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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Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically

I tried using Lambert W function the following way $$e^{k(t-1)} -t=0$$ $$e^{k(t-1)}=t$$ $$-ke^{-k} = -kte^{-kt}$$ $$W(-ke^{-k}) = W(-kte^{-kt})$$ $$-k = -kt \implies t = 1$$ but this only gives me one ...
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How to find the solution of $x(e^x)+\ln(x)+c = 0$, where $c$ is constant? [closed]

For $x(e^x)+\ln(x)+c = 0$, where $c$ is constant. I believe the solution is solved using Lambert $W$-function and derivatives, but I can't figured it out, I can't find any ideas can someone tell me ...
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Lagrange inversion theorem of $x^r(x+k)$ to generalize the W Lambert function

Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is expressible in terms of FoxH in Mathematica. $\text W_0(x)=\text W(x):$ $$-\lim_{a\to0}e^{\frac{(-x)^...
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2 votes
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$W_{-1} (x)$ series expansion?

By the Lagrange Inversion Theorem, one can derive the series expansion for the principal branch of $W_0(x)$: $$W_0(x)= \sum_{n=1}^{\infty} \frac{(-n)^{n-1}x^n}{n!}, \, |x| \leq \frac1e$$ For $x \in \...
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Formula for lower branch of Lambert W / Product log function

Is there some direct formula for the W-1 or lower branch of the Lambert W function? I saw that the Python package scipy and the function ...
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Projecting a point onto a convex set given by Log-Sum-Exp

Motivated by a wish to encode signal temporal logic specs (with linear predicates) as optimization problems w/o mixed integer approaches, I've been attempting to find a way to define the projection ...
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solving for exact solution of $x^{x^x} = 17$

Im trying to solve for the exact solution of $$ x^{x^x} = 17 $$ I understand that the previous solution, $x^x=17$, does not have an exact closed form solution and requires use of the Lambert W ...
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1 answer
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Find parameter to catenary interpolate a specific point

I'm working with the catenary equation and this equation is given by $$ f(x) = a \cdot \cosh\left(\dfrac{x}{a}\right) $$ I know this function pass at the point $(x_0, \ y_0)$ and therefore I want to ...
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2 votes
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What is the formula for higher integration of Lambert´s W function?

First I will present the notation for higher integration, which I copied from Danya Rose \begin{align} J_x^n(f(x)) = \int ...\int f(x) dx^n \end{align} Here are the higher integrations of the ...
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An equation involving the incomplete Gamma function

Fix $a, b>0$. For $x >0$, I need to solve the following equation in $y$: $$ \gamma(a, xy) = y^b $$ where $\gamma(s,t) = \int_0^t r^{s-1}e^{-r} dr$ is the incomplete Gamma function. How do I ...
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How to Interpret Lambert W Function?

I just used an online calculator to calculate the following values of t. 1.) I am confused about how I am supposed to interpret the $W_{-1}$ and $W_0$. Could anyone help me out? Here is my original ...
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How is $W_{-1}(-\frac{2}{3}\cdot e^{-\frac{2}{3}}) = -1.429$ if $W(x)$ is inverse of $f(x)=x\cdot e^x;x<-1$?

$$ f(x)=x\cdot e^x \space ;\space x<-1 \\ W_{-1}(x)=f^{-1}(x) \\ W_{-1}(x\cdot e^x)=f^{-1}(x\cdot e^x)=x $$ But $W_{-1}(-\frac{2}{3}\cdot e^{-\frac{2}{3}}) = -1.429$ not $-0.66 \dot 6$? It may be a ...
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Asymptotic formula for twice iterated factorial $(n!)!$

Being familiar with Stirling's formula for factorial: $$n!\sim\sqrt{2\pi n}\left(\frac n e\right)^n,\quad\color{gray}{n\to\infty}$$ I naïvely assumed that for twice iterated factorial we can simply ...
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1 vote
1 answer
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Lambert W of 2πi

$$2πi = 2πi, $$ $$2πi = 2πi \cdot e^{2πi}$$ Where $$e^{2πi} =1$$ $$W(2πi) = W(2πi \cdot e^{2πi}),$$ $$W(2πi) = 2πi$$ Where $$ W(xe^x) = x$$ When I check whether the last statement is valid in ...
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Question about the Lambert W function

I have the equation $u=e^{- \frac{1}{2}(x-(1+2u)t)^2}$ which I want to solve for $u(x,t)$, hopefully using the Lambert W function. I can rearrange it to get it in the forms $u^2e^{[2ut+t-x]^2}=1$, $u^...
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1 vote
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Are general degree 5 polynomials solvable only with only elementary functions and the Lambert W function?

I know it is not solvable in terms of radicals, and I don't know if it is known if it can be solved with only elementary functions of any kind. But can it be solved using only elementary functions and ...
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4 votes
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Approximation of harmonic numbers and their analytical inverse.

In the same spirit as DeTemple–Wang for a series expansion of harmonic numbers, I tried to approach the problem as $$H_n\sim\frac 12 \log(n^2+n+a)+\gamma-\frac 1{b(n^2+n+a)+\Delta}\tag 1$$ hoping to ...
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logarithm function in a polynomial equation

I faced the following polynomial equation which has a logarithm function inside of it: $\\ x^2 + \ln(x) - a = 0 $. I could manage to use approximation in small $x$, ($0<x \ll 1$) to solve this ...
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2 votes
1 answer
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Trying to solve $x^2=2^x$ : What's wrong below? [duplicate]

I just learned about Lambert W Function and I am trying to solve the above equation (and I have seen this article but I want to solve it myself). However, I seem not to be able to reach the correct ...
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2 votes
2 answers
103 views

Solving $k\left(e^{a(2k+1)}-1\right)=1$ for k?

I've been studying the structure of partial sums of the Dirichlet eta function and noticed that certain critical points occur when the summing from $n=1$ to $n\approx N(k)$ for $k\in\mathbb{N}$ where $...
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6 votes
2 answers
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When is it necessary to define a new function?

For example: Lambert $W$ is a non-elementary function that can be defined as a solution for $x$ to $x\cdot e^x$, but $\int{\frac{1}{\ln(t)}dt}$ is also supposed to be nonelementary. How do we know ...
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3 votes
3 answers
235 views

How to solve $e^x \ln(x) = a$?

I was wondering if it was possible to solve equations of the form $e^x \ln(x) = a, \;a > 0$ in terms of the Lambert $W$ function $W(x)$? I understand that fixed point iteration or the Newton-...
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Solve tangent of $y=\left(\log_{a}{x}\right)^2$ and $y=-ax+2$

How can I solve the tangent point and $a$ when $f(x)=\left(\log_{a}{x}\right)^2$ is tangent to $g(x)=-ax+2$? Although this can be solved by substituting $a=e^2$ and $x=e^{-2}$, then $f\left(e^{-2}\...
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Solving for $u$ after using Lambert W Function substitution.

I came across this question. I understand the given answer except for one particular point in the solution. The answerer uses a particular substitution: $x=-W(u)$ They then proceed to manipulate in a ...
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How to solve $x e^x + x = 1$?

I have seen a mathematics-related video here, which introduce a Lambert W function to the audience: $$f(x)=xe^x\\ W(x)=f^{-1}(x)$$ Then we can use $W(x)$ to solve some transcendental equation ...
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Approximating Lambert W-function as a fraction of logarithm

I need to solve an equation and I was able to find a solution in terms of Lambert W-function. However I need to solve the same equation for another set of parameters. I need to compare the root of ...
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29 votes
0 answers
413 views

An iterative logarithmic transformation of a power series

Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion: $$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$ Then, at each step ...
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1 answer
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Examples of closed forms of integrals with a power tower argument using W-Lambert function.

Here is a closed form of an integral that looks like: Integral form(s) of a general tetration/power tower integral solution: $$\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(an+b,cn+d)}{Γ(An+B)}$$ ...
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6 votes
0 answers
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Closed form of $\sum\limits_{n=1}^\infty \frac{γ(n+a,bn)n^k}{(n+c)!}$ with the Lower Gamma function?

$$\Large{\text{Goal:}}$$ One goal is to find better ways of expressing: $$\sum_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b)}$$ $$\Large{\text{Special Case:}}$$ Here are some closed form ...
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Is there a simplified Identity for $W(x)e^{(1/W(x))}$ if $W(x)$ is the lambert W function?

I have the expression $$ W_{-1}(x)e^\frac{1}{W_{-1}(x)} $$ where $-1 < x < 0$. I know that in general by the identity of $W(x)$ that $W(x)e^{W(x)}=x$, but is there a simplified form for the ...
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Solving equation in a form x(alnx + b) = c for x using Lambert W function

I don't know how to solve an equation in a form of x(alnx + b) = c for x using Lambert W function. a, b and c stand for constants. I tried using Wolfram Alpha, but it does not provide step by step ...
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0 answers
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What is the Taylor expansion of $f(x)=\sqrt {W\left( {\frac{c}{x}} \right)}$ around $x=0$?

I need to find the Taylor series of the following function around $x=0$, i.e., $$f(x)=\sqrt {W\left( {\frac{c}{x}} \right)}, $$ where $W$ is the so-called Lambert function and $c$ is a positive ...
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0 votes
1 answer
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Inverse of a log and linear function

How would you find the inverse of a function that is both linear and logarithmic? Take this for example: $f\left(x\right)=ln\left(x\right)+x+1$ Writing it as $y=ln\left(x\right)+x+1$ won't work, at ...
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0 answers
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Find $W_1$ branch of Lambert $W$-function

The given equation: $$\frac{8} {\ln(2)} \cdot \ln(x) = x.$$ Some algebra, and then we get the solutions: $$x = e^{W_0(-\frac{\ln(2)} {8})}\approx 1.1 ,\;\; x = e^{W_{-1}(-\frac{\ln(2)} {8})} \approx ...
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4 votes
1 answer
37 views

Can we use the Lambert W solution $y=-3W(K_2x^{-4/3})$ instead of $y =-3W(\frac 1 3\sqrt[3]{-\frac{K_1}{x^4}})$ if we choose an appropriate constant?

During the process of solving the separable differential equation $4y - x(y-3)y' = 0$, our solution acquires a constant when we go from $\frac 4 x = \frac{y-3}{y}y'$ to $\ln x + C_1 = y - 3 \ln y$. We ...
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$x^y = y^x$ what is $y$?

So today I came across an answer on MathStackExchange. I read it and found it well explained. But I stuck on a step where the user write "Solve using the properties of $W$ function". I don't ...
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1 vote
1 answer
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Is there any general formula that describes the sum and difference of Lambert W Function?

I know $\ln(a)+\ln(b)=\ln(ab)$ and $\ln(a)-\ln(b)=\ln(\frac{a}{b})$ or $\ln(a^{b}) = b\ln(a)$ and other rules for the ln (Natural Logarithm). But are there any sum or difference laws for the lambert ...
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0 votes
0 answers
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Solving for ${x}^{{x}^{3}}=5$

I am trying to solve $${x}^{{x}^{3}}=5$$ I took logarithms on both sides, and got $${x}^{3}\cdot \ln \left( x \right) -\ln \left( 5 \right) =0$$ Then I took the help of Maple, when I solve for x ...
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3 votes
0 answers
78 views

Solving an equation involving a matrix exponential

Suppose we have unknown scalars $x_1, x_2, ...,x_m \in \mathbb{R}$, known matrices $A_1, A_2, ...,A_m \in \mathbb{R}^{n\times n}$, and two known vectors $s_0, s_1\in\mathbb{R}^n$. I want to find $x_1,...
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Correct my sketch of proof about : $f(x)\geq -\gamma x$

Let $-1<x<1$ then define : $$\ln\left(\left(x\right)!\right)=f(x)$$ Then show that : $$f(x)\geq -\gamma x$$ Where we have the Euler-Mascheroni constant : Proof : I introduce the function : $$g(x)...
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7 votes
1 answer
152 views

How to derive this representation of the Lambert W function?

Lately I read this in some site about a closed-form representation of Lambert W function (all branch-cuts): $$\ln\bigg(\frac{W_k(z)-1}{ \ln(z)-1+2k\pi{i} }\bigg)=\frac{i}{2\pi}\int_0^\infty{\ln\bigg({\...
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-1 votes
2 answers
63 views

Solve for a with solution

Can somebody provide me the detailed solution for $$\log_{2021}a=2022-a$$ I know the answer is $2021$ but don't know, the solution. (got $2021$ by trial and error method)
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3 votes
2 answers
236 views

Conjecture: $\lim\limits_{x\to\infty}\operatorname{Re}\text W_x(x)\mathop=\limits^?-\ln(2\pi)$

The inspiration for the question is Closed form of $$\frac{d}{dk}\text W_k(z)$$ Derivative of W-Lambert function with respect to its branch cuts experiment. I also like making functions central to ...
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  • 5,244
2 votes
1 answer
139 views

Is there a way to write the $n^\text{th}$ super root in terms of the lambert W function?

Super root is one of the inverse functions of tetration defined as- $y=^nz$ $\implies z=\sqrt[n]{y}_s$ We can easily get an infinite series representation of $\sqrt[n]{z}_s$ using the Lagrange ...
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0 votes
0 answers
29 views

Question about complex number infinite tetrations

I was researching infinite tetrations recently and thought of a super interesting, although admittedly pointless, math question regarding it. I was wondering if anyone can help me solve it. Say we ...
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5 votes
1 answer
149 views

Closed form of $\frac{d}{dk}\text W_k(z)$. Derivative of W-Lambert function with respect to its branch cuts experiment.

For a change, I will ask a derivative question. Please consider the Generalized W-Lambert/Product Logarithm function $\text W_k(z)$. Let’s see what happens when we try to differentiate with respect to ...
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0 votes
1 answer
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Solving a finite linear recurrence relation of the form $s_{i+1}=\alpha s_i + \beta$

Suppose I have a sequence $s_0, s_1,...,s_N$, where $N$ is a finite number. This sequence obeys the recurrence relation $s_{i+1}=\alpha s_i+\beta$. The sequence also satisfies the following conditions:...
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1 vote
1 answer
117 views

Prove or disprove that $\left(f'(x)+g'(x)\right)\geq 2$

It's a result found with geogebra . Let us define : $$f(x)=\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$$ And : $$g(x)=\operatorname{W}\left(1-\frac{1}{x-1}\right)$$ Then it seems we have $2\leq x\leq \...
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0 votes
0 answers
25 views

Taylor expansion of the alternative branch of the lambert W function

I know that the Taylor expansion of the principal branch of the Lambert W function is, $$ W_0(z)=\sum\limits_{n=0}^\infty \frac{(-n)^{n-1}}{n!}z^n $$ But what is the Taylor expansion for $W_{-1}$?
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1 vote
0 answers
94 views

Inverse of $x\exp(-x)$

What would be the inverse of the function $y=x\exp(-x)$ when $1\leq x\leq \infty$ and $0\leq y\leq 1/e$. I tried to solve this with Lambert W function and the solution came out to be $-W(-y)$. But in ...
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