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

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40 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
1
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0answers
22 views

Is it possible to solve for $x$ using the lambert W function in the expression ${\ln\left(x\right)}=(t-x)^2$?

${\ln\left(x\right)}=(t-x)^2$ $\pm\sqrt{\ln\left(x\right)}+x=t$ $\mathrm{e}^{\sqrt{\ln\left(x\right)}+x}=e^t$ And that is as close as I can get it to the form $x\mathrm{e}^x$. What do I do next? ...
3
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0answers
38 views

$\varepsilon_\omega$ reachable using a Corless-Veblen $\varphi_1$?

$\newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ve}{\varepsilon} \newcommand{\om}{\omega} \newcommand{\al}{\alpha} \newcommand{\ph}{\varphi} \DeclareMathOperator{\W}{W} \DeclareMathOperator{\...
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3answers
49 views

Real values of a function involving the Lambert $W(x)$ function

I have the following function: $$y=-\dfrac{W\left(-\ln(k)\right)}{\ln(k)}$$ where $W(x)$ is the Lambert $W$ function defined as the solution of the equation: $$x=W(x)e^{W(x)}$$ If $k\in\mathbb{R}$ ...
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1answer
28 views

Limit including lower branch of Lambert function

I am trying to show that $\frac{1}{2\left(1-e^x\right)}-\frac{1}{x}W_{-1}\left[\frac{x}{2\left(1-e^x\right)}\exp\left(\frac{x}{2\left(1-e^x\right)}\right)\right]\geq 1,$ for $x>0$, where $W_{-1}$ ...
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0answers
30 views

On the inverse of the regularized upper incomplete gamma function

I'm interested to bound/approximate the the inverse of the regularized upper incomplete gamma function $Q^{-1}(a,z)$, where $Q(a,z) = \frac{\int_z^\infty t^{a-1} e^{-t} \mathrm{d} t}{\Gamma(a)} $. I ...
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1answer
67 views

Solve $(x+a)^{1/x} = b$ for $x$

Solve $(x+a)^{1/x} = b$ , for $x$ where $a$ & $b$ are real constant. Do not use Lambert W-function in solution. Instead of using Lambert W-function, there are solution steps look like "...
1
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2answers
47 views

Limit with Lambert-$W$ function

I have asked a similar question about this one particular limit: \begin{equation} A=\lim_{c\to 1}\exp\left[ -\left(\frac{1}{1-c}\right)\left(W_{0}\left[ B\left( 1+\frac{x}{rc}\right) \right]-W_{0}[B]\...
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2answers
53 views

How to calculate inverse of $y=3x+4\log(x+1)$?

How to calculate inverse of $y=3x+4 \log(x+1)$? Wolframalpha says that http://m.wolframalpha.com/input/?i=Inverse+3x%2B4+log%28x%2B1%29+&x=0&y=0
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1answer
52 views

Nasty Limit including Lambert-$W$ Function.

I would like to calculate the following limit: \begin{equation} A=\lim_{d\to 0^+}\exp\left[ -\left(\frac{d}{1-q}\right)\left(W_{0}\left[ B\left( 1+\frac{x}{rq}\right)^{\frac{1}{d}} \right]-W_{0}[B]\...
0
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1answer
70 views

If $|t| = |W(-\ln z)| = 1$ and $t^n =1$ then $z^{z^{z^{…}}}$ is convergent

Let $z \in \mathbb{C}$ and $W$ be the Lambert W function. In this post I was told if $|t| = |W(-\ln z)| = 1$ and $t^n =1$ for some $n \in \mathbb{N}$ than the iterated exponential $z^{z^{z^{...}}}$ ...
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1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
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0answers
35 views

Any way to characterize this family of polynomials?

I have a family of polynomials generated by the recurrence relation $P_{n+1}(w) = (1+w)P_n ^{\ \prime}(w) -(3n-1 +nw)P_n(w) \\ P_1(w) =1$ The family is related to the Lambert $W$-function by its ...
2
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1answer
78 views

Is there a relation between the branches of the Lambert function?

Is it possible to express $W_{-1}(z)$ exactly by a closed-form expression, allowing the principal branch function $W_0$ ? Update: I found this related post: http://mathoverflow.net/a/196321.
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2answers
41 views

Asymptotic evaluation of a quantity

Can we say that the following quantity (a recursion of logarithms): $W_{-1}(x)=\ln \cfrac{-x}{-\ln \cfrac{-x}{-\ln \cfrac{-x}{...}}}$ is $\Theta(\ln x)$? i.e., asimptotically upper and lower bounded?...
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0answers
25 views

$\log(x)$ as iteration-series: how can this be made correct?

I was tinkering with the question whether the logarithm $\log(x)$ can be expressed by some more useful series than by the Mercator series (in terms of (1+x)) for a certain question. One idea ...
2
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1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...
2
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1answer
64 views

Function inversion (analytical)

Can $t(x)$ be found from: $$A \, t + B\ln\frac{1-t}{t}=x \; ?$$ Here, $A>0, \; B < 0$ and $0 \lt t \lt 1$. The $t(x)$ should be given in analytical form (even if you use, say, Lambert's W - ...
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1answer
19 views

Sign of a quantity involving the Lambert function

Is the following quantity $$\frac{W(\ln x)}{x}\left(1-\frac{1}{1+W(\ln x)}\right)$$ positive for $x \geq 1$ ? Thank you very much.
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2answers
79 views

How do you solve x^2 = log^2(x)

I read a page that said that the limit as $x$ approaches infinity of (polynomial function)/(logarithmic function) = infinity and that the limit as $x$ approaches infinity of (logarithmic function)/...
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1answer
17 views

About lambert W function`s solution

I like to know solution of below equation. $$e^{-0.0042x}(1+0.0042x)=0.032$$ I use 'WolframAlpha' and get two real solution(x=-235.259 and 1256.97). General solution is $x=-\frac{5000}{21}(W_{n}(-\...
0
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2answers
20 views

Solving equation of form $x = -a/ln(bx)$

I have an equation that I am trying to solve, which can be reduced to the form $$ x = -\frac{a}{\ln(bx)}$$ where I am trying to solve for $x$. Mathematica says the solution is of the form $$x = \...
1
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1answer
64 views

Differential Equation involving Lambert W function

I was wondering whether there is an explicit solution to the following differential equation $$f'(x) = g'(t)\left(f(t)\left(\frac{a}{g(t)} -1 \right)-\frac{a}{g(t) \lambda}\left( 1+ W_{-1}(-e^{-1-\...
2
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0answers
29 views

Inverse of a Function including Lambert W

Given the function: \begin{equation} Λ_{c,d,r}(x)=rx^{c-1}\left[ 1-\frac{1-(1-c)r}{rd}\ln x \right]^{d} \end{equation} I would like to prove that its inverse function exists and is defined as: \begin{...
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3answers
538 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
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2answers
28 views

How do I solve for x? Do I need the Lambert W function?

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ y, d, b, and x are all real, positive numbers. How do I solve for x? Do use the lambert W function and if so how is that done? Thanks!
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2answers
76 views

Approximation of a quotient that involves the Lambert function.

I would like to find an asymptotic upper bound for $$\frac{-\ln n}{W(- \ln^{-c}n)}$$ where $c$ is positive and $W$ is the Lambert function. More precisely, I want something which dominates this ...
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1answer
31 views

Problem of simplification

When trying to solve the equation $y^y = \frac{\ln^{y(1+c)}n}{n}$ , I've found the result $$y=\frac{-\ln n}{W(-\ln^{-c}n)}$$ where $c$ is a positive constant and $W$ is the Lambert function. The ...
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1answer
30 views

Any advice on how to tackle this inequality? ($x^{a}e^{x+c}\leq b$ )

How might one go about solving the inequality: $x^{a}e^{x+c}\leq b$ where $a,b,c$ are arbitrary constants ($b\geq 0$ and $a\neq0$) for $x$. My first place would be to try and get all of the ...
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1answer
19 views

Is there a compendium of equations that are solvable in terms of Lambert W?

Such a compilation would list equations in the most general form possible along with their solutions using Lambert W, and a reference to the derivation. It would also mention equations for which no ...
3
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1answer
41 views

Explicit Representation of $x^{x^y}=y^{y^x}$.

How do you explicitly represent $x^{x^y}=y^{y^x}$ using the Lambert $W$ function? I started using logarithms to split it up and manipulate it to a form like xe^x. I do this semi-successfully. I go ...
1
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1answer
17 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
3
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1answer
34 views

Simplify $W(xa^x)$

I'm not sure if there is any way to simplify $W(xa^x)$. It's pretty clear that $a=e$ simplifies to $x$ or $W_k(xe^x)$, but any other value of $a$, other than trivial values like $a=0,1$, don't seem ...
0
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2answers
53 views

Lambert W function with natural log

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ I inserted this into Wolfram Alpha and it returned: $x = y \Bbb{W}[\frac{e^\frac{d-b}{y}d}{y})]$ y, d, b, and x are all real, ...
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0answers
16 views

What $p^{ax+b} = cx +d$ means in lambert w function?

Can somebody explain what $p^{ax+b} = cx +d$ means and how to apply the lambert w function onto this equation: $(16800)(1.1)^{n-1} = (18600)+ (1200)(n-1)$
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1answer
43 views

Solution in terms of Lambert $W$ function or infinite series

I've tried to solve $x\log x = 2(x-1)(1-ax)$ for $a\ge 0$. If $a=0$, I obtained $$x \log x = 2x-2\\ \to x(\log x-2)= -2 \\ \to x\log(xe^{-2}) = -2 \\ \to xe^{-2}\log(xe^{-2}) = -2e^{-2}\\ \to x =e^{...
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3answers
120 views

Solve for $x$ using the lambert W function $ \frac{\ln(1+bx)}{x} = a$

Question: Solve for $x$ using the lambert W function $$ \frac{\ln(1+bx)}{x} = a$$ I've got this far: $$ \frac{\ln(1+bx)}{x} = a$$ $$ \ln(1+bx) = ax $$ $$ 1+bx = e^{ax} $$ Stuck when ...
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2answers
31 views

How could I solve this equation: $ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1} $ for $x$?

I want to have a solution for $x$ in this equation. $$ n-ne^{x\ln(2)}+xe^{x\ln(2)}\ln(2)=ax^{n-1}$$ Thanks !
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2answers
39 views

How to derive $y^{y^n}=x$ explicit form with Lambert $W$ function

I think the answer is $y = \left( \frac{n \cdot \ln(x)}{W(n \cdot \ln(x))} \right)^{\frac{1}{n}}$, seems tricky. I'm a noob!
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1answer
30 views

is there a closed algebraic solution to x(x+a)e^x=b, a,b positive reals?

I am looking at the following equation which is solvable in terms of the Lambert-W function when $a=0$ (but it is strictly positive in my case, i.e. $a>0$): $x(x+a)e^x=b$ $(a,b>0)$ more ...
0
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1answer
52 views

How to derive the Lambert W function series expansion?

How do you use the Lagrange inversion theorem to derive the Taylor Series expansion of W(x)? How else can you derive a series expansion?
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1answer
37 views

How to derive inverse of x^x to be log(x)/W(log(x))

I understand the basics of the $W$ Lambert function, but I have problems working out some problems with it. I know the answer, but I don't know how to derive it. Help would be appreciated. $y^y=x$ is ...
0
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2answers
58 views

How do you solve this inequality? $\frac{1}{(n+1)4^{n+1}} < .001 $

$$\frac{1}{(n+1)4^{n+1}} < .001 $$ becomes $$ 1000 < (n+1)4^{n+1}$$ Where do you go from here? Am I supposed to plug in a table of values for n? n=1: 1000 < 32 n=2: 1000 < 192 n=3:...
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2answers
93 views

Solving $4x = e^x$ without graphing and looking for intersection

If I want to solve the equation $4x = e^x$, is there a way to solve for $x$ without graphing and looking for intersection?
3
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2answers
45 views

For what values does this method converge on the Lambert W function?

Someone from another question had noted that the following statement $$W(x)=\ln\left(\frac x{\ln\left(\frac{x}{\ln\left(\frac x{\ln(\dots)}\right)}\right)}\right)$$ Can be found from the identity $W(...
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2answers
72 views

Does Lambert W (Product Log) count as an explicit solution?

Say I have an equation that I can solve in $x$ as follows: $$ x = LambertW_{-1}(y)$$ Where LambertW is the product-log function. Can I say I have an explicit solution for $x$? It looks like that, ...
0
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1answer
46 views

Solve $x^y+y^x=a$ for $y$?

Just as I question states, I want to solve the equation for $y$, but that is proving difficult as you cannot simply just use algebraic methods. I suspect the Lambert W function might come into play.
2
votes
2answers
66 views

Finding solution with Lambert function

I have following equation to solve for $x$ $$\ln\left(1+\frac{bx}{a}\right)=\frac{4cx}{a}$$ where $a>0,b>0$ and $c>0$. In my own attempt I replaced $1+\frac{bx}{a}$ by $y$ and with this ...
2
votes
2answers
43 views

How to find solution for this equation

I have following equation$$x\ln(1+\frac{b}{x})=b$$ where $b>0$. How to find the solution for $x$. I know how to solve equation involving $x\ln(x)$ but I don't know how to solve equation where ...
2
votes
2answers
65 views

Has $e^x = ax^2$ a general solution for all $x$?

I was fiddling around with some math and stumbled upon $\exp(x) = a x^2$, finding myself unable to find a solution. Does it even have a general solution $a$ for all $x$? Some googling brought me to ...