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Questions tagged [lambert-w]

For questions related to the Lambert W or product log function, the inverse of $f(z)=ze^z$.

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Bounds on the Lamber W-function around $x \approx -(1/e)$.

I am looking for a bound on the principal branch of Lamber W-function $W(x)$ that works well when $x$ is approaching $-\frac{1}{e}$. There are several bounds like this bound \begin{align} W_{0}(x)\...
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Integral of $ \int x^{n-1}W(x)dx $

How to prove that : $$ \int x^{n-1}W(x)dx = \frac {x^ne^{[-nW(x)]}[-nW(-x)]^{-n}[n\Gamma(n+1, -nW(x)- \Gamma(n+2, -nW(x))]} {n^2} $$ Where $W(x)$ is the Lambert-W function https://en.wikipedia.org/...
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Solution to $e^{-\frac{x^2}{2}}\frac{1}{x} = y$ with Lambert W function

Basically, I have the problem to need a solution for $e^{-\frac{x^2}{2}}\frac{1}{x} = y$ with $y\in (0,\infty)$. Due to continuity and $\lim_{x\to 0} e^{-\frac{x^2}{2}}\frac{1}{x} = \infty$ and $\lim_{...
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Lambert involution

Let begin our discussion with the following curve $$ x=ye^{-y}.$$ Let choose a parametrization variable $z$ and we parametrize the above curve $$ x(z):=ze^{-z}, y(z)=z $$ We find the ramification ...
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Simulating the Lambert W Distribution…

I am trying to simulate the following distribution, with pdf given by: $$f(t) = \frac{2 \mu (1-\rho) e^{-2 \mu t} (1+\mu \rho t) \left(\rho ^{K+1} e^{\mu t}-\rho ^2 \left(e^{\mu t}-1\right)+\mu ...
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Approach to an Integral involving the Lambert function

I look for some help in solving the integral $$ \int\limits_{0}^{z_t} \mathrm{d} z\, \sqrt{a+\left[\left(\frac{1}{z}W_0(z\,e^{-z})+1\right)^2 -1 \right]}$$ involving the Lambert function $W_0$, where ...
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Proving that $\int x^{x^{x^{.^{…}}}} dx= \sum_{n=1}^{\infty}\frac {(-n)^{n-1}}{n!} \Gamma(n, -\ln x)$ [Proof Verification]

Please check if I solved this correctly and if there are any mistakes. Many of steps are well known properties , so I might have skipped them. To prove: $$\int x^{x^{x^{.^{.......}}}} dx= - \sum_{n=...
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2answers
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Proving that : $ \frac{W(x)}{xe^x}=\sum\limits_{n=0}^{\infty} \frac{(-1)^n}{n!}T(n)x^n $ [closed]

How to prove that: $$ \frac{W(x)}{xe^x}=\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}T(n)x^n $$ where $T(n)$ counts the number of forests of rooted labeled trees using labels in a subset of $\{1,\ldots,n\}$...
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Lambert W function equation

I was attempting to solve different types of equations using the Lambert W function. I stumbled across one that depending on how the signs were placed I was able to solve it or not. For instance, I’ve ...
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How would we evaluate the definite integral $ \int_0^{e^{\frac{1}{e}}}\frac {-W(-\ln(x))}{\ln(x)}dx $

How would we evaluate the definite integral: $$ \int_0^{e^{\frac{1}{e}}}\frac {-W(-\ln(x))}{\ln(x)}dx $$ Here $W$ is the Lambert W function. More information about this function can be found here. ...
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Solving an equations using LambertW function [closed]

I have just started learning the LambertW function, so if my question is very basic I am really sorry but I can't understand how solving for x in the below equation ...
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Series expansion of an expression that has productlog function about -1/e

I have this term: $-\frac{1}{p}W(-\frac{p}{e})\bigg[\log\bigg(-\frac{1}{p}W(-\frac{p}{e})\bigg)-1\bigg]$. In order to reduce complexity I've taken $-\frac{p}{e}=x$ which gives $f(x)=\frac{1}{xe}W(x)...
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1answer
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What is the relation between Wright Omega Function and Lambert W Function?

I have a logarithmic equation: $u = A\left(\ln(u)+1\right)$. I used MATLAB's symbolic toolbox to solve this equation for u. ...
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Proper way of notating nested expression whose value is 1

Trying to express $x=1$ in the following way, replacing its value recursively, but it's not clear that it "converges" properly $$x=-e^{\pi\sqrt{e^{\pi\sqrt{e^{\pi\sqrt{e^{\pi\sqrt{...}}}}}}}}$$ any ...
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How to solve $x \times e^{x^2}=1$?

I wanted to solve the following equation. $$x \times e^{x^2}=1$$ I heard about the $W$ Lambert function but what I can see on Wikipedia can only solve the equations of type $x \times e^{x}=\lambda$. ...
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Finding expression for variable x in an equation (use Lambert function?)

Let $a$, $b$ and $c$ be constants. How can one find an expression for variable $x$ in the following equation? $$\frac{a\cdot (b+x)}{c} = (1+\frac{a\cdot x}{c}) \cdot \ln(1+\frac{a\cdot x}{c})$$ From ...
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1answer
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Finding expression for variable x from equations like xlogx + y = 0

After spending hours of trying to find an expression for variable P1 from the equation in the link, I have not been successful. I would be very grateful if someone could show the steps in rearranging ...
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Solve $x^k \exp(-x)=a$ for $x$ and some inequalities

I am trying to characterize the set i \begin{align} A_{a,k}=\{ x \ge 0: x^k \exp(-x) > a \} \end{align} in terms of inequaliti on $x$ where $a$ and $k$ are some give positive numbers. That is I ...
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2answers
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$e^{-x}-x=0$ solution procedure

I realized that this exponential equation has to be solved using Lambert $w$ function, I also know that the result is $x= w(1)$, but I don't know how to get there. Would you mind helping me with this? ...
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1answer
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How can I solve $x = e^{a+bx} + c?$

I need to solve this implicit equation for a physical system. I know that the similar equation $x = xe^x$ (solved with the Lambert W-function) doesn't have real roots, but since this equation has ...
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2answers
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Smooth solutions of $u_t - x u u_x = 0$ deduced from characteristics

Consider the equation $u_t - x u u_x = 0$. with cauchy data $u(x,0) = x$. Solving this equation I see the characteristics are given by $x= r e^{-rt}$ for some $r$ and the solution is defined ...
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What is the solution of $x^a+(1+x)^b=0 $?

I am not a mathematician, but a theoretical physicist. I am faced with this equation coming from some plasma phenomenon and I am unable to 'recognise' it. Mathematica software cannot solve it. I ...
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2answers
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How to find all the roots of $xe^{-x}=A$

I am trying to find all the roots of the equation $xe^{-x}=A$. I tried using LambdertW function. I used $-xe^{-x}=-A$ Solution is $x = -LW(-A)$, where $LW$ is the Lambertw finction. As $-xe^{-x}$ ...
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1answer
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Sharp bounds for the principal branch of the Lambert W function?

I'm looking for references for bounds on the principal $W_0$-branch of the Lambert W-function, specifically in the range $[ -\frac 1e, 0)$. I'm trying to work with the expression $W(-xe^{-x})$ with $x ...
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The relationship between $\sum_{k\geq1}\frac{k^{k+a}}{k!(4e)^{k/2}}$ and the number of labeled rooted trees of subsets of an n-set

I was reading Jack D'Aurizio's website matemate.it and learnt that $$\sum_{k\geq1}\frac{k^{k}}{k!(4e)^{k/2}}=1\tag{1}$$ I worked out some other relatable values: $$\sum_{k\geq1}\frac{k^{k+1}}{k!(4e)^{...
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How to calculate compounding frequency for an loan (Using Lambert-W Function)

A bond will become worth 500 dollars when it becomes due in 5 years. If the bond was purchased today for 450 dollars at 2.13% per year, determine how frequently the interest was compounded. I tried ...
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Approximating measurement data with $ae^{bx}+c$

I have measurement data which looks like the sum of an exponential and a constant function. It is enough to see it as a continuous function, say $f(x)$. I am looking for the $a$, $b$, $c$ values for ...
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Closed form of an improper integral to solve the period of a dynamical system

This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period. In a special case, the integral is $$I=\int_{r_1}^{r_2}\frac{dr}{r\sqrt{\Phi^2(r,r_1)-1}}$...
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1answer
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Trying to take a numerical integral involving the Lambert W function in R

I'm trying to get an approximate answer to the following integral: $\lim_{n \to \infty} \frac{n+1}{2} \int_1^{\ln(n)}xe^{-x}(1-\exp(-x - W(-xe^{-x})))^2 dx $ I'm currently just interested in if it ...
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1answer
28 views

Verifying a limit with Lambert W function

Is the following limit computation correct:$$a = \lim\limits_{x\rightarrow 1} \exp\left\{\frac{W_{-1}\left(x\ln(x)\right)}{x}\right\} = \exp\left\{\frac{W_{-1}\left(1\cdot 0\right)}{1}\right\} = \exp(-...
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1answer
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Asymptotic behavior of roots of an equation involving exponential and logarithm

Prelude This Post is a continuation of this Original Post. The original problem asked is: How many solutions does the following equation have: $$ a^x = \log_a(x) \,,\quad a \in (0,1) \wedge x \in\...
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How to solve equations where the power of $x$ is a function of $x$?

I have been trying to find a solution for equations of the type $x^{px-c} = a$. I know how to use Lambert W. function to find solutions for $x^x = a$, but the function of $x$ at the exponent is making ...
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quadratic tetraic equation

I have recently found interesting thing: Lambert W funtion inverse of $f(x)=xe^x$ it was easy to find roots of $x^x=a$ but I wonder is possible to find formula for root of equation $ax^x+bx+c=0$ ????
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How to prove $\int_{-\infty}^\infty \frac{dx}{(e^x -x)^2 +\pi^2}=\frac{1}{1+W(1)}$? [duplicate]

Recently I've become aware of the result$$\int_{-\infty}^\infty \frac{dx}{(e^x -x)^2 +\pi^2}=\frac{1}{1+W(1)}$$ where $W(z)$ is the Lambert W Function. However, I do not know of any methods to prove ...
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Is the Lambert W function analytic? If not everywhere then on what set is it analytic?

I would appreciate if someone can help me answer the following questions. Although I read several papers and documents on the Lambert W function, I could not assess on what set is this function (or ...
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1answer
50 views

Solution for $ y(x) - ae^{y(x)} = f(x) $ involving Lambert W function

I need to solve an equation of the type: $$y(x) -ae^{y(x)} = f(x)$$ with $a>0$. Furthermore, the expression for $f(x)$ can't be evaluated analytically (it's the solution of a differential ...
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Solve ${W_{-1}}'(-1/x)=-1$

This problem arose when I tried to find the maximum of $f(x)=\ln(x\ln(x\ln(x\cdots)))-x$. This can be written as $f(x)=\exp(-W_{-1}(-1/x))/x -x$ by substituting the recursion into $f$. The negative ...
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Functional equation for the lambert w function?

I've been long with this question or trying to find something similar: Is there a functional equation of a reflection formula for the Lambert W function? The Lambert W function is the inverse ...
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What is the maximum value of $\ln(x+\ln(x+\ln(x+\cdots)))-\ln x$?

This question is somewhat similar to my last set of infinite nests (see here) but this time I would like to attain an upper bound instead of the area, as $\int_1^\infty\ln x\,dx$ does not converge. ...
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Help understanding Mathematica's solution for $x^x=4$

Solve for all $x$:$$x^x=4$$ If we take the $\ln$ of both sides, we get: $$x\ln x=\ln 4$$ Since, $x=e^{\ln x}$, we get: $$(\ln x)e^{\ln x}=\ln 4 \implies \ln x=W(\ln 4)\implies x=e^{W(\ln 4)}$$ Where $...
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Solving equations involving x↑↑3 [duplicate]

How would you solve for x in the equation: $$x^{x^x}=2$$ using the Lambert W-Function?
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Solve $x=d^{d}\log(d)$ using Lambert $W(x)$

We have $a=b^b$, so $$\log(a)=b\log(b)$$ $$x=\frac{x}{W(x)}\log\left(\frac{x}{W(x)}\right)$$ $$b=\frac{\log(a)}{W(\log(a))}$$ Next we have $c=d^{(d^{d})}$, so $$\log(c)=d^{d}\log(d)$$ In general $^{k}...
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Laplace's method with Lambert function

I need to find the following asymptotic expansion as $t\rightarrow \infty$ : $\int_{0}^{e^{-1}}e^{-t\sqrt{-y\ln y}}{\rm d}y. $ Introducing the new variable (related to the left branch of the ...
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Inverse Arc Length of a Parabola

Is the inverse for the arc length of a parabola (say, $f(x)=\dfrac{x^2}{2}$) not discovered, or not possible to express given elementary functions and product log ($W(x)$)? If the latter is so, is ...
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2answers
131 views

Laplace Transform of Lambert W function

Does there exist a Laplace transform of the Lambert W function (evaluated at $at$, where $a$ is a constant) that can be expressed in terms of elementary functions and the product log ($W(x)$)? The ...
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1answer
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Book on the Lambert W function.

The Lambert W function is the inverse of function $x\mapsto xe^x$. It is traditionally denoted by $W(x)$. The function $W(x)$ is bivalued in interval $(-\frac{1}{e},0)$. See Wikpedia and Wolfram for ...
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51 views

Inverse of $x^x$ using Lambert W function

I am currently looking into the Lambert W function. From my understanding it is defined as: $$f(x)=xe^x$$ $$W(x)=f^{-1}(x)$$ So in the application of this I am trying to define the inverse of the ...
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2answers
39 views

Special solution to $a+e^a\ln x = x+e^a\ln a = a+e^x\ln a$

I was messing around with the equations of the form $a+e^b\ln c$. I set two variables equal and graphed them and I noticed something that interested me enough to ask about. Let $x\in\mathbb{R}$. For ...
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2answers
50 views

Solve for $x$ implicitly when $x(0)=3$.

Let $$\frac{dx}{dt}=\frac{1}{16}x(4-x)^2.$$ Solve for $x$ implicitly when $x(0)=3$. Can anyone explain what this means or how to solve it. Thank you.
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1answer
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Computing the inverse of $Ax+Bx\log x$

I'm trying to inverse the function $f:x\mapsto Ax+Bx\log x$. I know from Wolfram Alpha that the result is: $f^{-1}(x)=\frac{A}{BW(\frac{Ae^{x/B}}{B})}$ where $W$ is the W-Lambert (or product ...