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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Lambert W-Function

Is there a standard name for the inverse of the Lambert W-Function, in the manner that the name "exponential function" is the name for the inverse function of the logarithmic function.
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1answer
87 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
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2answers
46 views

Bound for function involving Lambert W

Given that $W(x)$ is the Lambert W function, how can one prove that $$(2+W(x))e^{-W(x)}\leq 2 \frac{\log^2 x}{x}, \quad x\geq e^2$$ Is it possible to generalize this and find a function $f(x)$ such ...
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0answers
42 views

Complex number as function of real number

While seeking all solutions of $ Z ^ 2 = 2 ^ Z $ we have three real roots of $Z : z_1=2, z_2=4, $ and a third real root given in terms of LambertW function: $ z_3=-\frac{2 W\left(\frac{\log ...
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3answers
202 views

Request for help to solve an equation with LambertW: $ (x^2-4\,x+6) e^x =y$

I want to solve the following equation: $$ (x^2-4\,x+6) e^x =y \tag{1} $$ It looks a bit like the following equation: $$ x e^x =y \tag{2} $$ Since the solution of equation (2) is: x=LambertW(y), ...
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2answers
150 views

How do i solve $x^2/3=2/3^x$

How is this done? $${x^2\over3}={2\over3^x}$$ I'm out of ideas. It seems simple but it is quite not so the case in my perception of it.
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1answer
95 views

Can I solve this with a Lambert Function?

New to W-Functions and do not understand it properly. How do I solve this equation? I know about numerical solutions (or graph solution), but I'm interested in pure algebraic solution if it exists. ...
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2answers
47 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
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5answers
109 views

Graph of the function $x^y = y^x$, and $e$ (Euler's number).

Earlier, I was using the Desmos Graphing Calculator, and I wanted to remind myself of what the graph of the function $x^y = y^x$ looked like. If you have never seen what it looks like before, it is ...
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1answer
36 views

Prove graphically that the Lambert equation has exactly zero, one or two roots

I need some help on the below problem. Consider the Lambert equation: $xe^x = a$ for real values of x and a (a) Show graphically that the equation has exactly one root $ \xi(a) \ge 0 $ if $ ...
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1answer
56 views

Two kind of equations involving natural log and exponentiation

I know how to solve equations using Lambert's W function like $xe^x=k$ or $e^x+x=k$ But how can I solve this two kinds of equations involving natural log ? $e^x \ln(x)=k$ and $e^x+\ln(x)=k$ I ...
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1answer
74 views

Transcendental equations involving more than 2 terms

I now how to solve transcendental equations involving only two terms like: $xe^x=k$ $x=W(k)$ Where W(x) is the Lambert's Omega function. But how can I solve (for $x$) a more general case? Like: ...
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2answers
60 views

Logarithms and ratios.

This is the question: $$\log_b 64 = \frac{3}{b}$$ And have to find $b$. So I tried a bit and got this:$$\frac{b}{\log b} = \frac{\log 64}{3}$$ But have no idea what to do next. Thanks for your ...
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2answers
66 views

Closed form solution to $x\log_2(1+\frac{a}{x}) = b$ using Lambert W.

Is there an expression for the solution to \begin{equation} x\log_2(1+\frac{a}{x}) = b \end{equation} where $a$ and $b$ are constants, and $x$ is the variable? I am aware that there are no solutions ...
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1answer
30 views

Help solving non-trivial logarithmic inequality

I have the following equation: $$\dfrac{2\pi G\lambda M^4}{m^2}\ln\left(\dfrac{\phi_i}{\phi_e}\right)+2\pi G\left(\phi_i^2-\phi_e^2\right)\ge 65$$ which, for the purpose of this question, I'll ...
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1answer
44 views

How to solve the equation 5x=0.01^x [duplicate]

Hi I recently posted a this question earlier and got some excellent answers but to take it a little further I liked k170's answer however it contained a Lambert W Function in the answer and I was ...
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1answer
24 views

Simplify expression with lambert w-Function

I have an expression and i am almost sure what it equals: $ e^{-W_{-1}\left(-\frac{log\left(x\right)}{x}\right)} $ I only need a simplified version of this expression for $x\geq e$. I assume: ...
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1answer
170 views

Branch points of the Lambert W function

Let $W_{k}(z)$ be the kth branch of the Lambert W function. My question pertains to the branch point that $W_{0}(z)$ shares with $W_{-1}(z)$, and $W_{1}(z)$ at $z = - \frac{1}{e}$. By the inverse ...
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2answers
37 views

$\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$

How to prove $\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$ where W is the Lambert W function? Maple ...
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0answers
30 views

Closed form solution for DDEs?

I am solving the equation $X-A-B\exp(-Xy)-C\exp(-Xz)=0$ where $X, A, B$ and $C$ are 2x2matrices and $y$ and $z$ are scalars. What will be the closed form solution ...
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0answers
26 views

solving/approximating the transcendental inequality $c \le αx + β(b^x) + γx(b^x)$

I couldn't find a representation of $x$ using Lambert $W$ function and I doubt this is even possible. Assuming there is no clean solution and numerical methods must be used, is there a way to ...
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2answers
113 views

Finding solutions to $ x^x = 2x$

A friend claims it isn't possible to find a closed form for the smaller positive real solution of $x^x = 2x$. Numerically we have seen that $0.346...$ and $2$ are solutions, but are failing to do ...
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1answer
84 views

Finding inverse of $f(x) =\frac{\ln(x)}{x}$

How do you find the inverse of the following function $$f(x) = \frac{\ln (x)}{x}$$ What looked like a simple question made my head hurt during exam.
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0answers
56 views

Lambert W function, W(x), representation for entire domain

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express ...
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1answer
111 views

Solving $a=\Big(1+\frac{b}{x}\Big)^x$ for $x$

How to solve this equation for $x$? $$a=\Bigg(1+\frac{b}{x}\Bigg)^x$$ It's not a task that I was asked to solve by someone. I just have to solve it because it's a part of my project. If it's ...
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1answer
52 views

Montonicity of Lambert W

Is Lambert $W(x)$ function, an increasing function from $0\rightarrow\infty$? How about in negative axis and complex plane? Note $W(x)$ is given by $$W(x)e^{W(x)}=x.$$ Charts could help understand.
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0answers
49 views

The Lambert function has two real branches for $x∈[−1/e,0)$: the principal branch $W_0$ and the branch $W_-1$

I am trying to understand Lambert W function. I am new to this special function. What is the actual meaning of the word two real branches of Lambert W function for any real $x$. How to find the branch ...
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1answer
59 views

Solving equation involving self-exponentiation

How do I solve the equation $\displaystyle x=ay^2(by)^{\frac 1y}$ for $y$, where $a$ and $b$ are constants? I've been trying to manipulate this into a form on which I can use the Lambert W function, ...
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0answers
34 views

How to check if some equation can be solved using Lambert $\operatorname{W}$ function.

I'm very interested in Lambert $\operatorname{W}$ function and I want to know how to check if some equation can be solved using this function. Example $1$: $$e^xx=a$$ For this equation it is obviously ...
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1answer
47 views

Lambert W function identity from differential equation

For constants $v,K$ and a function $C(t)$, can you prove that if : $$ \frac{dc}{dt} = - \frac{vc(t)}{K + c(t)},~\text{with } c(0) = c_0 $$ Then the solution: $$ \left[ K \ln c(t) + c(t) ...
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1answer
22 views

Lambert W function multiplication with scalar

Let $W$ be the Lambert W function, $Y$ be a real valued function and $x \in \mathbb{R} $. Given $ Ye^Y = x \iff Y = W(x) $ is it true that $Y = kW(\frac{1}{k}x)$ for non-zero $k \in \mathbb{R} $ ? ...
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2answers
92 views

Can $x^{x^x}=k$ be solved using the W function?

The lambert W function is defined to be the inverse of $f(x)=xe^x$, and the equation $x^x=k$ can be solved fairly easily using the function: $$x^x=k$$ $$\ln(x^x)=\ln(k)$$ $$x\ln(x)=\ln(k)$$ ...
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1answer
27 views

$y = ln(p+qe^x)/x$, solve $x$

$y = \ln(p+qe^x)/x$ $p$ and $q$ are constants. Express $x$ in terms of $y$. I believe I have to use Lambert W function, but I'm stumped. Thinking help is needed. Thank you very much!
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1answer
45 views

Solving $ f'(x) =-\log( f(x) +a ) $

Can the solution of $$ f'(x) = -\log( f(x) + a ) $$ with $f(0)=0$ and $a \in (0,1)$ be well approximated by the Lambert W function for $x>0$? It seems that morally this might be the case (by ...
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1answer
35 views

Differential equation $x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x)$

In a proof of the series expansion of the Lambert-W-function, I need that it is the only non-zero function satisfying: $$ x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x) $$ Is it true?
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1answer
38 views

How do I apply the product log function (W(x)) to this equation?

I have the following: $$3(e^{hv\over kT}-1)v^2 = e^{hv\over kT}\frac {hv^3}{kT}$$ Which is the numerator of the derivative of Planck's energy distribution formula when the derivative is set to $0$. ...
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2answers
48 views

Can someone show me how to solve Lambert functions, such as the one here?

I would like to understand what process (steps) are required to arrive at the answer of 43.559... as shown in the following equation. I have looked at Wikipedia and I have also looked at the MathWorld ...
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2answers
214 views

How to solve this equation $x^{2}=2^{x}$?

How to solve this equation $$x^{2}=2^{x}$$ where $x \in \mathbb{R}$. Por tentativa erro consegui descobri que $2$ é uma solução, mas não encontrei um método pra isso. Alguma sugestão?(*) ...
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2answers
78 views

Solving an inequality with terms both within LambertW and outside of it.

$\newcommand{\LambertW}{\operatorname{LambertW}}$I am trying to solve the following inequality: $$100n^2<2^n, n\in\mathbb{R}$$ I have applied the following steps: \begin{align} & ...
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2answers
237 views

Lambert- W -Function calculation?

I have an equation of the form: $$ n \log n = x $$ Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for ...
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1answer
27 views

Solving $\log(x) = vx^α$ for $x$ via Lambert W function

Sure I can just get an answer from wolfram alpha, but I want to know the steps involved. I noticed the title equation while reading this: https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf The paper ...
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1answer
81 views

Limit of Lambert $W$ Product Log is the Natural Log?

In solving this equation $\large y=x^ne^x$ I get the result that $$n \cdot W\left( \frac{y^{1/n}}{n}\right)=x $$ So now it is apparent to me that when $n=0$ you would simply get $\ln(y)=x$ by ...
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1answer
103 views

Solve $2A{\frac{1-\sqrt{w}}{\log{w}}}=1$ in terms of Lambert W function.

I have tried it in this way: $$2A(1-\sqrt{w})=\log{w}$$ $$w\exp(2A\sqrt{w})=\exp{2A}$$ $$A^2w\exp(2A\sqrt{w})=(A\exp{A})^2$$ $$A^2w=W^2(A\cdot \exp{A})$$ $$w=A^-2W^2(A\cdot \exp{A})$$ Is this solution ...
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1answer
41 views

Integration of Lambert W function

I am interested in the integration of Lambert W function. Differentiation is ok but I am unable to integrate it. How to perform it?
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3answers
46 views

$\sum_{n=1}^{+\infty} \frac{n^{n-1}v^n}{n!}$ for what value of $v$ this series will be convergent? How to proceed for it?

I am interested in the convergence of the series $$\sum_{n=1}^{\infty} \left( \frac{n^{n-1}}{n!}v^{n} \right).$$ This series defines the tree function.
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2answers
49 views

Can anyone explain how to differentiate the Lambert W function?

I'm interested in the differentiation of the Lambert W function $y = xe^x$. I am unable to understand how to proceed for it.
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1answer
107 views

Is there a closed form for the inverse of $y=x^{x^x}$?

It's pretty well known, and easy to derive, that $y=x^x$ has the inverse $y=\frac{\ln x}{W(\ln x)}$. I've had no luck trying to work out the inverse of any larger power towers, though. Is there any ...
0
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1answer
80 views

Taylor series of Lambert W function $W_{-1}(x)$

Is possibile to find the closed form for the Taylor series of Lambert W function $W_{-1}(x)$? What do you think? On Wikipedia there is the Taylor series of $ W_0$ around $0$.
3
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1answer
53 views

Lambert function. Calculate $W(b)$ from $W(a)$.

The Lambert W function is defined as follows: $$z = W(z)e^{W(z)}$$ for any complex number z. Many equations involving exponentials can be solved using the W function. For example: $$ Y = X e ^ X ...
2
votes
1answer
61 views

How to use the Lambert W function instead of iterating

I am trying to calculate Id given the following equations: Vd = 5 - (Id * R) Id = Is * e^(Vd/.025) Is = 10^-15 R = 1000 By substitution: ...