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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45 views

How to find the inverse function of f(x)=x+sin(x)-a

The problem is how to find the inverse function of $$f(x)=x+\sin(x)-a$$ where $a$ is real parameter. I tried to write $\sin(x)$ as $\frac{i}{2}(e^{-ix}-e^{ix})$. Problem is how to solve this equation: ...
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2answers
106 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
42 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
70 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
62 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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2answers
42 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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0answers
31 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 ...
2
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1answer
29 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
80 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
38 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
33 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
50 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
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1answer
107 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
83 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 ...
4
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0answers
59 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
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1answer
109 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
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2answers
143 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
60 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 ...
1
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2answers
54 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
39 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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2answers
51 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
102 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
66 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
45 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
121 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
307 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
90 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
231 views
0
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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
184 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
66 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
votes
1answer
79 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 $q$?
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0answers
106 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
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2answers
165 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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0answers
699 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 ...
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2answers
121 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?
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3answers
165 views

Is there a way to solve this equation? (maybe with Lambert's W Function?)

I'd like to know if there is a way to solve the equation $$x\ln x=\alpha+\beta x$$ for known constants $\alpha,\beta\in\mathbb{R}$. I know that Lambert's W Function $W$ can be used to solve $$x\ln ...
2
votes
2answers
506 views

Andre LeClair, Riemann zeta zero approximation?

This sequence A177885 in the oeis seemingly relates imaginary parts of non-trivial Riemann zeta zeros with the LambertW function. The real and imaginary parts of the Riemann zeta function is the sum ...
1
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1answer
113 views

Finding Principal Branch Value of Lambert W function

I have this negative number $x<-1$ (thus in the lower branch of the Lambert W function). Is it possible to find $W_{0}(xe^x)$ in terms of $x$ in a useful/non-trivial form?
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0answers
94 views

Lambert W function?

I need to solve the next equation: $$ax^{bx+c}=d$$ where a, b, c and d are positive real values. Do I need to use Lambert W function, or there is some other method? Thanks!
3
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0answers
108 views

Does $^{\frac12}x=e^{W(\ln x)}$, or not?

Whenever I see tetration discussed here, I inevitably see it asserted that there's no consistent continuous definition for tetration. However, it seems to me that If we restrict ourselves to ...
10
votes
2answers
346 views

Approximation to the Lambert W function

If: $$x = y + \log(y) -a$$ Then the solution for $y$ using the Lambert W function is: $$y(x) = W(e^{a+x})$$ In a paper I'm reading, I saw an approximation to this solution, due to "Borsch and ...
5
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2answers
149 views

Naively estimating the factorial

A naive way to estimate the factorial is $n! \geq (a+1) (a+2) \dots n \geq a^{n-a}$ for any $a$. For example, it gives $n! \geq (n/2)^{n/2}$ and slightly better $n! \geq (n/3)^{2n/3}$. I am interested ...
3
votes
0answers
90 views

Sum formula for the $\Omega$ constant

I was looking a bit around, and was interested in the konstant $\Omega$. It is defined as number satisfying the equation $$ x e^x = 1 $$ Now, Wikipedia, gives an reccurence relation for the constant ...
4
votes
4answers
257 views

Lambert function approximation $W_0$ branch

I am looking for a simple, inexpensive and very accurate approximation of the Lambert function ($W_0$ branch) ($-1/e < x < 0$).
0
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1answer
55 views

Does $i = -\frac{(2\;W({\pi\over2}))}{\pi}$

Let $x = -\frac{(2\;W({\pi\over2}))}{\pi}$, where $W$ denotes the Lambert W-function. As $${\log(i^2)\over i} = \pi$$ and $${\log(x^2)\over x}=\pi$$ Does $x = i$?
1
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1answer
401 views

what is exactly analytic continuation of the product log function

When I solve in wolfram equation like this $xe^x=z$ they give me the solution $x=W_n(z)$ I know about $x=W_0(z) $ and $x=W_{1}(z)$ but for $n$ I searched in the internet but I didn't find anything ...
1
vote
1answer
76 views

Question about Lambert W function

I'm looking for a series for $W_0(x)$ for x $\in [\frac{-1}{e},\infty [$ but every time i found only for $x\in [\frac{-1}{e}, \frac{1}{e}]$ and what about a series for $W_-1(x)$ if it is no series ...