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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vote
2answers
45 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
votes
1answer
44 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
vote
1answer
50 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
46 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 ...
0
votes
1answer
25 views
solve non linear differential equation: $y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0$
Could somebody help me to solve the non linear differential equation, where $y$ is a function of the time and starts with $y(0)=0$
$$
y'\cdot\alpha+y+\beta\cdot e^{\delta\cdot y}+\theta = 0
$$
It will ...
0
votes
1answer
55 views
An aproximation of the lambertw function for a complex number
Here is my problem,
I used the fact that $W(x)=\ln(x)-\ln(W(x))$, replacing $W(x)$ by $\ln(x)-\ln(... $ a lot amount of times and it seems to works for simple $x$ but when I try with, for example, ...
9
votes
1answer
92 views
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 ...
8
votes
1answer
145 views
Evaluation of an integral involving the Lambert W function
Wikipedia claims that
$$\int_0^\infty W\left(\frac{1}{x^2}\right) \,\text dx=\sqrt{2\pi}$$
and a numerical computation seems to confirm this.
How can this result be proven?
