Questions tagged [lambert-w]
For questions related to the Lambert W or product log function, the inverse of $f(z)=ze^z$.
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Does $\ln(\frac{2x}{\ln(2x+1)})\sim W(x)$?
The function $f(x)= \ln(\frac{2x}{ln(2x+1)})$ when plotted is similar to the plot of the Lambert W function. The Wolfram Alpha says that the limit $\lim_{x\rightarrow\infty} \frac{W(x)}{f(x)}=1$, but ...
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Residue involving Lambert function: $\underset{z=\frac{2}{\pi}W(\frac{\pi}{2})}{\text{Res}}\dfrac{1}{(e^{-\pi x}-x^2)^n}$
Context
I was tring to find a way to evaluate the infinite tetration of $i$ defined as:
$${}^{\infty}i:=i^{i^{i^{.^{.^{.}}}}}$$
1° attempt
Considering $z=i^{i^{i^{.^{.^{.}}}}}$ we can notice that $z=...
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Second derivative of Lambert W function [closed]
I am aware that the derivative of Lambert W function: dW(x)/dx= W(x)/x[1+W(x)] but I cannot solve for d^2W(x)/dx^2. It would be a great help for me if anybody could provide the solution to my problem
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How to algebraically derive the inverse of $x + e^x = y$?
Okay, so here's an approach I took:
$$ x + e^x = y $$
$$ e^{[x + e^x]} = e^y $$
$$ e^x e^{e^x} = e^y $$
$$ e^x = W(e^y) $$
$$ x = \ln{W(e^y)} $$
Where $W(z)$ is Lambert W function.
This works, but if ...
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A geometry problem involving the Lambert $W$ function [closed]
For what $x$ value can $W(x)$, $W(2x)$ and $W(3x)$ be the sides of a right triangle where $W$ denotes the Lambert $W$ function?
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Are there any complex solutions to the equation ${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$? [closed]
Thought of this question after learning about the Lambert W function and wanted to challenge myself. Are there any complex solutions to the equation $${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$$
Tried to work ...
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1
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1
answer
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Does Lambert W function works for vector valued equations
I faced an algebraic equation that I'm not sure if there's a closed form solution for it.
The equation is
$$
\bf{A} Z + b \exp(\mu + 1^T.Z) = 0
$$
Where $\bf{A}$ is a diagonal matrix and $\bf{b}, 1, Z$...
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1
answer
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generalizations of Lambert function for the solution of equation
Is there a general method to get the solution of
$$ xf(x)=a,$$
where $f(x)$ is an smooth analytic function?
So the solution can be expressed as $ g(a)$ and $g(a)f(g(a))=a
$ for some function $g(x)$ ...
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1
answer
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Can $mx + n + e^x = a$ be solved with Lambert W function?
So I've encountered this exercise where it asks you to find given $a, m, n \in \mathbb{R}$ an $x \in \mathbb{R^+}: mx + n + e^x = a$. I was told it can be algebraically solved using Lambert's W ...
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Solve for $m(t)$ in the integral transform $\int_0^1(1-t^n) m(t) dt=\frac{(n+1)^n}{n^{n-1}} $ for $n>0$.
Background
(You can skip this part, but maybe you find it interesting.)
Is $ \displaystyle f_1(x,v) = \sum_{n=0}^{\infty} \frac{x^n}{(n!)^v } > 0 $ for all real $x$ and $0<v<1$ ?
Lets start ...
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Is there a way to simplify this expression $ (i\cdot T+ e^{-i\cdot T} -1)$
I've been using Maple to solve some problems at hand. In particular, when I solve this integral
$$ \int_{0}^{T} {(e^{i\cdot (t - T)} - 1)\cdot (N-a) \over b} dt = S $$
I get the following solution:
$$ ...
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Exercise for student about $y\simeq P$
Inspired by On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$? I made an exercise for the student ...
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How can I solve the equation $y=e^{\cos(x)}\sin(x)$?
I was reading about the Lambert W function, and I want to know if it is possible to extend the ideas to solve the given equation for real values of x. $$y=\sin(x)e^{\cos(x)} $$
I know that the W ...
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How would you solve $3^x = 2x + 3$ using the Lambert $W$ function
Could someone provide a solution to the equation
$$
3^x = 2x+3.
$$
Our teacher told us to solve it graphically, but I was curious what the exact answers might be and just plugged it into Wolfram ...
0
votes
1
answer
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Solving $-z e^{-z} = (x-z) e^{x-z}$ with $x,z \neq 0$
I have an equation $$-z e^{-z} = (x-z) e^{x-z}$$ where $z=\frac{r S}{P}$ and $x=\frac{r y (P-S)}{P}$. I know that $-z \in (-\infty,-1)$ and $x-z \in (-1,0)$ with $x \neq 0$. Is there any way to solve ...
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How can I approximate this this equation $(ax+b)\exp(-cx) = (fx+d)$ to real-number?
I am solving this equation
$$(ax+b)\exp(-cx) = (fx+d)$$ using generalized Lambert W function and $a,b,c,d,f,d$ are all real-valued.
I drew this equation's graph using Matlab, and I confirmed this ...
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1
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How do you find solutions to the equation $\frac{x}{y^\frac{x}{y}} = 1$, which involves the Lambert W function?
How do you solve for $x$ in the following equation:
$$
\frac{x}{y^\frac{x}{y}} = 1
$$
By graphing it, I know there are two real solutions. I tried doing it in the following way:
$$
x = y^\frac{x}{y} \\...
0
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1
answer
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How to solve equation $(ax+b)\exp(x) = (cx+d)$ using Lambert W function [closed]
Are there any closed form solutions to
$$(ax+b)\exp(x)= cx+d$$
for real-valued $a,b,c$ and $d$?
4
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3
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views
Is there an exact solution to $x \sinh\Big(\frac{1}{x}\Big) = a$?
Is there an exact formula for solutions to the equation $x \sinh\Big(\frac{1}{x}\Big) = a$ where $a,x \in \mathbb{R}^+$? And if not, why?
I tried to rearrange to apply Lambert W somewhere to no avail.
...
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0
answers
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How many solutions for a Lambert W function
So by definition, for $ye^y=c$, $W(c)$ is the set(let's say A) of all solutions which satisfy y. I want to find what is $n(A)$
Here's what I got:
Let $f(x)e^{f(x)}=c$, then $n(B)$ where $B$ is $A\cap \...
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Equations similar to Lambert-$W$ with quadratic exponents
I've seen solutions saying that an equation in the format:$$ \ln(x) - \frac{bx}{a} = - \frac{bc}{a} $$
can be solved using the Lambert W function and I am comfortable doing so. My equation however is ...
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1
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Using Lambert W to solve for time of flight of a projectile with air resistance
picture from Wikipedia page about projectile motion, under air resistance section
Wikipedia: Projectile motion - Derivation of the time of flight:
$$c_1t+c_2+c_3e^{c_4t}=0$$
I understand how the ...
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Does the second positive solution (besides $x = 1$) of the equation $e^{x^2-1}=x^3-x\ln x$ have a closed form?
Does the equation $$e^{x^2-1}=x^3-x\ln x$$ have a closed form solution ?
The given equation has $2$ positive real roots. Graphically
It is not hard to see that $x=1$ is a rational solution. The ...
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2
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Inverse function for $f(x)=x/(1-e^{-x})$
I'm looking for the inverse function for $f(x)=x/(1-e^{-x})$, over the domain $x>0$.
Wolfram says that the answer is $f^{-1}(x)=x+W(-xe^{-x})$, for $x>1$, where $W(x)$ is the Lambert W function. ...
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Interpreting/understanding the lambertW on Maple software
I decided to use the Maple software to help me solve dynamic optimization problems, and I found this final solution for T. (example in the picture).
example
What does the LambertW mean for the ...
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Help me find the inverse of this function
While working on a recreational math problem I have come to the point were I need to find the inverse of one of the two following functions;
$$\frac{W_0(x)(W_{-1}(x)+1)}{(W_0(x)+1)W_{-1}(x)}$$
$$\frac{...
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3
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Finding a real closed-form solution to a tricky transcendental equation
One of my friends posed the following question with a prize of free bubble tea for anyone who could find a closed-form solution to the problem (not necessarily in terms of elementary functions):
What ...
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1
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Equation involving $e^x$ and $x$
I want to solve this equation with parameter $a>0$ and find non-zero values of $x$, for $x<\frac{-1}{e}$:
$\frac{a+x}{a}=e^x$.
Lambert function is not applicable since it gives only $0$ (because ...
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2
answers
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Is Lambert W-function possible for this type of equation? [closed]
I am trying to find the Lambert W solution of the following equation, but I am not able to proceed further because I am not able to put the equation in the form of $x=W(x)\exp(W(x))$.
The equation is:
...
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1
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Solving an equation using Lambert function
Could anyone help me solve this equation in terms of $x$ using the Lambert $W$ function or even other methods:
$$x^{-a}e^{ - kx}
=c(n-x)^{-a}e^{ - k(n-x)}$$
Where $k$, $c$, and $n$ are constants, and $...
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6
answers
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How can we find Lambert W solution to $\dfrac {x\ln x}{\ln x+1}=\dfrac{e}{2}$?
Find all real solutions: $$\frac {x\ln x}{\ln x+1}=\frac{e}{2}$$
Cross multiplication gives $$2x\ln x=\ln (x^e)+e$$ I didn't see any useful thing here. I tried solving this equation in WA. The ...
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Approximations and limits for Lambert $W$ of decreasing functions
I'm aware of the following approximation that approaches the Lambert W Function:
$$
W(f(x))\approx\ln\left(f(x)\right)-\ln\left(\ln(f(x))\right)
$$
However, this approximation fails to capture the ...
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How to solve $x + 3^{x} = 4$ using Lambert W Function.
As stated in the title I am trying to solve the equation $$x + 3^{x} = 4$$ using Lambert W Function and which led me to the result $$x = 4 - \frac{W(3^{4} \ln{3})}{\ln{3}}$$ and driven by the belief ...
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Evaluate $\sum\limits_{n=0}^\infty\operatorname W(e^{e^{an}})x^n$ with Lambert W function
$\def\W{\operatorname W} \def\Li{\operatorname{Li}} $
Interested by $\sum_\limits{n=1}^\infty\frac{\W(n^2)}{n^2}$, here is an example where Lagrange reversion applies to a Lambert W sum:
$$\W(x)=\ln(...
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If $1 < x < e < y$ and $x^{1/x} = y^{1/y}$ then $y > e^2/x$.
If $1 < x < e < y$
and
$x^{1/x} = y^{1/y}$
then
$y > e^2/x$.
Since
$x^{1/x}$
is decreasing for
$x > e$,
this is equivalent to
$x^{1/x}
\lt (e^2/x)^{x/e^2}
$for $1 < x < e$.
Also, ...
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Upper bound on the Lambert function
Is there an upper bound on the Lambert function $$W(-\frac{k}{e})$$ for $0 < k < 1$? Or should the condition be $0 < k \leq 1$?
I know that $W(-\frac{1}{e}) = -1$; would it be okay to say $W(-...
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Reference books on Lambert function
Does anyone have a good reference book on the Lambert W function? I need it as reference for my thesis particularly, as reference for this
The general solution to $$x=a+be^{cx}$$ is $$x=a-\frac{1}{c}...
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How to solve $a \cdot x^{2} + b \cdot x + c + \exp\left( d \cdot x + e \right) = 0$ for $x$ where $x \in \mathbb{C} \cup \{ \hat{\infty} \}$?
General
Question: How to solve $a \cdot x^{2} + b \cdot x + c + \exp\left( d \cdot x + e \right) = 0$ for $x$ where $x \in \mathbb{C} \cup \left\{ \hat{\infty} \right\} \wedge \left\{ a,\, b,\, c,\, d,...
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Solving $B a^t + C a^{2t} + D t a^t + E t^2 + F t + G = 0$
I'm working my way throught a complex system of equations, and I reached a point where what I need to solve for $t$ may be written as follows (everything else is constant):
$$B a^t + C a^{2t} + D t ...
2
votes
3
answers
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Showing that $\ln(x)^2+2(x-1)\ln x-3x+1=0$ has only $2$ real solutions
Is there any elementary way to show that $\ln (x)^2+2(x-1)\ln x-3x+1=0$ has $2$ real solutions on $(0,\infty)$?
I did it by this way.
Let $f(x)=(\ln x)^2+2(x-1)(\ln x)-3x+1$.
signs of $f(\frac{1}{2})$...
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1
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On combinatoric equality $\sum_{i=1}^{n-1}{n\choose i} i^{i-1} (n-i)^{n-i-1}=2(n-1)n^{n-2}$
This originates from Cayley's formula of tree/forest counting. I'm using notations from reference: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/...
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How to rewrite $e^{-x} = -\ln(x)$ such that the Lambert W function can be applied?
Original problem:
Find the real solution of $$e^{-x} = -\ln(x)$$
The first thing I did was divide both sides by $e^{-x}$.
With a few more steps I arrived at
$$e^x\ln(x) = -1$$
This was great, but ...
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0
answers
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views
Evaluate the definite integral $\int_0^\infty \frac{W{(x)}}{xe^x}\,dx$
It is a well known identity that
$$
\int_0^\infty \frac{\log{(x)}}{e^x}\,d{x} = -\gamma
$$
This shows how the Euler–Mascheroni constant is directly connected to the exponential function and its ...
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Is the Lambert W function the Newton flow of the exponential function?
Is this right?
The Lambert W function, denoted by $W(z)$, is defined as the inverse function of $f(z) = ze^z$. In other words, if $w = W(z)$, then we have $z = w e^w$.
The continuous Newton's method ...
user1057203
2
votes
1
answer
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Show that $x/e^x<1.5/x^2$ for $x>0$
I have trouble with proving the following inequality
$$\frac{x}{e^x}<\frac{1.5}{x^2}$$ for $x>0$.
My attempt:
$$1.5e^x-x^3>0$$
Let $f(x)=1.5e^x-x^3$
$$f'(x)=1.5e^x-3x^2 $$
$$f'(x)=0$$
$$e^x=...
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2
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How to solve $x^{y^z}=z$
Initially I isolated the y in $x^y=y$, but I just wanted to expand the infinite power tower to two letters in the tower, but I can't solve for z in the equation $x^{y^z}=z$. I tried to use Lambert W ...
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how to verify the inverse compensator of an exponential Hawkes process?
I derived an expression for the inverse compensator of the Hawkes process with a sum of exponential kernels that I wrote about at https://vixra.org/abs/1211.0094 and want to verify that it is correct.
...
user1057203
0
votes
1
answer
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A equation about Lambert $W$ function
$$(\Delta X)(\Delta P)=\frac{\hbar}{2}\exp\left[\frac{\alpha^{2n}l_{PI}^{2n}}{\hbar^{2n}}\left(\langle\hat{P}\rangle^2+(\Delta P)^2\right)^n\right]$$
I want to solve the equation to get $\Delta P$ ...
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0
answers
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Solving a problem related to the Lambert-w function
For $f(x), g(x), x>0$, if
$$ g(x)>\frac{f(x)}{\log f(x)},$$
what is an upper-bound for $f(x)$? A clear approximation is $f(x) \sim g(x) \log g(x)$, but I do not see how to go further without ...
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1
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Find $\int_a^b(f(x)-g(x))\mathrm{d}x$ if $f(x)=x^{x^{1-x}}$ and $g(x)$ is the linear equation from the extrema $(a,f(a))$, $(b,f(b))$ of $f(x)∀x≥0$.
Find $\int_a^b(f(x)-g(x))\mathrm{d}x|$ if $f(x)=x^{x^{1-x}}$ and $g(x)$ is the linear equation from the extrema $(a,f(a))$, $(b,f(b))$ of $f(x)∀x≥0$.
Find the exact area.
The area under $g(x)$ as ...
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