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

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Alternative representations for $\int_1^e e^{e^{t W_{-1}\left(\frac{-\ln(t)}{t}\right)}}dt$ and $\int_{e^{-e}}^{e^{\frac1e}}e^{t+W_0(-\ln(t))}dt$?

Some time ago I posted an answer in this post. The original question was about evaluating $$ \lim_{n\to\infty} \int_{0}^{\infty} \frac{e^x}{^{2n}x} \mathrm{d}x $$ where $^{2n}x$ is the power tower of ... user avatar Robert Lee

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Lambert W of a constant multiple

$ \ln(cx) $ can be expressed as $ f(c) + \ln(x) $, where $ f(c)=\ln(c) $. Does the lambert W function have a similar property? (Can $ W(cx) $ be expressed as $ f(c)+W(x) $ for some function $ f $). user avatar Sky Jaheim

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Looking for $B$ such that $\Gamma(ax+b) \prod_{i=1}^n (ax+e_i)\sim \Gamma(ax+B)$

Quite often, I need to solve for $x$ $$y=\Gamma(ax+b)\prod_{i=1}^n (c_ix+d_i)\tag 1$$which, numerically does not make much problems looking for the zero of function $$f(x)=\log\Big[\Gamma(ax+b)\Big]+\... user avatar Claude Leibovici

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Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically

I tried using Lambert W function the following way $$e^{k(t-1)} -t=0$$ $$e^{k(t-1)}=t$$ $$-ke^{-k} = -kte^{-kt}$$ $$W(-ke^{-k}) = W(-kte^{-kt})$$ $$-k = -kt \implies t = 1$$ but this only gives me one ... user avatar Adam Shegan

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How to find the solution of $x(e^x)+\ln(x)+c = 0$, where $c$ is constant? [closed]

For $x(e^x)+\ln(x)+c = 0$, where $c$ is constant. I believe the solution is solved using Lambert $W$-function and derivatives, but I can't figured it out, I can't find any ideas can someone tell me ... user avatar gerard hotaro

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Lagrange inversion theorem of $x^r(x+k)$ to generalize the W Lambert function

Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is probably expressible in terms of FoxH in Mathematica or through another general hypergeometric ... user avatar Tyma Gaidash

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$W_{-1} (x)$ series expansion?

By the Lagrange Inversion Theorem, one can derive the series expansion for the principal branch of $W_0(x)$: $$W_0(x)= \sum_{n=1}^{\infty} \frac{(-n)^{n-1}x^n}{n!}, \, |x| \leq \frac1e$$ For $x \in \... user avatar Mailbox

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Formula for lower branch of Lambert W / Product log function

Is there some direct formula for the W-1 or lower branch of the Lambert W function? I saw that the Python package scipy and the function ... user avatar testerp2034

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Projecting a point onto a convex set given by Log-Sum-Exp

Motivated by a wish to encode signal temporal logic specs (with linear predicates) as optimization problems w/o mixed integer approaches, I've been attempting to find a way to define the projection ... user avatar Zach

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solving for exact solution of $x^{x^x} = 17$

Im trying to solve for the exact solution of $$ x^{x^x} = 17 $$ I understand that the previous solution, $x^x=17$, does not have an exact closed form solution and requires use of the Lambert W ... user avatar Jacob

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Find parameter to catenary interpolate a specific point

I'm working with the catenary equation and this equation is given by $$ f(x) = a \cdot \cosh\left(\dfrac{x}{a}\right) $$ I know this function pass at the point $(x_0, \ y_0)$ and therefore I want to ... user avatar Carlos Adir

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What is the formula for higher integration of Lambert´s W function?

First I will present the notation for higher integration, which I copied from Danya Rose \begin{align} J_x^n(f(x)) = \int ...\int f(x) dx^n \end{align} Here are the higher integrations of the ... user avatar Weeeeeooooo

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An equation involving the incomplete Gamma function

Fix $a, b>0$. For $x >0$, I need to solve the following equation in $y$: $$ \gamma(a, xy) = y^b $$ where $\gamma(s,t) = \int_0^t r^{s-1}e^{-r} dr$ is the incomplete Gamma function. How do I ... user avatar gm01

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How to Interpret Lambert W Function?

I just used an online calculator to calculate the following values of t. 1.) I am confused about how I am supposed to interpret the $W_{-1}$ and $W_0$. Could anyone help me out? Here is my original ... user avatar Benjamin Little

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How is $W_{-1}(-\frac{2}{3}\cdot e^{-\frac{2}{3}}) = -1.429$ if $W(x)$ is inverse of $f(x)=x\cdot e^x;x<-1$?

$$ f(x)=x\cdot e^x \space ;\space x<-1 \\ W_{-1}(x)=f^{-1}(x) \\ W_{-1}(x\cdot e^x)=f^{-1}(x\cdot e^x)=x $$ But $W_{-1}(-\frac{2}{3}\cdot e^{-\frac{2}{3}}) = -1.429$ not $-0.66 \dot 6$? It may be a ... user avatar UWU gg

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