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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 ... calculus integration sequences-and-series definite-integrals lambert-w- 5,225
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 $). logarithms functional-equations lambert-w- 83
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 $$f(x)=\log\Big[\Gamma(ax+b)\Big]+\... approximation gamma-function newton-raphson lambert-w- 215k
Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically
I tried using Lambert W function the following way $$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$$ but this only gives me one ... functions roots lambert-w- 11
How to find the solution of $x(e^x)+\ln(x)+c = 0$, where $c$ is constant? [closed]
For $x(e^x)+\ln(x)+c = 0$, where $c$ is constant. I believe the solution is solved using Lambert $W$-function and derivatives, but I can't figured it out, I can't find any ideas can someone tell me ... calculus linear-algebra derivatives logarithms lambert-w- 77
Lagrange inversion theorem of $x^r(x+k)$ to generalize the W Lambert function
Motivation: $2$ branches of Lambert $\text W_k(z)$ is a limit of the inverse of $x^n(x+c)$ which is probably expressible in terms of FoxH in Mathematica or through another general hypergeometric ... limits roots special-functions hypergeometric-function lambert-w- 5,332
$W_{-1} (x)$ series expansion?
By the Lagrange Inversion Theorem, one can derive the series expansion for the principal branch of $W_0(x)$: $$W_0(x)= \sum_{n=1}^{\infty} \frac{(-n)^{n-1}x^n}{n!}, \, |x| \leq \frac1e$$ For $x \in \... power-series lambert-w- 347
Formula for lower branch of Lambert W / Product log function
Is there some direct formula for the W-1 or lower branch of the Lambert W function? I saw that the Python package scipy and the function ... definite-integrals improper-integrals lambert-w- 1
Projecting a point onto a convex set given by Log-Sum-Exp
Motivated by a wish to encode signal temporal logic specs (with linear predicates) as optimization problems w/o mixed integer approaches, I've been attempting to find a way to define the projection ... convex-optimization projection gradient-descent lambert-w karush-kuhn-tucker- 56
solving for exact solution of $x^{x^x} = 17$
Im trying to solve for the exact solution of $$ x^{x^x} = 17 $$ I understand that the previous solution, $x^x=17$, does not have an exact closed form solution and requires use of the Lambert W ... exponential-function lambert-w- 21
Find parameter to catenary interpolate a specific point
I'm working with the catenary equation and this equation is given by $$ f(x) = a \cdot \cosh\left(\dfrac{x}{a}\right) $$ I know this function pass at the point $(x_0, \ y_0)$ and therefore I want to ... exponential-function hyperbolic-functions lambert-w hyperbolic-equations- 175
What is the formula for higher integration of Lambert´s W function?
First I will present the notation for higher integration, which I copied from Danya Rose \begin{align} J_x^n(f(x)) = \int ...\int f(x) dx^n \end{align} Here are the higher integrations of the ... integration lambert-w- 21
An equation involving the incomplete Gamma function
Fix $a, b>0$. For $x >0$, I need to solve the following equation in $y$: $$ \gamma(a, xy) = y^b $$ where $\gamma(s,t) = \int_0^t r^{s-1}e^{-r} dr$ is the incomplete Gamma function. How do I ... gamma-function lambert-w- 59
How to Interpret Lambert W Function?
I just used an online calculator to calculate the following values of t. 1.) I am confused about how I am supposed to interpret the $W_{-1}$ and $W_0$. Could anyone help me out? Here is my original ... lambert-w- 27
How is $W_{-1}(-\frac{2}{3}\cdot e^{-\frac{2}{3}}) = -1.429$ if $W(x)$ is inverse of $f(x)=x\cdot e^x;x<-1$?
$$ 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 ... lambert-w- 43
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