# Questions tagged [lambert-w]

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

676 questions
Filter by
Sorted by
Tagged with
43 views

### 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 ...
94 views

### 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 ...
1 vote
47 views

37 views

### Can we use the Lambert W solution $y=-3W(K_2x^{-4/3})$ instead of $y =-3W(\frac 1 3\sqrt{-\frac{K_1}{x^4}})$ 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 ...
1 vote
104 views

### $x^y = y^x$ 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 ...
1 vote
65 views

### 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 ...
53 views

### 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 ...
78 views

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 $x_1,... 0 votes 0 answers 30 views ### Correct my sketch of proof about :$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 : $$g(x)... 7 votes 1 answer 152 views ### How to derive this representation of the Lambert W function? Lately I read this in some site about a closed-form representation of Lambert W function (all branch-cuts):$$\ln\bigg(\frac{W_k(z)-1}{ \ln(z)-1+2k\pi{i} }\bigg)=\frac{i}{2\pi}\int_0^\infty{\ln\bigg({\... -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) 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 ... 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 ... 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 ... 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 ... 0 votes 1 answer 47 views ### 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:... 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 \...
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 $W_{-1}$?
### 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 ...