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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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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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59 views

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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1answer
32 views

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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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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39 views

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

How would you solve for x in the equation: $$x^{x^x}=2$$ using the Lambert W-Function?
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29 views

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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41 views

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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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
41 views

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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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
36 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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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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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 ...
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1answer
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How to solve for a in the below equation:

I have the following equation $$\int_0^1 \frac{a^x -1}{a-1} dx = r,$$ where the integral evaluates as: $$\frac{1}{1-a} + \frac{1}{\log(a)} = r.$$ I would like to solve for a but this is proving ...
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Inequality in formula solved with Lambert W

If I have an equation of the form $a = b \cdot e^b$ then the Lambert W function tells me that $W(a) = b$. But what if I instead have $a \geq b \cdot e^b$ ? Do I then have $W(a) \leq b$ or $W(a) \geq ...
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Solve for $x$ in $\mathbf{a} - \mathbf{b} \sigma(c + \boldsymbol{\theta}^\intercal \mathbf{x}) - \mathbf{D}\mathbf{x} = \mathbf{0}$

Exactly as the title states, I'm looking for a closed-form solution $\mathbf{x} \in \mathbb{R}^n$ to \begin{align*} \mathbf{a} - \mathbf{b} \sigma(c + \boldsymbol{\theta}^\intercal \mathbf{x}) - \...
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3answers
41 views

Solving an equation with Lambert's W function? Or by any other means?

I am trying to solve the following equation for x in terms of $y$ and $c$ (with $x,y \in [0,1]$) \begin{equation} \log\left(\frac{x}{1-x-y}\right) + \frac{x}{1-x-y} + \frac{y}{1-x-y} = c \end{...
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Is my solution to $a^x=x+b$ correct?

I was trying to find a general solution to the equation $a^x=x+b$. First, I used a substitution: $$u=a^x\Longleftrightarrow x=\log_au$$ Then, it went as follows: $$u=\log_a u+b$$ $$-b=\log_a u-u$$ $...
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1answer
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Integrating a function involving Lambert W

I want to solve the following integral, where $W$ is the Lambert W function. \begin{equation} \int \frac{W(e^{4x-3})}{1+W(e^{4x-3})}dx \end{equation} I assume $x \in [0, 1]$. Can someone please ...
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1answer
38 views

Trying to solve an equation using Lambert-W function

My equation is: $$ \frac{w}{2\sqrt{\Pi \sigma^2}} e^{\frac{-(x-\delta)^2}{4 \sigma^2}}=x $$ I am struggling to solve it fo $x$ knowing that: $ \sigma, w,\delta$>0. Probably, it could be solved ...
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80 views

Can we prove $B (n) = \frac{1}{4} G (n - 1) G (n)$ is an indicator function that takes on the value 1 for 'bad' and 0 for “good” Gram points?

Let \begin{equation} g (n) = 2 \pi e^{1 + W \left( \frac{8 n + 1}{8 e} \right)} \end{equation} be the approximate value of the $n$-th Gram point. Let \begin{equation} G (n) = \frac{Z (g (n))}{...
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What is the function $y = (1+1/x)^x$ solved for $x$?

I came across this function in algebra ($e$ being its limit as $x$ goes to infinity) while studying compounded interest. Since this function is a little modified from the real interest formula $y=(1+1/...
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2answers
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Where did I make a mistake in simplifying this?

So I found general solution to $x^y=y^x$ for positive values on the Internet via Lambert $W$ function and it goes like this: $$y=\frac{-x\cdot W\left(\frac{-\log(x)}{x}\right)}{\log(x)}.$$ Now there ...
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1answer
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How to solve following equation $a(x-2)e^x-c x-d=0$ (Maybe using Lambert W function)

How to solve following equation $a(x-2)e^x-c x-d=0$. I know that the equation like $a(x-2)e^x-d=0$ can solve using Lambert W function, but with this equation I'm confused, and I can solve.
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1answer
67 views

Power series expansion involving Lambert-W function

Need to power series expand this term: $ -\frac{e}{2z} W\left(\frac{-2z}{e^2} \right) \left( \log \left( - \frac{e}{2z} W \left( \frac{-2z}{e^2}\right) \right) -1 \right)$ I tried expanding this ...
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Are there other solutions of $x^{x^3-x}=2^{x^2+x}$ than $x=-1 $ and $x=2$ in $\mathbb{R}$?

I have tried to solve that equation $x^{x^3-x}=2^{x^2+x}$ in $\mathbb{R}$ , I have got only two integers solutions which they are : $x=-1$, $x=2$ , are there others ? Note: if we try to study this :...
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Integrals involving the fractional part function and the W-Lambert function

I am trying make interesting integrals involving the fractional part function and special functions. I wondered if it is possible to deduce a series representation (in the atempt to get a closed-form ...
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1answer
42 views

Taylor expansion of $x/W(x)$

For $x>e$, express $x/W(x)$ with respect to $\ln(x)$ and $\ln \ln(x)$, where $W(\cdot)$ is the Lambert-W function. In Wikipedia, we can find the expression of $W(x)$ with respect to $\ln(x)$ and $...
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Solution of $W_0(x)-W_{-1}(x)=1$

In my answer to this question, $W(.)$ being Lambert function, I indirectly showed that $$W_0(x)-W_{-1}(x)=1 \implies x=-\frac {1} {e-1}\, \exp \left( \frac {-1} {e-1}\right)$$ Is there any way to ...
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63 views

Numerical Analysis of Lambert Function

For a mathematical project I want to analyse the following function: $R_{\infty} = 1 + \frac{W(\epsilon e^{-\epsilon})}{\epsilon}$, in which the $W$ stands for the Lambert function. I have no ...
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derivative of log-lambert function

How can I calculate the derivative of the following function w.r.t $x$ $$f(x)=\log_2(1+W(ax))$$ where $W$ is the lambert function.
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If you had an equation where x^x = y where x is a positive number that is not 0 or 1, can there ever be more than 1 value of x?

If $x^x = 1$, both $0$ and $1$ would satisfy $x$. Does this mean $0 = 1$?. If you had an equation where $x^x = y$ where $x$ is a positive number that is not $0$ or $1$, can there ever be more than $1$...
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156 views

Looking for solution of a 9th grader's problem

My 9th grade (USA) son was given the following problem as part of an exam (which flummoxed him and me): Solve $3(x-2)^2+5 = 3^{x+2} +5$. This can be reduced to something like $y^2 3^y = 3^3$, but ...
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112 views

Isolating x analytically in this equation : $(xe^{ax})^x=f(y)$

I have come upon the equation $(xe^{ax})^x=c$, where a and c can be regarded as real constants, and am looking for an analytical expression for x. The value is constrained to $x>0, \in \mathbb{R}$. ...
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1answer
59 views

How do I solve the following equation?

$$ Ae^{Bx} + Cx = D $$ Solve for $x$, where $A, B, C, D$ are real constants. This popped up in a run of the mill high school first-order DE question. No matter what I try with logarithms it doesn't ...
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1answer
76 views

Analytical solution to mixed distribution fit to failure time data — Lambert W perhaps?

I have a set of $n$ device failure times $\{t_i>0\}$ for $i=1...n$ and $N-n$ devices which have not yet failed. Using maximum likelihood I am attempting to find a closed-form analytical solution ...
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0answers
70 views

Transcendental Equation. [duplicate]

I was able to solve $ \ln x = - x $ using Lambert's function. I was wondering how does one solve $ \ln x = x$ Does the solution even exist for this equation?
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1answer
58 views

Solving improper integral involving product logarithm and exponentials

It is possible to verify or show numerically that $\displaystyle \int^{\infty}_{-\infty}W(e^{x-e^{x}})dx=\frac{\pi^2}{12}$ $W(x)$ is the Lambert W function. This integral does not seem to have a ...
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1answer
97 views

Solve $x+\log(x-1)\cdot (1-x) = 0$

This is my first post here, so I'll try my best. I want to solve the root for this equation $$x+\log(x-1)\cdot (1-x) = 0$$ I know I require the Lambert $W$ function, but I always get to this form $$(...
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0answers
94 views

Solution to Vector Lambert W function type Equation

I was wondering if anyone has any ideas for a closed-form solution to the equation $$Ax + \exp(x) +b =0$$ where $x,b \in \mathbb{R}^n$, $A$ is a symmetric positive definite matrix and $\exp$ denotes ...
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2answers
51 views

Indefinite integral involving the Lambert W or product log function.

How to calculate the integral $\displaystyle \int \frac{1-x}{x W\left(\frac{1-x}{x}\right)} \, dx$ ? I tried making a substitution but it doesn't seem to work. Does this integral have a symbolic ...
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2answers
42 views

How to program formula involving cyclotomic polynomials and Lambert series?

I want to know how to program this formula (https://people.math.gatech.edu/~mschmidt34/images/sum-of-divisors-exact-formula.png) but I can't understand the math behind or the several variables used ...