Questions tagged [lambert-w]
For questions related to the Lambert W or product log function, the inverse of $f(z)=ze^z$.
815 questions
3
votes
1
answer
79
views
Definite Integration with Lambert W-Function $\int_0^\infty e^{-W(x)^2} \, dx$
I am interested in (elegantly) solving the „Lambert-Gaussian-Integral“ ;) $$I:=\int_0^\infty e^{-W(x)^2} \, dx$$ I tried the substitution $u = W(x)$, so $x = u e^u$, $dx = e^u (u + 1) \, du$.
The ...
- 1,179
10
votes
1
answer
211
views
Definite integral with the Lambert W Function
I am trying to find the integral
$$
\int^{1}_{0} \frac{\mathrm{W} \left( -x\log x\right) }{-x\log x} \mathrm{d} x=\frac{\pi^{2} }{12}
$$
By expanding the Lambert W-Function into a series, one ...
- 1,179
1
vote
4
answers
179
views
Simple approximations to the Lambert W function principal branch $W_0(x)$
I am wonder if there are simple approximation to the $W_0(x)$ function, in Wikipedia there is shown some that are not exactly simple (power series, some for very big numbers), and in Google I didn't ...
- 1,985
3
votes
0
answers
50
views
How can I formally prove this real integral representation as an analytic continuation of the Lambert W function?
I have been investigating the following real-valued definite integral form that appears to act as an analytic continuation of the Lambert W function into the complex plane (except along the negative ...
0
votes
1
answer
94
views
$|u| \ll \log z$ implies that $e^u \sim \frac{1}{\log z}$?
In this paper on page $347$ (PDF page $19$) the authors state that $$ \left( 1 + \frac{u}{\log z} \right) e^u = \frac{1}{\log z} $$ and the assumption that $|u| \ll \log z$ implies that $e^u \sim \...
- 2,155
2
votes
3
answers
178
views
$\int_0^1 \frac{dx}{1 + x \exp(x)} = A = \omega ?$
Consider
$$\int_0^1 \frac{\mathrm dx}{1 + x \exp(x)} = A = ?$$
Now my first guess was this is the omega constant $\omega = W(1)$ where $W$ is the lambert W function.
And I guess some substition and ...
- 17.9k
2
votes
2
answers
180
views
Why does the given integral expression for $u_{0}$ solve $u\left ( 2u- 1 \right )\ln u= 1$ analytically, and what is the math behind it?
I am trying to maximize the function $f\left ( x \right )= xe^{-2^{x}\left ( 2^{x}- 1 \right )}$ for $x> 0$. To find the critical points, I computed the derivative and set it to zero, which led to ...
- 272
1
vote
1
answer
96
views
Are fixed points of $x = a^x$ structurally linked to inverse prime-counting formulas?
I've been exploring two seemingly different equations:
The inverse of the Prime Number Theorem suggests:
$$
x \sim \frac{n}{W(n)}
$$
as an approximation to solve $\pi(x) = n$.
The fixed point ...
- 23
3
votes
1
answer
210
views
Solving in closed form: $e^{x^3}\ln x + x^2 = 0$
My task is to solve $$e^{x^3}\ln x + x^2 = 0$$ with Lambert $W$ function.
I could not derive the form $ue^u=a$ .
$$e^{x^3}\ln x + x^2 = 0$$
$$\ln x=-\frac {x^2}{e^{x^3}}$$
$$x=e^{-\frac {x^2}{e^{x^3}}}...
user1615370
7
votes
2
answers
189
views
Asymptotic behavior of the recurrence $x_{n+1} = x_n + a^{x_n}, 0 < a < 1$
Consider the series $x_n$ such that $x_1 = 1$ and for some fixed $a \in (0, 1)$ this series satisfies the recurrence $$x_{n+1} = x_n + a^{x_n}.$$ What is the asymptotic behavior of $x_n$, in ...
0
votes
2
answers
142
views
Can the Lambert $\mathrm W$ function be used to solve $x^{x-1} = 75?$ [duplicate]
I have this equation to solve for $x$:
$$x^{x-1} = 75$$
I can't solve this and I don't know if Lambert $\operatorname W$ function can work on this.
Until now, I wrote this:
$$(x-1)\ln x = \ln 75$$
$$\...
2
votes
2
answers
159
views
What algebraic technique can solve for x in $c= \frac{x}{\log_{10}\left ( a+ (bx)^{-d} \right )}$?
I encountered this problem in hydraulics and my textbook states that no closed form solution for $V$ exists in the Darcy-Weisbach equation (with Swamee-Jain's friction equation, $f$),
\begin{align*}
...
5
votes
3
answers
204
views
A series regarding omega constant and the Lambert $W$ function
I find that $$a_n:= \sum_{j=1}^{n-1} \binom{n} {j} \frac{(-j )^{j-1}} {( n+j )^{j}}$$
then $\lim_{n\to \infty} a_n = W(1) = \Omega$
Further if we make the series a ploynomial of $x$ i.e.
$$
f_n(x):= \...
- 111
2
votes
2
answers
105
views
Solving Exponential Function using the W Lambert Function - Clarification
I have a function I've been trying to solve of the form: $e^{\alpha t}(\alpha + D) = C$, where $\alpha$ is my unknown, $t$ is a time parameter, and $D$ and $C$ are constants. Based on what I ...
- 21
2
votes
1
answer
201
views
Preferential sampling and balls form the urn
Consider an urn with an initial state of $n \ge 1$ red balls and $n$ white balls. Draw a ball from the urn, uniformly at random, and note its color. If the ball is white, do not replace it; if the ...
- 111