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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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Solve for $y$ in $ye^{\frac{1}{2}(y-\frac{1}{y})}=x$

I'd like to solve the following equation for $y$ in terms of $x$ $$ye^{\frac{1}{2}(y-\frac{1}{y})}=x$$ This equation is close in nature to the definition of the Lambert $W$ function, but different ...
fewfew4's user avatar
  • 881
1 vote
2 answers
58 views

Solving $\frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0$ for $M \in \mathbb{R}$

I am trying to solve the following nonlinear equation analytically: $$ \frac{M e^{-M}}{1-e^{-1}} - \epsilon = 0 \, , $$ where $ M \in \mathbb{R} $ and $ 0 < \epsilon \ll 1 $. A solution can be ...
Siegfriedenberghofen's user avatar
0 votes
1 answer
87 views

Compose Lambert W and exponential in a numerically stable manner

I'm looking for a numerically stable way to evaluate $$ W_0(a \exp(b)), $$ where $W_0$ is the main branch of the Lambert W function, and $a > 0$. When $b$ is large, computing $\exp(b)$ can be ...
Alex Shtoff's user avatar
3 votes
2 answers
94 views

Are there other ways to express $W_0(Ax\exp{x})$?

I'm working on a physics problem where the Lambert W function arises. I get very close to being able to cancel it, as its argument is almost its inverse function, but not quite. The term I end up with ...
robmac's user avatar
  • 31
0 votes
1 answer
54 views

General formula for approach speed of a limit that approaches ln(x)

I recently learned of the limit $$\lim_{n\rightarrow\infty}n\left(\sqrt[n]{x}-1\right) = \ln(x),\;\; x\in\mathbb{R}^+$$ and played around with plotting $y=x\left(\sqrt[x]{a}-1\right)$ and $y=\ln(a)$ ...
joelproko's user avatar
5 votes
1 answer
104 views

How could we approximate $\int \frac{W(t) }{1+W(t)}\,\, \frac {\sin(t)} t \, dt$?

In a now deleted post appeared an interesting integral. $$I=\int\frac 1 x \,\,\sin \left(\frac{\log (x)}{x}\right) \,dx$$ which does not make (too much) problems from a numerical point of view. As one ...
Claude Leibovici's user avatar
0 votes
0 answers
39 views

Using the r-Lambert function to solve a system of transcendental equations

I am trying to use the r-Lambert function applied to a vector in order to solve a system of transcendental equations, however, I am facing some difficulties when trying to obtain the right expression ...
Ignacio Canabal's user avatar
0 votes
1 answer
136 views

How to use the Lambert's W function on this exponential equation: $3-x^2=2^x$? [closed]

How to use the Lambert's W function on this exponential equation: $3-x^2=2^x$? I am new to the Lambert's W function and there is barely anything I could find on Google.
Toshiv's user avatar
  • 21
0 votes
0 answers
160 views

How do I solve the following equation exactly: $\sin(x)^{\cos(x)}=2$? [duplicate]

I solved the following equation yesterday for fun and I have the exact result (But these are just some solutions, not all): $$\sin(x)^{\sin(x)}=2 \iff x=\frac{\pi}{2}-i\ln (\exp^{W(\ln(2))}\pm{\sqrt{\...
Mordor_07's user avatar
1 vote
2 answers
72 views

Solving an equation analytically of the type $n \cdot x^{n - 1} \cdot y + \frac{x^n}{z} = w$ for n

I am currently faced with solving the following equation for my thesis. I am looking for a symbolic answer to n $$ k \left( n \cdot d_{\text{rz}}^{n-1} \cdot \frac{\partial d_{\text{rz}}}{\partial r} +...
WissenschaftLink's user avatar
1 vote
1 answer
44 views

Bounding the solution of a logarithmic equation

Given a small number $\varepsilon >0$ and a constant $1/3\le \alpha < 1 $, I am looking for the smallest possible number $x^*$ such that for all real $x\ge \max\{x^*,3\}$, we have $$\frac{x}{(\...
Stratos supports the strike's user avatar
0 votes
2 answers
99 views

Proof using the Lambert W function that 1 = 0 - What went wrong?

All values that satisfy $x^2=2^x$ would satisfy $\ln(2)x^3 = x\ln(2)e^{x\ln(2)}$, and would therefore satisfy the relationship $W(\ln(2)x^3) = x\ln(2)$. The problem is that when I graph these ...
Alexandra's user avatar
  • 453
1 vote
2 answers
259 views

Solve $x^x-5x+6=0$ using Lambert W function.

How do I solve $x^x - 5x + 6 = 0$ using the Lambert W function? EDIT: I solved equations $2^x - 5x + 6 = 0$ and $3^x - 4x - 15 = 0$ using Lambert W function, but not able to solve $x^x - 5x + 6 = 0$ ...
Prashant Kumar G's user avatar
2 votes
1 answer
73 views

Asymptotics of Lambert W function

I want to show $$ e^{W(n+1)}-e^{W(n)} = o(1) $$ for $n\to\infty$. Since I'm not that used to the Little-o notation I'm wondering if my reasoning is correct. I tried to prove it in the following way ...
Ellenier's user avatar
  • 117
1 vote
2 answers
137 views

How to solve $\ln(x) = 3\left(1-\frac{1}{x}\right)$?

I have been working for this problem for a while: $$\ln(x) = 3\left(1-\frac{1}{x}\right)$$ and by graphing, plugging and chugging values and rigorously doing the math, I can clearly see that one of ...
SMK's user avatar
  • 69
2 votes
1 answer
85 views

Asymptotics of Lambert W

Equations (145)-(148) of this paper claim that under certain conditions the function $y(\epsilon)$, given by solving $f(y)=\epsilon$ where $f(y)=a\left(\frac{b}{\sqrt{y}}\right)^y$, has the following ...
mavzolej's user avatar
  • 1,482
0 votes
1 answer
75 views

How to find the complex roots of $2^z = z^2$?

Plotting the function on the real axis reveals that it has three solutions Therefore the solution set is: $$ z = \left\{-\frac{2}{\ln2}W\left(\frac{\ln2}{2}\right),2,4\right\} $$ But plotting in the ...
Aster's user avatar
  • 1,230
0 votes
1 answer
112 views

2nd order ODE with Lambert W function term

I developed a mathematics model which can be described by 2nd order equation with constant coefficients as shown below. $$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} + A\frac{\mathrm{d} y}{\mathrm{d} x}- B\ W(...
matt7's user avatar
  • 3
-3 votes
1 answer
109 views

Multiple roots in equation in $e^x = x^{nx}$ [closed]

When I was observing the increment trend difference between exponential functions and power functions, I decided to try solving the roots of the two equations. The most apparent way is by Lambert W ...
Forest Lam's user avatar
0 votes
1 answer
112 views

Why Lambert answer doesn't satisfy the original equation in WolframAlpha

I solved the differential equation $y(y'+a)=b$ and found the answer $$ay+b\ln(y-\frac{b}{a})=-a^2(t+c)$$ as I wanted the explicity form of my solution, by giving it to WolframAlpha at address https://...
next_mokami's user avatar
1 vote
1 answer
58 views

How to plot $W(\exp(-x))$ in wolframalpha or sage

I tried to plot the function $W(\exp(-x))$ in both WolframAlpha and Sage and I got: No result in Wolframalpha (empty 3d box or 2d chart). Empty set in SAGE. Any help?
riemannium's user avatar
  • 1,069
3 votes
2 answers
199 views

Need your help in solving $\log_\sqrt[34]{2} x = 4x^4-3x^3-2x^2+x$

I have been playing with graphs until made a nice equation $$\log_\sqrt[34]{2} x = 4x^4-3x^3-2x^2+x$$ The real answers are 1 and 2. But how to solve it? And is it possible to stretch it to complex ...
A PIG's user avatar
  • 39
10 votes
3 answers
1k views

Finding the smallest set of values that make an exponent of a known value. [closed]

I need some help with figuring out an equation or process? Not sure what the correct term is? Full disclosure: Unfortunately I’m not even remotely a mathematician so I’m grateful for your patience. If ...
Oblivious George's user avatar
0 votes
1 answer
88 views

How to solve a functional equation using the $W$ function?

The equation to solve: $$a^x+bf(x)+c = 0$$ Where $f(x)$ is a polynomial equation of degree $n$ without a constant term as it is covered by $c$. I have solved the case where $f(x) = x$: $$a^x+bx+c = 0$$...
Red Mermaid's user avatar
4 votes
3 answers
170 views

Compute $\int_{0}^{e}\frac{\ln(1-W(x))}{x}dx$

So I made this integral for funzies, and I'm having trouble solving it. $$\int_0^e\frac{\ln(1-W(x))}x dx$$ I may have made a mistake. I let $u=W(x)$, then the integral becomes $$\int_{0}^{1}\frac{(u+1)...
Silver's user avatar
  • 861
0 votes
1 answer
45 views

Implicit functions and exponentials

I am working with real numbers and trying to express u as function of x, given the equation: $x^2-u\,e^u = 4$. I am a bit lost because of the exponential part. I read something about the Lambert ...
hbillie's user avatar
2 votes
1 answer
151 views

How to solve the equation:$(2x)^{x}=14+x^{x}$?

How to solve the equation: $(2x)^{x}=14+x^{x}$ ? I am trying to solve this equation in this way: $(2x)^{x}=14+x^{x}\implies 2^{x}x^{x}=14+x^{x}\implies x^{x}(2^{x}-1)=14\implies x^{x}=\frac{14}{2^{x}-...
Syamaprasad Chakrabarti's user avatar
3 votes
2 answers
106 views

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 ...
Tio Zuca's user avatar
  • 380
3 votes
0 answers
83 views

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=...
Math Attack's user avatar
1 vote
3 answers
164 views

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 ...
Yuki Endo's user avatar
  • 217
0 votes
1 answer
51 views

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?
Mr. D's user avatar
  • 11
0 votes
1 answer
134 views

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 ...
number eight's user avatar
1 vote
1 answer
84 views

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 $\bf{b}, 1, Z$...
Nosrat Mohammadi's user avatar
0 votes
1 answer
48 views

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 $g(x)$ ...
Jose Perez's user avatar
0 votes
1 answer
73 views

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 ...
kantianmarxist's user avatar
5 votes
0 answers
143 views

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 ...
mick's user avatar
  • 16.4k
1 vote
2 answers
106 views

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: $$ ...
Meg's user avatar
  • 15
1 vote
3 answers
262 views

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 ...
Soham Saha's user avatar
  • 1,444
0 votes
2 answers
123 views

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 ...
Norbert Domokos's user avatar
0 votes
1 answer
51 views

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 ...
user42398423's user avatar
-1 votes
1 answer
115 views

How can I approximate 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 ...
SEUNGMIN SIM's user avatar
0 votes
1 answer
67 views

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} \\...
RJ Onyx Moonshadow's user avatar
0 votes
1 answer
134 views

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 $d$?
SEUNGMIN SIM's user avatar
4 votes
3 answers
213 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. ...
LeaG's user avatar
  • 45
0 votes
0 answers
43 views

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 \...
æîōü's user avatar
5 votes
2 answers
190 views

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 ...
Charles Brook's user avatar
0 votes
1 answer
85 views

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 ...
Samwit 7's user avatar
2 votes
2 answers
133 views

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 ...
hardmath's user avatar
  • 654
3 votes
2 answers
155 views

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. ...
CJstats's user avatar
  • 33
0 votes
1 answer
83 views

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 ...
Meg's user avatar
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