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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1answer
16 views

Lambert W function - bounding the additive error of each iteration

I proved that for any $x\in \left(0,\frac{1}{e}\right]$: $$\ln(x)-\ln\left(\ln\left(\frac{1}{x}\right)\right)-\ln\left(1+\frac{1}{\sqrt 2}\right)< W_{-1}(-x)\leq \ln(x)-\ln\left(\ln\left(\frac{1}{x}...
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0answers
27 views

Is this equation solvable with Lambert function or some other way?

I'm new to Lambert function and was wondering if this was the correct method of solving the following equation. I have generalized it to $e^{\frac{a+bx}{cx^2}}(x^2+d_1x+ d_2)= k$ If it can't be solved ...
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1answer
62 views

Prove that $\Gamma(\operatorname{W}(x))$ is convex $\forall x>0$

Background : At the begining I was studing a function wich increases slowly and maybe have some property useful in number theory .Particulary I have found : Let $0<x\,$ define the function : $$f(x)...
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1answer
68 views

Limit of an expression that includes the Lambert W function

What is $\lim\limits_{k\rightarrow\infty}\ \sqrt{-(k+1)W_{-1}\left(\frac{-1}{(1+\frac{1}{k})\exp\left(\frac{1+\left(\frac{2}{k}\right)\ln\left(\frac{\sqrt{k}\Gamma(k/2)}{\sqrt{2}\Gamma((k+1)/2)}\right)...
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2answers
33 views

Confusion regarding usage of Lambert function

I stumbled upon an equation that goes like: $$e^{\pi x} - \frac{x}{k} = -1$$ I learnt that Lambert function is useful when dealing with such equations where it can take the form $f(x) = xe^x$. So, the ...
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1answer
61 views

Solving equation $y^2-2\ln(y)=x^2$

enter image description hereCan someone help me to solve this equation: $$y^2-2\ln(y)=x^2$$ I want to find y. I have tried to solve the problem but I couldn't. Thanks in advance.
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2answers
80 views

How to solve $1-x = \frac{ax}{e^{ax}-1}$

I am looking for an explicit solution to the equation $$1-x = \frac{\alpha x}{e^{\alpha x} -1}$$ I tried this with the Lambert-W function but can't get a sensible solution. How would you approach this?...
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1answer
84 views

Evaluating $\int _0^{\infty }W\left(\frac{1}{x^3}\right)\:\mathrm{d}x$

How can i evaluate $\displaystyle\int _0^{\infty }W\left(\frac{1}{x^3}\right)\:\mathrm{d}x$ in an easy manner i managed to end up with this $$3\int _0^{\infty }\frac{W\left(\frac{1}{x^3}\right)}{W\...
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1answer
46 views

How can I solve for $a$ in $0=\frac{\sqrt{\frac{a}{2}}-\sqrt{2a}}{a^{2}}+\frac{E_{\alpha}-E_{\beta}}{4}\exp\left(-\frac{a}{4}\right)$

I am trying to find the local maximum of $\frac{\sqrt{2a}}{a}-\left(E_{\alpha}-E_{\beta}\right)\exp\left(-\frac{a}{4}\right)$, where $E_{\alpha}$ and $E_{\beta}$ are constants: $$\displaystyle\frac{d}{...
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22 views

General Form for Intersection of Quadratic with Natural Log Function

Given $x. a_0, a_1, a_2, b\in\mathbb{R}$ is there a general solution to the equation: $a_0+a_1x+a_2x^2=b\ln(x)$ I am aware of the connection between this sort of equation and the Lambert W function, ...
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1answer
56 views

How to find $x$ for $-\frac{1}{\sqrt{2}x^{\frac{3}{2}}}=-\frac{1}{4}e^{-\frac{x}{4}}\left(A-B\right)$

The title pretty much explains it; I've had trouble with this because when taking the logarithm of both sides $x$ can never be isolated. I have been looking into the Lambert W function, but I've never ...
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0answers
60 views

On the average order of a multiplicative function defined in terms of particular values of Lambert $W$ function

We denote the main/principal branch of the Lambert $W$ function as $W(x)$, and we define the multiplicative function $$f(n) = \begin{cases} 1, & \text{if $n=1$} \\ \prod_{p\mid n}e^{-W(p^{e_p})}, ...
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1answer
103 views

Solve $(x^{2}+1)e^{-x}=u$ for $x$?

I need to solve the following equation for $x$ $(x^{2}+1)e^{-x}=u$ I tried Lambert's W function but couldn't find solution for $x$.
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1answer
264 views

For which $\alpha>0$ does $x\le|W(-cx^2)|^{-\alpha}$

Let $x\in(0,1)$. I want to know for which $\alpha>0$ it's true that $$ x\le|W(-cx^2)|^{-\alpha},\label{1}\tag{$\ast$} $$ where $W$ is the Lambert W-function and $c>0$ is some constant. In my ...
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2answers
57 views

Is there a way to solve: $\space x - a = b\cdot x \cdot e^x$ in terms of the Lambert-W function?

Is there a way to solve the following equation in terms of the Lambert-W function? I'm unable to cast it into a form suitable for using the Lambert-W. $$x - a = bxe^x$$ I'm intentionally not looking ...
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0answers
25 views

Is there an identity for $z$ if $z=2^z$ outside the Lambert-W function?

If $z=2^z$, this implies $z=-\frac{W_n(-\ln2)}{\ln2} \forall n \in \mathbb{Z}$ Taking into account the Lambert-W function having many special values surrounding the natural log, are there any unique ...
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0answers
22 views

solution of a transcendental equation

Can the following equation be solved (for y) in terms of the Lambert W function? $$ y \left(W \left(- e^{y-1} \right) +1\right)=-\frac{1}{x} $$
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2answers
47 views

Solve $x = a(1-c \exp(-bx)) $

I am trying to solve the following equation \begin{align} x = a(1-c \exp(-bx)) \end{align} for some $a>0$, $c \in (0,1)$ and $b\in (0,1)$. One can find an exact solution for this equation in terms ...
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0answers
31 views

Integral related to Lambert W function

I have stumbled upon the following integral while trying to solve an ordinary differential equation: $$ I=-\int {\left. \frac{1}{\xi \left( {W}\left( -{{ e}^{\frac{\xi}{g}-\frac{{m}}{g}-1}}\right) +1\...
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1answer
63 views

Integrate $W(e^u)$

Please help to integrate $$\int_{e+1}^{2+e^2} W(e^u) \, du,$$ where $W$ is the Lambert W function? Shall feyman technique of integration should be used here? I am not able to do it by integration by ...
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2answers
40 views

Equations via Lambert's W function

I'm studying Lambert's W function and I came across the equation $2^x = 2x$. Upon inspection it is easy to see that $x = 1$ and $x = 2$ are the real solutions to the equation. Solving for Lambert's W ...
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10 views

$\frac{a}{X} \frac{b+c}{b}+\ln{\left(\frac{d}{X} \right)} \gtreqless \lambda \ln{\left(\frac{a}{X} \right)}$?

How do we make the following comparison? $$\frac{a}{X} \frac{b+c}{b}+\ln{\left(\frac{d}{X} \right)} \gtreqless \lambda \ln{\left(\frac{a}{X} \right)}$$ where $X=a\left(1-\frac{c}{(\lambda -1)b} \right)...
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0answers
16 views

Analytical approximation of Lambert W Function

Consider the following function $$B\frac{a}{x}=\ln{\left(\frac{a}{x}\right)}$$ where $B=\left[\frac{c}{a(b-1)} + \frac{e}{(b-1)d} \right]$ with the conditions: $0<B<1, \quad 0<c<x<a,\...
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1answer
31 views

Solving this equation that involves Lambert W function is challenging

I'm trying to find the solution for $x$ from the following equation, which I believe involves some sort of of Lambert $W$ function or product logarithm. $$\frac{a}{x}\left[\frac{c}{a(b-1)} + \frac{e}{(...
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1answer
71 views

How to solve $x^x-x=1$?

I was recently posed the question "solve for $x$ in $x^x-x=1$". The intended answer was $x=0$, assuming that $0^0=1$, but I used brute force and determined another solution, $x\approx1....
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1answer
46 views

Prove the convergence of the integral $\int_{0}^{\infty}\frac{x^n}{\Gamma\Big(\operatorname{W}(x)\Big)}dx$

Prove that the following integral is convergent $\forall \,n\geq 1$ a natural number : $$\int_{0}^{\infty}\frac{x^n}{\Gamma\Big(\operatorname{W}(x)\Big)}dx$$ Where we have the Gamma function and the ...
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1answer
74 views

Study of the function $f(x)=x^{\frac{1}{x}}$ with $x\geq 1$ without derivative and using of Lambert's function

Well we want to study the function : $$f(x)=x^{\frac{1}{x}}$$ with $x\geq 1$ Using the definition of an increasing/decreasing function we have : $$x^{\frac{1}{x}}\leq^{?} y^{\frac{1}{y}}$$ Or $$\Big(\...
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2answers
62 views

A recurrence relation which approximates the Lambert W Function

I have the following recurrence relation, which seems to approximate the Lambert W Function pretty well: $W(x)\approx\log(f_n(x))$ for a sufficiently large value of $n$, where: $f_0(x) = x$ $f_{k+1}(...
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1answer
47 views

Analytical solution to $1-x=k\ln(x)$

Find an analytical real solution to $1-x=k\ln(x)$, in which $k$ is real and $x\ne 1$. I notice that (please correct if I am wrong): when $k>=0$, we have only one real solution of $x=1 \Rightarrow ...
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1answer
171 views

inverse function of $f(x)=x^{x^x}$

I have easily found the inverse of $f(x)=x^x$ using the following: $$y=x^x\Rightarrow \ln(y)=x\ln(x)\Rightarrow W(\ln(y))=\ln(x)\therefore x=e^{W(\ln(y))}$$ However I am struggling to do the same for $...
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4answers
100 views

limit related to the Lambert function

I am trying to evaluate the following limit $$ L=\lim_{x \rightarrow 0^+}\frac{2 \operatorname{W}\left( -{{ e}^{-x-1}}\right) \left( {{\operatorname{W}\left( -{{e}^{-x-1}}\right) }^{2}}+2 \...
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1answer
21 views

Minimax rational approximation of $W(x)/\ln(1+x)$ with polynomial degree $1$

I need to compute the minimax rational approximation of $W(x)/\ln(1+x)$ on the range $(1/e,e]$, with numerator and denominator of degree not larger $1$, where $W$ is the Lambert W Function. If this ...
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0answers
28 views

Definite integrals in terms of $W_0,W_{-1}$, which have the opposite signs

$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$ Consider definite integral \begin{align} I_n&= \int_0^1 \...
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1answer
8 views

How to determine the integer k as lambert branch to knowing the solutions of an equation?

Actually this is my first time to self-study about lambert w function, and i have interest on it, so forgive me if this question sounds stupid. I can derive it manually with algebra if the solution ...
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1answer
31 views

Better approximation for Lambert W Function near zero

I've looking for a polynomial approximation for Lambert W Function around zero. I am aiming at the range of $0\leq x\leq e$, and if possible then even $-e\leq x\leq e$. The asymptotic expansion (...
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2answers
95 views

How do you solve $2^x-x=3$?

Maybe it's a simple question, but I can't figure it out. How do you solve $2^x-x=3$? Using logarithms? I could write $\log_2(x-3)=x$, but then what? Thank you!
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1answer
49 views

Calculating Lambert W Function

I'm trying to evaluate Lambert W Function , I used the formula $$ W(z)e^{W(z)} = z \Rightarrow W(z) = \frac{z}{W(z)} $$ $$ W(z) \approx ln(z)-ln(ln(z)-ln(...)) $$ But the result is very bad If I used ...
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0answers
26 views

Numerical Approximation for Negative Lambert Function?

Backpacking off my previous question, Finding root of function, possible Lambert function?, of finding the root of the following equation: $c_1-\frac{2}{c_2}(x+2)e^{-x/2}=0$, Where $0<c_1\le1$ and $...
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1answer
30 views

Finding root of function, possible Lambert function?

So this is my function, and I'm trying to find the root where f(x)=0: $c_1-\frac{2}{c_2}(x+2)e^{-x/2}=0$ where $0< c_1\le1$ and $c_2\ge2$ This is what I got thus far: $c_1c_2-2(x+2)e^{-x/2}=0$ ...
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0answers
43 views

Exploring extentions of Tetration

Recently I've been kind of curious about tetration, specifically why it doesn't introduce any new inverse functions in the way lower operations do- addition needs subtraction, multiplication needs ...
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1answer
37 views

Explicit solutions to $x^y=y^x$ using Lambert $W$

A Flammable Maths video gives the solutions to the title equation by $y=-\frac{x}{\ln x}W(-\frac{\ln x}{x})$. This makes a lot of sense, given that Wikipedia gives $W_0(-\frac{\ln x}{x})=-\ln(x)$ for $...
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1answer
26 views

Calculate the convergence radius of an asymptotic expansion

I'm reading about the asymptotic expansion of the Lambert W function: $$W(x)=\frac{1^0}{1!}x^1-\frac{2^1}{2!}x^2+\frac{3^2}{3!}x^3-\frac{4^3}{4!}x^4+\dots$$ And it says that the convergence radius ...
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1answer
28 views

Determine limit of $W_0(a_n b_n \text{exp}(b_n +c)) - (b_n+c)$

Suppose we have some sequences of positive real numbers $a_n$, $b_n$, with $a_n \to a \in \mathbb{R}$, $b_n \to \infty$. Furthermore, suppose some constant $c \in \mathbb{R}$. How could I determine ...
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2answers
82 views

How can I calculate $W(-x)$ using $W(x)$?

The question is with regards to Lambert W Function: Given $W(x)$, I need to calculate $W(-x)$. Is there any way to do that? I've searched through the function identities, but couldn't find anything ...
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1answer
90 views

Upper bound on $x$ where $2^x \leq (ax)^4$

We have a number $a > 1$ and we know the following inequality: $$2^x \leq (ax)^4$$ And need to find an upper bound on $x$. I thought of trying to calculate where $2^x$ intersects $(ax)^4$ and ...
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0answers
28 views

Solving equations containing sum of inverse of Lambert W function

Lambert W function, W(z), is defined as the inverse relation of the function $f(w)=we^w$, i.e. if $we^w = z$ then $W(z) = w$. This function is implemented in several software libraries. If I wish to ...
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0answers
24 views

Prove that $\frac{3}{50}<\int_{0}^{1}\exp\Big(-\operatorname{W^2(x)}\Big)\operatorname{W^e(x)}dx$

At the beginning I was thinking to the Laplace transform of the Lambert's function as there is no easy to way express this I propose this similar problem : $$\frac{3}{50}<\int_{0}^{1}\exp\Big(-\...
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1answer
32 views

Upper bound $x\mapsto\frac{W\left(\frac{2\alpha}{x^2}\right)}{2\alpha}$ for $x\in[0,1]$ Lambert function

Let $W$ be the Lambert function. Let $\alpha>0$. I am looking for an upper bound on the function $x\mapsto\frac{W\left(\frac{2\alpha}{x^2}\right)}{2\alpha}$ for $x\in[0,1]$. Ideally, the bound ...
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1answer
41 views

Transcendental Equation with Quadratic Part (Can it be solved via Lambert W function)?

There I hope to minimize an optimization problem: $$ \min_{x \in \mathbb{R}_+} f(x) = x^2 -ye^{-x^2} + r(x -d )^2,$$ where $y, d \in \mathbb{R}$ and $r \in \mathbb{R}_+$. For this equation, the most ...
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1answer
31 views

Lambert-Function Solution of $\ln(x) = x-2$

Found an equation on AP Calculus:$$ \ln(x) = x-2.$$ Two solutions: 0.15859, 3.14619. Got the first solution thus: $e^{\ln(x)} = e^{x-2}$ $x = e^x e^{-2}$ $x (e^{-x}) = e^{-2}$ $-...

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