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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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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20 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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29 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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15 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
29 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
12 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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60 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
22 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
43 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
34 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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24 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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41 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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70 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
75 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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20 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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66 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
95 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
32 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
44 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
35 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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94 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 ...
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1answer
48 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 $0$.
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42 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 ...
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1answer
48 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: ...
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1answer
52 views

Expressing an formula in term of another one

I have this formula $$-\frac 1\lambda\left[\lambda D+1+W_{-1}\left(-r\exp(-\lambda D-1)\right)\right]$$ with $r$ , $\lambda$ and $D$ >0. Where $W$ is the Lambert W function ...
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117 views

LambertW: $ x=W(x\cdot e^{x}) $ for $ x \ge -1$ but not for $x \lt-1$. How do I express my formula/my text?

I just found by numerical heuristics for some systematic numbers $q(x)_\text{heuristical}$ depending on $x=1,2,3,4,\ldots$ using WolframAlpha the suggested interpretation in terms of the ...
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1answer
55 views

How to use the Lambert W function in an equation like this?

I was thinking of $\frac{4}{3}$ and found that $(\frac{4}{3})^4$ roughly equals $3$ (very roughly), and I though I would try to find the pairs of numbers where the equality suffices. So given the ...
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77 views

Explicit expression for root of equation

Is it possible to find an explicit expression for the root(s) (except $x=0$) for the following function $$f(x)= x-2 + 2b^x$$ where $0\leq b \leq 1$. Numerically this is no problem at all. But what ...
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1answer
73 views

Solutions to $x+e^x=k$

So I am trying to solve $x+e^x=k$ and here is what I have done: $$x+e^x=k$$ $$e^{x+e^x}=e^k$$ $$e^xe^{e^x}=e^k$$ Now, if we use the lambert W function which has the identity such that if $y=xe^x$ ...
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45 views

Conversion of an implicit to an explicit funtion [closed]

${1 \over a}({1 \over x} - {1 \over x_0}) - b \ln({x \over x_0}) = t$; where $a, x_0$ and $b$ are constants. Find an explicit function $t \mapsto x(t)$. I was suggested the use of Lambert W function ...
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40 views

Express everything in term of $\lambda$ with $\lambda$ as an argument of Lambert W function.

This question is related to the one that I have asked this morning: Root of equation, solvability This start from the same equation as before, but now the denominator is t+D and (t+D)^2 instead of ...
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1answer
51 views

Is it possible to solve this equation using lambert function

Can the following equation be solved using Lambert function? $$x(1+e^x)=a$$
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92 views

Minimum Argument Difference to Make the Lower Bound > the Upper Bound

Assume $g$ is a function that grows asymptotically as $$ g(n) \in\frac n {log(n)} + O(\sqrt n),\,n \in \Bbb N\tag1 $$ I wish to find $h(n)$ such that $$ g(n) \le g(n+h(n)). $$ i.e. Given the bounds ...
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48 views

Lambert W function inequality

I'm trying to approximate the number of lottery cards where the probability of the jackpot according to the birthday paradox is not lower than the number of cards divided by the number of all probable ...
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47 views

More on the Lambert W. Function

Fair warning: some of this might not be correct, but please correct me if I am wrong. I know that the Lambert W. function is equivalent to the inverse function of y = xe^x. I also know (or at least ...
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68 views

Lambert Function Question

Given $y$, for the Lambert $W$ function $y=xe^x\implies W(y)=x$, can you determine $x$? Specifically, how would you evaluate $e^{W(-1/3)}$ without a calculator?
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124 views

Lambert W function

I wan to show that, for $x \in [-\frac{1}{\exp(1)},0)$, $W_0(x) + W_{-1}(x) \leq -2$. Can any body help or give a suggestion? It seems trivial from the graph of functions but I need a rigorous proof. ...
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1answer
101 views

Solve exponential equation $(1−ax^2)∗e^{bx^2}=c$ for x. transcendental algebraic equation?

is it possible to solve an equation with the given form analytically? $$ (1-ax^2)*e^{bx^2}=c $$ $$ e^{bx^2}-ax^2e^{bx^2}=c $$ I've already tried it using a logarithmic function but I cannot manage ...
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75 views

Another interesting integral related to the Omega constant

Another interesting integral related to the Omega constant is the following $$\int^\infty_0 \frac{1 + 2\cos x + x \sin x}{1 + 2x \sin x + x^2} dx = \frac{\pi}{1 + \Omega}.$$ Here $\Omega = {\rm ...
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201 views

Matlab and lambertw function

I have Matlab 7.9.0 (R2009b) I want use the W lambert function. does someone know if Lambertw is only available to updated version of Matlab ? or if i need to do something before ? I am not very good ...
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148 views

The integral $\int\ln(x-\ln(x))~dx$

The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind. I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one? If it cannot be done in closed ...
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Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
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60 views

Does this limit imply that a function is “close” to Lambert W?

Suppose I am given the following limit involving function $f(n)\geq 0$: $$\lim_{n\rightarrow\infty}\log n-f(n)-\log f(n)=c$$ where $c$ is a constant. I am wondering if that implies that $f(n)$ is ...
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1answer
41 views

Solve $\; 0.75^x(x+3)\le0.3 $ (Lambert-W-Function?)

I currently have a problem with solving the following equation: $$0.75^x(x+3)\le0.3$$ It looks like it might be solvable using the Lambert-W function, but the x+3 throws me off. Wolframalpha is able ...
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63 views

Dose Lambert W function tends to infinity?

Does $W(x)\to\infty$ as the real number $x\to+\infty$? I find the equation (4.19) in paper https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf. It shows $$W(x)=\log x-\log\log x+\cdots$$. Assuming ...
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114 views

Power series to calculate LambertW up to infinity?

Is this an allowed operation to calculate the Lambert W function as a power series up to infinity, or is there some trouble in defining it this way? Mathematica programs: ...
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72 views

Are there other power series for the Lambert W function than this one?

Are there other known power series for the Lambert W function, other than this one: $$W(x) = x-x^2+\frac{3 x^3}{2}-\frac{8 x^4}{3}+\frac{125 x^5}{24}-\frac{54 x^6}{5}+\frac{16807 ...
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2answers
159 views

Solving an ODE without Lambert W function

I have a question regarding the possibility of solving the following ODE: $$\left[2x(t)+t\right]x^{\prime}(t)=1$$ such that $x(0)=-1$. If we make the substitution $w(t)=2x(t)+t$, we obtain the ...
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342 views

How find this integral $I=\int_{0}^{1}\sqrt{1-W^2(x)}dx$

How find this nice integral $$I=\int_{0}^{1}\sqrt{1-W^2(x)}dx$$ where, $W(x)$ is Lambert W function My try: let $$\sqrt{1-W^2(x)}=u\Longrightarrow W(x)=\sqrt{1-u^2}$$ and since ...
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1answer
112 views

Does this follow from the definition of the LambertW function?

The LambertW function $W(s)$ also called ProductLog seems to satisfy this relation: $$-W(s) = \underbrace{-s e^{-s e^{\cdot^{\cdot^{-s e^{-W(s)}}}}}}_n$$ Or truncated: $$-W(s) = -se^{-s e^{-s e^{-s ...