Questions tagged [lambert-w]

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

Equations involving exponentials multiplied to linear terms, such as $x+2=e^x$, cannot be solved using elementary functions in general. However, they may be solved in closed form using the Lambert W function, the inverse of $f(z)=ze^z$. In the given example: $$x+2=e^x$$ $$(-x-2)e^{-x-2}=-e^{-2}$$ $$-x-2=W(-e^{-2})$$ $$x=-W(-e^{-2})-2$$ There are two real branches of the Lambert W, $W_0(z)$ for $z\in[-1/e,\infty)$ and $W_{-1}(z)$ for $z\in[-1/e,0)$, as well as infinitely many complex ones.

As discussed in the Corless et al. reference, the use of $W$ follows early Maple usage. Exact solutions to some mathematical models in the natural sciences, such as Michaelis–Menten kinetics, use this function.

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Interesting integral related to the Omega Constant/Lambert W Function

I ran across an interesting integral and I am wondering if anyone knows where I may find its derivation or proof. I looked through the site. If it is here and I overlooked it, I am…
Cody
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An iterative logarithmic transformation of a power series

Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion: $$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$ Then, at each step we apply the following…
Vladimir Reshetnikov
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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 Supan"(?): $$y(x) = W(e^{a+x}) \approx x\left(1 -…
nbubis
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Closed form of $\sum_{n = 1}^{\infty} \frac{n^{n - k}}{e^{n} \cdot n!}$

When seeing this question I noticed that $$ \sum_{n = 1}^{\infty} \frac{n^{n - 2}}{e^{n} \cdot n!} = \frac{1}{2}. $$ I don't know how to show this, I tried finding a power series that matches that but no avail. Hints are very much appreciated. But…
Ramanujan
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Prove that $\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$ where $\Omega e^\Omega=1$

Whilst reading this Math SE post, I saw that the OP mentioned the integral $$\int_0^\infty \frac{1+2\cos x+x\sin x}{1+2x\sin x +x^2}dx=\frac{\pi}{1+\Omega}$$ where $\Omega$ is the unique solution to the equation $$xe^x=1$$ However, the question was…
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Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, primarily by exploring various manipulations…
Archaick
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Does $\int_0^{\infty} \left( p + q W \left( r e^{- s x + t} \right) + u x \right) e^{- x} d x$ have a closed-form expression?

Does $\int_0^{\infty} \left( p + q W \left( r e^{- s x + t} \right) + u x \right) e^{- x} d x$ (with 6 variables) where W is the Lambert W function (also known as ProductLog in Mathematica) have a closed-form expression? If we drop the variable $s$…
crow
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Complete monotonicity of a sequence related to tetration

Let $\Delta$ denote the forward difference operator on a sequence: $$\Delta s_n = s_{n+1} - s_n,$$ and $\Delta^m$ denote the forward difference of the order $m$: $$\Delta^0 s_n = s_n, \quad \Delta^{m+1} s_n = \Delta\left(\Delta^m s_n\right).$$ We…
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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 $\Omega = {\rm W}_0(1) = 0.56714329\ldots$ is the Omega constant while…
omegadot
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Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation $$n!=a^n\, 10^k$$ I used for $n!$ Stirling…
Claude Leibovici
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Closed form of an improper integral to solve the period of a dynamical system

This improper integral comes from a problem of periodic orbit. The integral evaluates one half of the period. In a special case, the integral…
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Closed form of an integral involving Lambert function

I'm trying to compute the following integral explicitly. $$I=\int_{0}^{+\infty} dx \left(1+\frac{1}{x}\right) \frac{\sqrt{x}}{e^{-1}+xe^x}$$ The best I managed to do is to do a change of variable $x=W(y)$, where W is the Lambert function. The…
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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 that $x=\operatorname{W}_k(a)$ where…
user164524
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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 it's evaluation. Any ideas as to how I can…
techriften
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Are those two numbers transcendental?

Suppose, $u$ solves the equation $$u^u=\pi$$ and $v$ solves the equation $$v\cdot e^v=\pi$$ So, we have $u=e^{W(\ln(\pi))}$ and $v=W(\pi)$. $u$ and $v$ should be the real solutions (in this case, they are unique). If someone can prove that $u$ and…
Peter
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