^{-x-1}=-{\frac {1}{6}}&({\mbox{multiply by }}{-}1/3)\\\Leftrightarrow \ &(\ln 3)(-x-1)\ e^{(\ln 3)(-x-1)}=-{\frac {\ln 3}{6}}&({\mbox{multiply by }}\ln 3)\end{aligned}}

This last equation has the desired form and the solutions for realx are:

(\ln 3)(-x-1)=W_{0}\left({\frac {-\ln 3}{6}}\right)\ \ \ {\textrm {or}}\ \ \ (\ln 3)(-x-1)=W_{-1}\left({\frac {-\ln 3}{6}}\right)

and thus:

x=-1-{\frac {W_{0}\left(-{\frac {\ln 3}{6}}\right)}{\ln 3}}=-0.79011\ldots \ \ {\textrm {or}}\ \ x=-1-{\frac {W_{-1}\left(-{\frac {\ln 3}{6}}\right)}{\ln 3}}=1.44456\ldots

Generally, the solution to

x=a+b\,e^{cx}

is:

x=a-{\frac {1}{c}}W(-bc\,e^{ac})

wherea,b, andc are complex constants, withb andc not equal to zero, and theW function is of any integer order.

Super root

[edit]

Along with the topic of the so calledSophomore's dream thetetration function $f(x)=x^{x}$ became a well known function. Its inverse function is a special case of the so called super root and it can be determined and displayed as follows:

x^{x}=y

The power rule gives following expression:

\exp[x\ln(x)]=y

The natural logarithm of that is taken:

x\ln(x)=\ln(y)

The Lambert W function is used now:

\ln(x)=W_{0}[\ln(y)]

And in the final step the second last equation will be divided by the last equation:

x={\frac {\ln(y)}{W_{0}[\ln(y)]}}

A calculation example is made:

x^{x}=2

x=\ln(2)\div W_{0}[\ln(2)]\approx 1.559610469462369349970388768765

Tree counting and combinatorics

[edit]

Cayley's formula states that the number oftree graphs onn labeled vertices is $n^{n-2}$ , so that the number of trees with a designated root vertex is $n^{n-1}$ . Theexponential generating function of this counting sequence is:

$T(x)=\sum _{n=0}^{\infty }{\frac {n^{n-1}}{n!}}x^{n}.$

The class of rooted trees has a natural recurrence: a rooted tree is equivalent to a root vertex attached to a set of smaller rooted trees. Using theexponential formula for labeled combinatorial classes,^[26] this translates into the equation:

$T(x)=xe^{T(x)},$

which implies $-T(-x)e^{-T(-x)}=x$ and

$W_{0}(x)=-T(-x)$ .

Reversing the argument, theMaclaurin series of $W_{0}(x)$ around $x=0$ can be found directly using theLagrange inversion theorem:

W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n},

and this gives the standardanalytic proof of Cayley's formula. But the Maclaurin series radius of convergenceis limited to $|x|<1/e$ because of the branch point at $x=-1/e$ .

Inviscid flows

[edit]

Applying the unusual acceleratingtraveling-wave Ansatz in the form of $\rho (\eta )=\rho {\big (}x-{\frac {at^{2}}{2}}{\big )}$ (where $\rho$ , $\eta$ , a, x and t are the density, the reduced variable, the acceleration, the spatial and the temporal variables) the fluiddensity of the correspondingEuler equation can be given with the help of the W function.^[27]

Viscous flows

[edit]

Granular anddebris flow fronts and deposits, and the fronts of viscous fluids in natural events and in laboratory experiments can be described by using the Lambert–Euler omega function as follows:

H(x)=1+W\left((H(0)-1)e^{(H(0)-1)-{\frac {x}{L}}}\right),

whereH(x) is the debris flow height,x is the channel downstream position,L is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.

Inpipe flow, the Lambert W function is part of the explicit formulation of theColebrook equation for finding theDarcy friction factor. This factor is used to determine the pressure drop through a straight run of pipe when the flow isturbulent.^[28]

Time-dependent flow in simple branch hydraulic systems

[edit]

The principal branch of the LambertW function is employed in the field ofmechanical engineering, in the study of time dependent transfer ofNewtonian fluids between two reservoirs with varying free surface levels, using centrifugal pumps.^[29] The LambertW function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes: ${\begin{aligned}Q_{\text{turb}}&={\frac {Q_{i}}{\zeta _{i}}}W_{0}\left[\zeta _{i}\,e^{(\zeta _{i}+\beta t/b)}\right]\\Q_{\text{lam}}&={\frac {Q_{i}}{\xi _{i}}}W_{0}\left[\xi _{i}\,e^{\left(\xi _{i}+\beta t/(b-\Gamma _{1})\right)}\right]\end{aligned}}$ where $Q_{i}$ is the initial flow rate and $t {\displaystyle t}$ is time.

Neuroimaging

[edit]

The LambertW function is employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brainvoxel, to the corresponding blood oxygenation level dependent (BOLD) signal.^[30]

Chemical engineering

[edit]

The LambertW function is employed in the field of chemical engineering for modeling the porous electrode film thickness in aglassy carbon basedsupercapacitor for electrochemical energy storage. The LambertW function provides an exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.^[31]^[32]

Crystal growth

[edit]

In thecrystal growth, the negative principal of the Lambert W-function can be used to calculate the distribution coefficient, ${\textstyle k}$ , and solute concentration in the melt, ${\textstyle C_{L}}$ ,^[33]^[34] from theScheil equation:

{\begin{aligned}&k={\frac {W_{0}(Z)}{\ln(1-fs)}}\\&C_{L}={\frac {C_{0}}{(1-fs)}}e^{W_{0}(Z)}\\&Z={\frac {C_{S}}{C_{0}}}(1-fs)\ln(1-fs)\end{aligned}}

Materials science

[edit]

The LambertW function is employed in the field ofepitaxial film growth for the determination of the criticaldislocation onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise theelastic energy stored in the films. Prior to application of LambertW for this problem, the critical thickness had to be determined via solving an implicit equation. LambertW turns it in an explicit equation for analytical handling with ease.^[35]

Semiconductor devices

[edit]

It was shown that a W-function describes the relation between voltage, current and resistance in a diode.^[36]

The use of the LambertW Function to analytically and exactly solve the terminals' current and voltage as explicit functions of each other in a circuit model of a diode with both series and shunt resistances was first reported in the year 2000.^[37]

The LambertW Function was introduced into compact modeling of MOSFETs in 2003 as a useful mathematical tool to explicitly describe the surface potential in undoped channels.^[38]

The LambertW function-based explicit analytic solution of the illuminated photovoltaic solar cell single-diode model with parasitic series and shunt resistance was published in 2004.^[39]

Porous media

[edit]

The LambertW function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneous tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the −1 branch applies if the displacement is unstable with the heavier fluid running underneath the lighter fluid.^[40]

Bernoulli numbers and Todd genus

[edit]

The equation (linked with the generating functions ofBernoulli numbers andTodd genus):

Y={\frac {X}{1-e^{X}}}

can be solved by means of the two real branchesW₀ andW₋₁:

X(Y)={\begin{cases}W_{-1}\left(Ye^{Y}\right)-W_{0}\left(Ye^{Y}\right)=Y-W_{0}\left(Ye^{Y}\right)&{\text{for }}Y<-1,\\W_{0}\left(Ye^{Y}\right)-W_{-1}\left(Ye^{Y}\right)=Y-W_{-1}\left(Ye^{Y}\right)&{\text{for }}-1<Y<0.\end{cases}}

This application shows that the branch difference of theW function can be employed in order to solve other transcendental equations.^[41]

Statistics

[edit]

The centroid of a set of histograms defined with respect to the symmetrizedKullback–Leibler divergence (also called the Jeffreys divergence^[42]) has a closed form using the LambertW function.^[43]

Pooling of tests for infectious diseases

[edit]

Solving for the optimal group size to pool tests so that at least one individual is infected involves the LambertW function.^[44]^[45]^[46]

Exact solutions of the Schrödinger equation

[edit]

The LambertW function appears in a quantum-mechanical potential, which affords the fifth – next to those of theharmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and theinverse square root potential – exact solution to the stationary one-dimensionalSchrödinger equation in terms of theconfluent hypergeometric functions. The potential is given as

V={\frac {V_{0}}{1+W\left(e^{-{\frac {x}{\sigma }}}\right)}}.

A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to^[47]

z=W\left(e^{-{\frac {x}{\sigma }}}\right).

The LambertW function also appears in the exact solution for thebound state energy of the one dimensional Schrödinger equation with aDouble Delta Potential.

Exact solution of QCD coupling constant

[edit]

InQuantum chromodynamics, thequantum field theory of theStrong interaction, thecoupling constant $\alpha _{\text{s}}$ is computed perturbatively, the order n corresponding toFeynman diagrams including n quantum loops.^[48] The first order,n = 1, solution is exact (at that order) and analytical. At higher orders,n > 1, there is no exact and analytical solution and one typically uses aniterative method to furnish an approximate solution. However, for second order,n = 2, the Lambert function provides an exact (if non-analytical) solution.^[48]

Exact solutions of the Einstein vacuum equations

[edit]

In theSchwarzschild metric solution of the Einstein vacuum equations, theW function is needed to go from theEddington–Finkelstein coordinates to theSchwarzschild coordinates. For this reason, it also appears in the construction of theKruskal–Szekeres coordinates.

Resonances of the delta-shell potential

[edit]

The s-wave resonances of the delta-shell potential can be written exactly in terms of the LambertW function.^[49]

Thermodynamic equilibrium

[edit]

If a reaction involves reactants and products havingheat capacities that are constant with temperature then the equilibrium constantK obeys

\ln K={\frac {a}{T}}+b+c\ln T

for some constantsa,b, andc. Whenc (equal to⁠ΔC_p/R⁠) is not zero the value or values ofT can be found whereK equals a given value as follows, whereL can be used forlnT.

{\begin{aligned}-a&=(b-\ln K)T+cT\ln T\\&=(b-\ln K)e^{L}+cLe^{L}\\[5pt]-{\frac {a}{c}}&=\left({\frac {b-\ln K}{c}}+L\right)e^{L}\\[5pt]-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}&=\left(L+{\frac {b-\ln K}{c}}\right)e^{L+{\frac {b-\ln K}{c}}}\\[5pt]L&=W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\\[5pt]T&=\exp \left(W\left(-{\frac {a}{c}}e^{\frac {b-\ln K}{c}}\right)+{\frac {\ln K-b}{c}}\right).\end{aligned}}

Ifa andc have the same sign there will be either two solutions or none (or one if the argument ofW is exactly−⁠1/e⁠). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.

Phase separation of polymer mixtures

[edit]

In the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to theEdmond-Ogston model, the solutions for binodal and tie-lines are formulated in terms of LambertW functions.^[50]

Wien's displacement law in aD-dimensional universe

[edit]

Wien's displacement law is expressed as $\nu _{\max }/T=\alpha =\mathrm {const}$ . With $x=h\nu _{\max }/k_{\mathrm {B} }T$ and $d\rho _{T}\left(x\right)/dx=0$ , where $\rho _{T}$ is the spectral energy energy density, one finds $e^{-x}=1-{\frac {x}{D}}$ , where $D {\displaystyle D}$ is the number of degrees of freedom for spatial translation. The solution $x=D+W\left(-De^{-D}\right)$ shows that the spectral energy density is dependent on the dimensionality of the universe.^[51]

AdS/CFT correspondence

[edit]

The classical finite-size corrections to the dispersion relations ofgiant magnons, single spikes andGKP strings can be expressed in terms of the LambertW function.^[52]^[53]

Epidemiology

[edit]

In thet → ∞ limit of theSIR model, the proportion of susceptible and recovered individuals has a solution in terms of the LambertW function.^[54]

Determination of the time of flight of a projectile

[edit]

The total time of the journey of a projectile which experiences air resistance proportional to its velocitycan be determined in exact form by using the LambertW function.^[55]

Electromagnetic surface wave propagation

[edit]

The transcendental equation that appears in the determination of the propagation wave number of an electromagnetic axially symmetric surface wave (a low-attenuation single TM01 mode) propagating in a cylindrical metallic wire gives rise to an equation likeu lnu =v (whereu andv clump together the geometrical and physical factors of the problem), which is solved by the LambertW function. The first solution to this problem, due to Sommerfeldcirca 1898, already contained an iterative method to determine the value of the LambertW function.^[56]

Orthogonal trajectories of real ellipses

[edit]

The family of ellipses $x^{2}+(1-\varepsilon ^{2})y^{2}=\varepsilon ^{2}$ centered at $(0,0)$ is parameterized by eccentricity $\varepsilon$ . The orthogonal trajectories of this family are given by the differential equation $\left({\frac {1}{y}}+y\right)dy=\left({\frac {1}{x}}-x\right)dx$ whose general solution is the family $y^{2}=$ $W_{0}(x^{2}\exp(-2C-x^{2}))$ .

Generalizations

[edit]

The standard LambertW function expresses exact solutions totranscendental algebraic equations (inx) of the form:

e^{-cx}=a_{0}(x-r)

wherea₀,c andr are real constants. The solution is $x=r+{\frac {1}{c}}W\left({\frac {c\,e^{-cr}}{a_{0}}}\right).$ Generalizations of the LambertW function^[57]^[58]^[59] include:

An application togeneral relativity andquantum mechanics (quantum gravity) in lower dimensions, in fact a link (unknown prior to 2007^[60]) between these two areas, where the right-hand side of (1) is replaced by a quadratic polynomial inx:
$e^{-cx}=a_{0}\left(x-r_{1}\right)\left(x-r_{2}\right),$ 2
wherer₁ andr₂ are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argumentx but the terms liker_i anda₀ are parameters of that function. In this respect, the generalization resembles thehypergeometric function and theMeijerG function but it belongs to a differentclass of functions. Whenr₁ =r₂, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standardW function. Equation (2) expresses the equation governing thedilaton field, from which is derived the metric of theR =T orlineal two-body gravity problem in 1 + 1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanicaldouble-well Dirac delta function model forunequal charges in one dimension.
Analytical solutions of the eigenenergies of a special case of the quantum mechanicalthree-body problem, namely the (three-dimensional)hydrogen molecule-ion.^[61] Here the right-hand side of (1) is replaced by a ratio of infinite order polynomials inx:
$e^{-cx}=a_{0}{\frac {\displaystyle \prod _{i=1}^{\infty }(x-r_{i})}{\displaystyle \prod _{i=1}^{\infty }(x-s_{i})}}$ 3
wherer_i ands_i are distinct real constants andx is a function of the eigenenergy and the internuclear distanceR. Equation (3) with its specialized cases expressed in (1) and (2) is related to a large class ofdelay differential equations.G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).^[62]

Applications of the LambertW function in fundamental physical problems are not exhausted even for the standard case expressed in (1) as seen recently in the area ofatomic, molecular, and optical physics.^[63]

Plots

[edit]

Plots of the LambertW function on the complex plane
z = Re(W₀(x +iy))
z = Im(W₀(x +iy))
z = |W₀(x +iy)|
Superimposition of the previous three plots

Numerical evaluation

[edit]

TheW function may be approximated usingNewton's method, with successive approximations tow =W(z) (soz =we^w) being

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

Faster convergence may be obtained usingHalley's method,

w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{w_{j}e^{w_{j}}+e^{w_{j}}-{\dfrac {\left(w_{j}+2\right)\left(w_{j}e^{w_{j}}-z\right)}{2w_{j}+2}}}}

given in Corless et al.^[4] Because the computation time is dominated by the exponential function, this is only slightly more expensive than Newton's method.

For real $x\geq -1/e$ , it may be approximated by the quadratic-rate recursive formula of R. Iacono and J.P. Boyd:^[12]

w_{n+1}(x)={\frac {w_{n}(x)}{1+w_{n}(x)}}\left(1+\log \left({\frac {x}{w_{n}(x)}}\right)\right).

Lajos Lóczi proves^[64] that by using this iteration with an appropriate starting value $w_{0}(x)$ ,

For the principal branch $W_{0}:$
For the branch $W_{-1}:$
- if $x\in (-1/4,0):$ $w_{0}(x)=\log(-x)-\log(-\log(-x)),$
- if $x\in (-1/e,-1/4]:$ $w_{0}(x)=-1-{\sqrt {2}}{\sqrt {1+ex}},$

one can determine the maximum number of iteration steps in advance for any precision:

if $x\in (e,\infty )$ (Theorem 2.4): $0<W_{0}(x)-w_{n}(x)<\left(\log(1+1/e)\right)^{2^{n}},$
if $x\in (0,e)$ (Theorem 2.9): $0<W_{0}(x)-w_{n}(x)<{\frac {\left(1-1/e\right)^{2^{n}-1}}{5}},$
if $x\in (-1/e,0):$
- for the principal branch $W_{0}$ (Theorem 2.17): $0<w_{n}(x)-W_{0}(x)<\left(1/10\right)^{2^{n}},$
- for the branch $W_{-1}$ (Theorem 2.23): $0<W_{-1}(x)-w_{n}(x)<\left(1/2\right)^{2^{n}}.$

Toshio Fukushima has presented a fast method for approximating the real valued parts of the principal and secondary branches of theW function without using any iteration.^[65] In this method theW function is evaluated as a conditional switch of minimax rational functions on transformed variables: $W_{0}(z)={\begin{cases}X_{k}(x),&(z_{k-1}\leq z<z_{k},\quad k=1,2,\ldots ,17),\\U_{k}(u),&(z_{k-1}\leq z<z_{k},\quad k=18,19),\end{cases}}$ $W_{-1}(z)={\begin{cases}Y_{k}(y),&(z_{k-1}\leq z<z_{k},\quad k=-1,-2,\ldots ,-7),\\V_{k}(v),&(z_{k-1}\leq z<z_{k},\quad k=-8,-9,-10),\end{cases}}$ whereu,v,x, andy are transformations ofz:

u=\ln {z},\quad v=\ln(-z),\quad x={\sqrt {z+1/e}},\quad y=-z/(x+1/{\sqrt {e}})

Here $U_{k}(u)$ , $V_{k}(v)$ , $X_{k}(x)$ , and $Y_{k}(y)$ are rational functions whose coefficients for differentk-values are listed in the referenced paper together with the $z_{k}$ values that determine their subdomains. With higher degree polynomials in these rational functions the method can approximate theW function more accurately.

For example, when $-1/e\leq z\leq 2.0082178115844727$ , $W_{0}(z)$ can be approximated to 24 bits of accuracy on 64-bit floating point values as $W_{0}(z)\approx X_{1}(x)={\frac {\sum _{i}^{4}P_{i}x^{i}}{\sum _{i}^{3}Q_{i}x^{i}}}$ wherex is defined with the transformation above and the coefficients $P_{i}$ and $Q_{i}$ are given in the table below.

Coefficients for the $X_{1}$ subfunction
$i {\displaystyle i}$	$P_{i}$	$Q_{i}$
0	−0.9999999403954019	1
1	0.0557300521617778	2.275906559863465
2	2.1269732491053173	1.367597013868904
3	0.8135112367835288	0.18615823452831623
4	0.01632488014607016	0

Fukushima also offers an approximation with 50 bits of accuracy on 64-bit floats that uses 8th- and 7th-degree polynomials.

Software

[edit]

The LambertW function is implemented in many programming languages. Some of them are listed below:

Language	Function name	Required library
C/C++	`gsl_sf_lambert_W0` and`gsl_sf_lambert_Wm1`	Special functions section of the GNU Scientific Library (GSL)^[66]
	`lambert_w0`,`lambert_wm1`,`lambert_w0_prime`, and`lambert_wm1_prime`	Boost C++ libraries^[67]
	`LambertW`	LambertW-function^[68]
GP	`lambertw`
Julia	`lambertw`	`LambertW`^[69]
Maple	`LambertW`^[70]
Mathematica	`ProductLog` (with`LambertW` as a silent alias)^[71]
Matlab	`lambertw`^[72]
Maxima	`lambert_w`^[73]
Octave	`lambertw`	`specfun`^[74]
PARI	`glambertW, lambertWC, glambertW_i, mplambertW, lambertW`
Perl	`LambertW`	`ntheory`^[75]
Python	`lambertw`	`scipy`^[76]
R	`lambertW0` and`lambertWm1`	`lamW`^[77]
Rust	`lambert_w`,`lambert_w0` and`lambert_wm1`	`lambert_w`^[78]

Notes

[edit]

^Lehtonen, Jussi (April 2016), Rees, Mark (ed.), "The Lambert W function in ecological and evolutionary models",Methods in Ecology and Evolution,7 (2):1110–1118,Bibcode:2016MEcEv...7.1110L,doi:10.1111/2041-210x.12568,S2CID 124111881
^Euler, Leonhard (1783),"De serie Lambertina plurimisque eius insignibus proprietatibus",Acta Academiae Scientiarum Imperialis Petropolitanae (in Latin),1779 (II):29–51
^Chow, Timothy Y. (1999), "What is a closed-form number?",American Mathematical Monthly,106 (5):440–448,arXiv:math/9805045,doi:10.2307/2589148,JSTOR 2589148,MR 1699262.
^^a ^b ^c ^d ^eCorless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996)."On the LambertW function"(PDF).Advances in Computational Mathematics.5:329–359.doi:10.1007/BF02124750.S2CID 29028411.
^Lambert J. H.,"Observationes variae in mathesin puram",Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III, 128–168, 1758.
^Euler, L."De serie Lambertina Plurimisque eius insignibus proprietatibus".Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L.Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921.
^Scott, TC; Babb, JF; Dalgarno, A; Morgan, John D (Aug 15, 1993). "The calculation of exchange forces: General results and specific models".J. Chem. Phys.99 (4). American Institute of Physics:2841–2854.Bibcode:1993JChPh..99.2841S.doi:10.1063/1.465193.ISSN 0021-9606.
^Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J. (1993). "Lambert's⁠ $W {\displaystyle W}$ ⁠ function in Maple".The Maple Technical Newsletter.9:12–22.CiteSeerX 10.1.1.33.2556.
^Mező, István (2022).The Lambert W Function: Its Generalizations and Applications.doi:10.1201/9781003168102.ISBN 978-1-003-16810-2.S2CID 247491347.
^Bronstein, Manuel; Corless, Robert M.; Davenport, James H.; Jeffrey, D. J. (2008)."Algebraic properties of the Lambert⁠ $W {\displaystyle W}$ ⁠ function from a result of Rosenlicht and of Liouville"(PDF).Integral Transforms and Special Functions.19 (10):709–712.doi:10.1080/10652460802332342.S2CID 120069437.Archived(PDF) from the original on 2015-12-11.
^^a ^bA. Hoorfar, M. Hassani,Inequalities on the LambertW Function and Hyperpower Function, JIPAM, Theorem 2.7, page 7, volume 9, issue 2, article 51. 2008.
^^a ^bIacono, Roberto; Boyd, John P. (2017-12-01). "New approximations to the principal real-valued branch of the Lambert W-function".Advances in Computational Mathematics.43 (6):1403–1436.doi:10.1007/s10444-017-9530-3.ISSN 1572-9044.S2CID 254184098.
^Stewart, Seán M. (2009)."On certain inequalities involving the Lambert W function".Journal of Inequalities in Pure & Applied Mathematics.10 (4) 96.ISSN 1443-5756.
^Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation".IEEE Communications Letters.17 (8):1505–1508.arXiv:1601.04895.Bibcode:2013IComL..17.1505C.doi:10.1109/LCOMM.2013.070113.130972.S2CID 10062685.
^"Lambert function: Identities (formula 01.31.17.0001)".
^"Lambert W-Function".
^https://isa-afp.org/entries/Lambert_W.html Note: although one of the assumptions of the relevant lemma states thatx must be > 1/e, inspection of said lemma reveals that this assumption is unused. The lower bound is in fact x > 0. The reason for the branch switch ate is simple: forx > 1 there are always two solutions, −ln x and another one that you'd get from thex on the other side ofe that would feed the same value toW; these must crossover atx =e:[1] W_n cannot distinguish a value of ln x/x from anx <e from the same value from the otherx >e, so it cannot flip the order of its return values.
^"Webpage of István Mező PhD - Miscellaneous" (in German). Retrieved2023-01-30.
^"Papers with Code - An integral representation for the Lambert W function". Retrieved2023-01-30.
^David Jeffrey (2011-01-01),Stieltjes, Poisson and other integral representations for functions of Lambert $ W$, retrieved2023-01-30
^Finch, S. R. (2003).Mathematical constants. Cambridge University Press. p. 450.
^Kalugin, German A.; Jeffrey, David J.; Corless, Robert M. (2012)."Bernstein, Pick, Poisson and related integral expressions for LambertW"(PDF).Integral Transforms and Special Functions.23 (11):817–829.doi:10.1080/10652469.2011.640327.MR 2989751. See Theorem 3.4, p. 821 of published version (p. 5 of preprint).
^Dubinov, A. E.; Dubinova, I. D.; Saǐkov, S. K. (2006).The LambertW Function and Its Applications to Mathematical Problems of Physics (in Russian). RFNC-VNIIEF. p. 53.
^Robert M., Corless; David J., Jeffrey; Donald E., Knuth (1997). "A sequence of series for the LambertW function".Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97. pp. 197–204.doi:10.1145/258726.258783.ISBN 978-0-89791-875-6.S2CID 6274712.
^"The LambertW Function". Ontario Research Centre for Computer Algebra.
^Flajolet, Philippe; Sedgewick, Robert (2009-01-15). "Chapter II".Analytic Combinatorics. Cambridge University Press.doi:10.1017/cbo9780511801655.ISBN 978-0-521-89806-5.
^Barna, I.F.; Mátyás, L. (2013)."Analytic solutions for the one-dimensional compressible Euler equation with heat conduction closed with different kind of equation of states".Miskolc Mathematical Notes.13 (3):785–799.arXiv:1209.0607.doi:10.18514/MMN.2013.694.
^More, A. A. (2006). "Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes".Chemical Engineering Science.61 (16):5515–5519.Bibcode:2006ChEnS..61.5515M.doi:10.1016/j.ces.2006.04.003.
^Pellegrini, C. C.; Zappi, G. A.; Vilalta-Alonso, G. (2022-05-12)."An Analytical Solution for the Time-Dependent Flow in Simple Branch Hydraulic Systems with Centrifugal Pumps".Arabian Journal for Science and Engineering.47 (12):16273–16287.doi:10.1007/s13369-022-06864-9.ISSN 2193-567X.S2CID 248762601.
^Sotero, Roberto C.; Iturria-Medina, Yasser (2011)."From Blood Oxygenation Level Dependent (BOLD) Signals to Brain Temperature Maps".Bulletin of Mathematical Biology.73 (11):2731–47.doi:10.1007/s11538-011-9645-5.PMID 21409512.S2CID 12080132.
^Braun, Artur; Wokaun, Alexander; Hermanns, Heinz-Guenter (2003)."Analytical Solution to a Growth Problem with Two Moving Boundaries".Appl Math Model.27 (1):47–52.doi:10.1016/S0307-904X(02)00085-9.
^Braun, Artur; Baertsch, Martin; Schnyder, Bernhard; Koetz, Ruediger (2000). "A Model for the film growth in samples with two moving boundaries – An Application and Extension of the Unreacted-Core Model".Chem Eng Sci.55 (22):5273–5282.doi:10.1016/S0009-2509(00)00143-3.
^Asadian, M; Saeedi, H; Yadegari, M; Shojaee, M (June 2014). "Determinations of equilibrium segregation, effective segregation and diffusion coefficients for Nd+3 doped in molten YAG".Journal of Crystal Growth.396 (15):61–65.Bibcode:2014JCrGr.396...61A.doi:10.1016/j.jcrysgro.2014.03.028.https://doi.org/10.1016/j.jcrysgro.2014.03.028
^Asadian, M; Zabihi, F; Saeedi, H (March 2024). "Segregation and constitutional supercooling in Nd:YAG Czochralski crystal growth".Journal of Crystal Growth.630 127605.Bibcode:2024JCrGr.63027605A.doi:10.1016/j.jcrysgro.2024.127605.S2CID 267414096.https://doi.org/10.1016/j.jcrysgro.2024.127605
^Braun, Artur; Briggs, Keith M.; Boeni, Peter (2003). "Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films".J Cryst Growth.241 (1–2):231–234.Bibcode:2002JCrGr.241..231B.doi:10.1016/S0022-0248(02)00941-7.
^Banwell, T.C.; Jayakumar, A. (2000)."Exact analytical solution for current flow through diode with series resistance".Electronics Letters.36 (1):29–33.Bibcode:2000ElL....36..291B.doi:10.1049/el:20000301.
^Ortiz-Conde, Adelmo; Garcı́a Sánchez, Francisco J; Muci, Juan (2000-10-01)."Exact analytical solutions of the forward non-ideal diode equation with series and shunt parasitic resistances".Solid-State Electronics.44 (10):1861–1864.Bibcode:2000SSEle..44.1861O.doi:10.1016/S0038-1101(00)00132-5.ISSN 0038-1101.
^Ortiz-Conde, A.; Garcı́a Sánchez, F. J.; Guzmán, M. (2003-11-01)."Exact analytical solution of channel surface potential as an explicit function of gate voltage in undoped-body MOSFETs using the Lambert W function and a threshold voltage definition therefrom".Solid-State Electronics.47 (11):2067–2074.Bibcode:2003SSEle..47.2067O.doi:10.1016/S0038-1101(03)00242-9.ISSN 0038-1101.
^Jain, Amit; Kapoor, Avinashi (2004-02-06)."Exact analytical solutions of the parameters of real solar cells using Lambert W-function".Solar Energy Materials and Solar Cells.81 (2):269–277.Bibcode:2004SEMSC..81..269J.doi:10.1016/j.solmat.2003.11.018.ISSN 0927-0248.
^Colla, Pietro (2014). "A New Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium".PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering, Stanford University. SGP-TR-202.([2])
^D. J. Jeffrey and J. E. Jankowski, "Branch differences and LambertW"
^Flavia-Corina Mitroi-Symeonidis; Ion Anghel; Shigeru Furuichi (2019). "Encodings for the calculation of the permutation hypoentropy and their applications on full-scale compartment fire data".Acta Technica Napocensis. 62, IV:607–616.
^F. Nielsen, "Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms"
^https://arxiv.org/abs/2005.03051 J. Batson et al., "A COMPARISON OF GROUP TESTING ARCHITECTURES FOR COVID-19TESTING".
^A.Z. Broder, "A Note on Double Pooling Tests".
^Rudolf Hanel, Stefan Thurner (2020)."Boosting test-efficiency by pooled testing for SARS-CoV-2—Formula for optimal pool size".PLOS ONE. 15, 11 (11) e0240652.Bibcode:2020PLoSO..1540652H.doi:10.1371/journal.pone.0240652.PMC 7641378.PMID 33147228.
^A.M. Ishkhanyan, "The LambertW barrier – an exactly solvable confluent hypergeometric potential".
^^a ^bDeur, Alexandre; Brodsky, Stanley J.; De Téramond, Guy F. (2016). "The QCD running coupling".Progress in Particle and Nuclear Physics.90:1–74.arXiv:1604.08082.Bibcode:2016PrPNP..90....1D.doi:10.1016/j.ppnp.2016.04.003.S2CID 118854278.
^de la Madrid, R. (2017). "Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential".Nucl. Phys. A.962:24–45.arXiv:1704.00047.Bibcode:2017NuPhA.962...24D.doi:10.1016/j.nuclphysa.2017.03.006.S2CID 119218907.
^Bot, A.; Dewi, B.P.C.; Venema, P. (2021)."Phase-separating binary polymer mixtures: the degeneracy of the virial coefficients and their extraction from phase diagrams".ACS Omega.6 (11):7862–7878.doi:10.1021/acsomega.1c00450.PMC 7992149.PMID 33778298.
^Cardoso, T. R.; de Castro, A. S. (2005)."The blackbody radiation in aD-dimensional universe".Rev. Bras. Ens. Fis.27 (4):559–563.doi:10.1590/S1806-11172005000400007.hdl:11449/211894.
^Floratos, Emmanuel; Georgiou, George; Linardopoulos, Georgios (2014). "Large-Spin Expansions of GKP Strings".JHEP.2014 (3): 0180.arXiv:1311.5800.Bibcode:2014JHEP...03..018F.doi:10.1007/JHEP03(2014)018.S2CID 53355961.
^Floratos, Emmanuel; Linardopoulos, Georgios (2015). "Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes".Nucl. Phys. B.897:229–275.arXiv:1406.0796.Bibcode:2015NuPhB.897..229F.doi:10.1016/j.nuclphysb.2015.05.021.S2CID 118526569.
^Wolfram Research, Inc."Mathematica, Version 12.1". Champaign IL, 2020.
^Packel, E.; Yuen, D. (2004)."Projectile motion with resistance and the Lambert W function".College Math. J.35 (5):337–341.doi:10.1080/07468342.2004.11922095.
^Mendonça, J. R. G. (2019). "Electromagnetic surface wave propagation in a metallic wire and the LambertW function".American Journal of Physics.87 (6):476–484.arXiv:1812.07456.Bibcode:2019AmJPh..87..476M.doi:10.1119/1.5100943.S2CID 119661071.
^Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General relativity and quantum mechanics: Towards a generalization of the LambertW function".Applicable Algebra in Engineering, Communication and Computing.17 (1):41–47.arXiv:math-ph/0607011.Bibcode:2006math.ph...7011S.doi:10.1007/s00200-006-0196-1.S2CID 14664985.
^Scott, T. C.; Fee, G.; Grotendorst, J. (2013)."Asymptotic series of Generalized LambertW Function".ACM Communications in Computer Algebra.47 (185):75–83.doi:10.1145/2576802.2576804.S2CID 15370297.
^Scott, T. C.; Fee, G.; Grotendorst, J.; Zhang, W.Z. (2014)."Numerics of the Generalized LambertW Function".ACM Communications in Computer Algebra.48 (1/2):42–56.doi:10.1145/2644288.2644298.S2CID 15776321.
^Farrugia, P. S.; Mann, R. B.; Scott, T. C. (2007). "N-body Gravity and the Schrödinger Equation".Class. Quantum Grav.24 (18):4647–4659.arXiv:gr-qc/0611144.Bibcode:2007CQGra..24.4647F.doi:10.1088/0264-9381/24/18/006.S2CID 119365501.
^Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion".Chemical Physics.324 (2–3):323–338.arXiv:physics/0607081.Bibcode:2006CP....324..323S.CiteSeerX 10.1.1.261.9067.doi:10.1016/j.chemphys.2005.10.031.S2CID 623114.
^Maignan, Aude; Scott, T. C. (2016). "Fleshing out the Generalized LambertW Function".ACM Communications in Computer Algebra.50 (2):45–60.doi:10.1145/2992274.2992275.S2CID 53222884.
^Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007)."The Nodal Surfaces of Helium Atom Eigenfunctions"(PDF).Phys. Rev. A.75 (6) 060101.Bibcode:2007PhRvA..75f0101S.doi:10.1103/PhysRevA.75.060101.hdl:11383/1679348.Archived(PDF) from the original on 2017-09-22.
^Lóczi, Lajos (2022-11-15)."Guaranteed- and high-precision evaluation of the Lambert W function".Applied Mathematics and Computation.433 127406.doi:10.1016/j.amc.2022.127406.hdl:10831/89771.ISSN 0096-3003.
^Fukushima, Toshio (15 May 2013)."Precise and fast computation of Lambert W-functions without transcendental function evaluations"(PDF).Journal of Computational and Applied Mathematics.244:77–89.doi:10.1016/j.cam.2012.11.021. Preprint available atFukushima, Toshio (2020-11-25)."Precise and fast computation of Lambert W function by piecewise minimax rational function approximation with variable transformation".doi:10.13140/RG.2.2.30264.37128.
^"Lambert W Functions - GNU Scientific Library (GSL)".
^"Lambert W function - Boost libraries".
^Mező, István (2017-12-05),LambertW-function, retrieved2024-12-13
^"Lambert W function and associated omega constant".GitHub.
^"LambertW - Maple Help".
^ProductLog - Wolfram Language Reference
^lambertw – MATLAB
^Maxima, a Computer Algebra System
^"lambertw - specfun on Octave-Forge". Retrieved2024-09-12.
^ntheory - MetaCPAN
^"lambertw — SciPy v1.16.1 Manual".docs.scipy.org.
^Adler, Avraham (2017-04-24),lamW: LambertW Function, retrieved2017-12-19
^Sörngård, Johanna (2024-07-28),lambert_w, retrieved2025-03-27

References

[edit]

Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.;Knuth, Donald (1996)."On the LambertW function"(PDF).Advances in Computational Mathematics.5:329–359.doi:10.1007/BF02124750.ISSN 1019-7168.S2CID 29028411. Archived fromthe original(PDF) on 2010-12-14. Retrieved2007-03-10.
Chapeau-Blondeau, F.; Monir, A. (2002)."Evaluation of the LambertW Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2"(PDF).IEEE Trans. Signal Process.50 (9).doi:10.1109/TSP.2002.801912. Archived fromthe original(PDF) on 2012-03-28. Retrieved2004-03-10.
Francis; et al. (2000). "Quantitative General Theory for Periodic Breathing".Circulation.102 (18):2214–21.CiteSeerX 10.1.1.505.7194.doi:10.1161/01.cir.102.18.2214.PMID 11056095.S2CID 14410926. (Lambert function is used to solve delay-differential dynamics in human disease.)
Hayes, B. (2005)."WhyW?"(PDF).American Scientist.93 (2):104–108.doi:10.1511/2005.2.104.Archived(PDF) from the original on 2022-10-10.
Roy, R.; Olver, F. W. J. (2010),"LambertW function", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.
Stewart, Seán M. (2005)."A New Elementary Function for Our Curricula?"(PDF).Australian Senior Mathematics Journal.19 (2):8–26.ISSN 0819-4564.Archived(PDF) from the original on 2022-10-10.
Veberic, D., "Having Fun with LambertW(x) Function" arXiv:1003.1628 (2010);Veberic, D. (2012). "LambertW function for applications in physics".Computer Physics Communications.183 (12):2622–2628.arXiv:1209.0735.Bibcode:2012CoPhC.183.2622V.doi:10.1016/j.cpc.2012.07.008.S2CID 315088.
Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation".IEEE Communications Letters.17 (8):1505–1508.arXiv:1601.04895.Bibcode:2013IComL..17.1505C.doi:10.1109/LCOMM.2013.070113.130972.S2CID 10062685.

External links

[edit]

Wikimedia Commons has media related toLambert W function.

National Institute of Science and Technology Digital Library – LambertW
MathWorld – LambertW-Function
Computing the LambertW function
Corless et al. Notes about LambertW research
GPLC++ implementation with Halley's and Fritsch's iteration.
Special Functions of theGNU Scientific Library – GSL
[3]

Retrieved from "https://en.wikipedia.org/w/index.php?title=Lambert_W_function&oldid=1338997076"

Category:

Special functions

Hidden categories:

[8]ページ先頭

Movatterモバイル変換

Terminology

History

Elementary properties, branches and range

Inverse

Transcendence

Calculus

Derivative

Integral

Asymptotic expansions

Integer and complex powers

Bounds and inequalities

Principal branch

Secondary branch

Identities

Special values

Representations

Other formulas

Definite integrals

Indefinite integrals

Applications

Solving equations

General case

Super root

Tree counting and combinatorics

Inviscid flows

Viscous flows

Time-dependent flow in simple branch hydraulic systems

Neuroimaging

Chemical engineering

Crystal growth

Materials science

Semiconductor devices

Porous media

Bernoulli numbers and Todd genus

Statistics

Pooling of tests for infectious diseases

Exact solutions of the Schrödinger equation

Exact solution of QCD coupling constant

Exact solutions of the Einstein vacuum equations

Resonances of the delta-shell potential

Thermodynamic equilibrium

Phase separation of polymer mixtures

Wien's displacement law in aD-dimensional universe

AdS/CFT correspondence

Epidemiology

Determination of the time of flight of a projectile

Electromagnetic surface wave propagation

Orthogonal trajectories of real ellipses

Generalizations

Plots

Numerical evaluation

Software

See also

Notes

References

External links