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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3answers
49 views

Solve for scalar $k$ the equation $s=2\cdot \text W\left(\frac{e^{-k/2}}{2}\right)+k$

I want to find $k$ which is a scalar. How to solve this?
1
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1answer
27 views

Lambert- W -Function calculation?

I have an equation of the form nlogn=x . Upon searching , i came across the term "Lambert- W -Function",but couldn't find proper method for evauation, and ...
0
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1answer
15 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 ...
3
votes
1answer
55 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 ...
2
votes
1answer
83 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 ...
1
vote
1answer
26 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?
2
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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
32 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.
3
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1answer
68 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
34 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
37 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
44 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: ...
1
vote
1answer
48 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 ...
0
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2answers
115 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 ...
2
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1answer
46 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 ...
2
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0answers
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 ...
5
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1answer
66 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 $y=xe^x$ ...
0
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2answers
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 ...
0
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0answers
37 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 ...
3
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1answer
34 views

Is it possible to solve this equation using lambert function

Can the following equation be solved using Lambert function? $$x(1+e^x)=a$$
2
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1answer
88 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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0answers
42 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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0answers
40 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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2answers
64 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?
3
votes
1answer
117 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. ...
0
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1answer
89 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 ...
5
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0answers
70 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 ...
0
votes
1answer
146 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 ...
3
votes
2answers
145 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 ...
3
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0answers
62 views

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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2answers
56 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 ...
1
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1answer
40 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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vote
2answers
54 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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0answers
106 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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0answers
70 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 ...
0
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0answers
49 views

Does $a x+b=\cos(x)$ have a special-functions solution analogous to the Lambert W function?

The Lambert W function is defined as the solution to the equation $z=w e^w$, in the sense that for all $z\in\mathbb C\setminus(-\infty,-1/e]$ we can find a complex number $W(z)$ which obeys ...
3
votes
2answers
133 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 ...
7
votes
2answers
314 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:see http://en.wikipedia.org/wiki/Lambert_W_function My try: let ...
0
votes
1answer
102 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 ...
2
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1answer
264 views
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1answer
70 views

Can we get an analytical solution to this equation involving the Lambert W function?

Can we get an analytical solution to the variable $t$: $$H\left(1+W\left(A\exp\left(Bt\right)\right)\right)=1+W\left(X\exp\left(Yt+Z\right)\right)$$ $W(x)$ is the Lambert W function.$A$ $B$ $X$ $Y$ ...
2
votes
1answer
189 views

Inverse function of $x\mapsto \sqrt[x]x$ on $\left[0,e^{-1}\right]$

Why is it, that the inverse of $\sqrt[x]x$ is given by the infinite power tower in $x\in[\frac1e;e]$, but not in $x\in[0;\frac1e]$? I know that the power tower diverges on that interval, but even if ...
2
votes
1answer
70 views

What is the asymptote for the positions of the largest Stirling numbers of the second kind?

The infinite lower triangular array of Stirling numbers of the second kind starts: $$\begin{array}{llllllll} 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} ...
4
votes
1answer
79 views

Maximize $W(x) - (\ln(x) - \ln{\ln{x}})$

How can you maximize $f(x) = W(x) - (\ln(x) - \ln{\ln{x}})$ for $x\geq 2$? Numerically the answer seems to be at around $x \approx 41$ where you get $f(41) \approx 0.31$. Mathematica suggests the ...
0
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1answer
26 views

Why does $\exp\left[W\left(b\left(\ln{n}\right)^2\right) - \ln{b} - \ln{\ln{n}}\right] = \frac{\ln{n}}{W(b\ln^2{n})}$?

Why does $$\exp\left[W\left(b\left(\ln{n}\right)^2\right) \; - \; \ln{b} - \ln{\ln{n}}\right] = \frac{\ln{n}}{W(b\ln^2{n})}\;?$$ $W$ is the Lambert-W function and all variables are real and positive. ...
0
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1answer
82 views

How to solve $q= \frac{\ln{n}}{\ln{b} + \ln{q}+\ln\ln{n}}$

I have come across this equation recently. All the variables are positive and real too. $$q= \frac{\ln{n}}{\ln{b} + \ln{q}+\ln\ln{n}}.$$ Under what conditions can this be solved for $q$?
0
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0answers
112 views

Given a power series, how do I find an integral representation of the function that the power series represents?

I have this power series: $$f(x) = \sum _{n=1}^{\infty } \frac{x n^{k n} (-x)^n (k!)^n}{(n+1) (k n)!}+x+1$$ I know that for $k=1$: $$f(x) = \text{x/LambertW(x)}$$ and that for $k=0$ and $x=1$ ...
0
votes
2answers
169 views

What is the radius of convergence for this power series f(x)?

The following power series: $$f(x)=\sum _{n=1}^{\infty } \frac{x n^{k n} (-x)^n (k!)^n}{(n+1) (k n)!}+x+1$$ is equal to x/LambertW(x) when expanded at $x=0$ and when k=1. As pointed out by Andre ...
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vote
0answers
804 views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
-1
votes
2answers
124 views

Solving for $x$ in logarithmic equation

Could anyone please show me how to solve for $x$ this equation: $$1 = ax e^{-bx}$$ If solved in terms of the Lambert $W$-function, would that be considered a concrete answer?