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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Solving $x^y = y^x$ analytically in terms of the Lambert $W$ function

I'm interested in deriving the solution for $y$ in terms of $x$ given $x^y = y^x$ using the Lambert $W$ function. Wolfram Alpha states: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
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0answers
17 views

Connections between Lambert ProductLog and Bernoulli numbers

The Bernoulli numbers have many (as demonstrated here). Here is one property/characteristation: $$\frac{t}{e^t-1}=\sum_{k=0}^\infty \frac{B_k}{k!} t^k$$ Conspicuously missing from the MathOverflow ...
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1answer
28 views

Help Obtaining Numerical Approximation of Lambert W Solution

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function ...
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1answer
33 views

Lambert-W Function Integral

Stumbled upon this integral while doing some research. I'd love to see the different methods used to prove this subtle integral! $$\Large{\int_{0}^{\infty}{\dfrac{ \operatorname{W}(x)}{x\sqrt{x}} ...
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3answers
67 views

How to solve for $y$ the equation $x= y^y$? [duplicate]

I need an equation where I receive a number that when raised to itself equals the input. Formally: in $x=y^y$ solve for $y$. Intro to Calculus level knowledge. If the Lambert function is necessary, ...
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1answer
27 views

A problem in generalizing the Lambert's W function

The Lambert's Omega function has 2 real branches denoted by $W_{-1}(x)$ and $W_0(x)$ and it represents the solution(s) of the equation $xe^x=a$. I learned that this function can be generalized and ...
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2answers
143 views

Solving $\ln{x}=\tan{x}$ with infinitely many solutions

Lets take $f(x)=\ln{x}$ and $g(x)=\tan{x}$ When $f(x)=g(x)$ that is $\ln{x}=\tan{x}$, we see that the graph is like: Hence we see that there are infinitely many solutions to $x$ but the two ...
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1answer
120 views

How to solve the exponential inequality $x+3^x<4$

How to solve the inequality $$x+3^x<4$$ This problem is found in Spivak's calculus, ch 1 - the highly praised work - which is supposed to be a gentle introduction for beginners in mathematics. ...
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0answers
9 views

Minimum of function involving exponentials

I am trying to prove that this function involving exponentials: $g(x)=\frac{\sqrt{2 \pi } \left(1-2 e^{-2 \pi ^2 x ^2}\right) x }{2 e^{-\frac{1}{8 x ^2}}+\sqrt{2 \pi } x -1}$, when $x\geq1/2$ Is ...
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3answers
114 views

How to solve $x^2 = e^x$

The question is to find $x$ in: \begin{equation*} x^2=e^x \end{equation*} I know Newton's method and hence could find the approx as $x\approx -0.7034674225$ from \begin{equation*} ...
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1answer
28 views

Differential equation solution with Lambert $W$ function.

Solving the differential equation: $y'x\log y =1$ we easly find : $$ y(\log y-1)=\log x +c $$ I search an explicit solution $y=f(x)$ and WolframAlpha gives: $$ y=\dfrac{\log x+c}{W\left( \dfrac{\log ...
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2answers
115 views

How to solve equation $ x=W(a+bx^{n})+1 $?

How i can resolve the equation $x=W(a+b x^n)+1$, where $W$ is the Lambert $W$ function? thanks
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2answers
229 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
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1answer
43 views

Lambert W-Function

Is there a standard name for the inverse of the Lambert W-Function, in the manner that the name "exponential function" is the name for the inverse function of the logarithmic function.
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1answer
93 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
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2answers
49 views

Bound for function involving Lambert W

Given that $W(x)$ is the Lambert W function, how can one prove that $$(2+W(x))e^{-W(x)}\leq 2 \frac{\log^2 x}{x}, \quad x\geq e^2$$ Is it possible to generalize this and find a function $f(x)$ such ...
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0answers
46 views

Complex number as function of real number

While seeking all solutions of $ Z ^ 2 = 2 ^ Z $ we have three real roots of $Z : z_1=2, z_2=4, $ and a third real root given in terms of LambertW function: $ z_3=-\frac{2 W\left(\frac{\log ...
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3answers
264 views

Request for help to solve an equation with LambertW: $ (x^2-4\,x+6) e^x =y$

I want to solve the following equation: $$ (x^2-4\,x+6) e^x =y \tag{1} $$ It looks a bit like the following equation: $$ x e^x =y \tag{2} $$ Since the solution of equation (2) is: x=LambertW(y), ...
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2answers
150 views

How do i solve $x^2/3=2/3^x$

How is this done? $${x^2\over3}={2\over3^x}$$ I'm out of ideas. It seems simple but it is quite not so the case in my perception of it.
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1answer
103 views

Can I solve this with a Lambert Function?

New to W-Functions and do not understand it properly. How do I solve this equation? I know about numerical solutions (or graph solution), but I'm interested in pure algebraic solution if it exists. ...
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2answers
51 views

Solving $z=w/2-\sin(tw)/(2t)$ for $w$

Is it possible to solve $$z=\frac{w}{2}-\frac{\sin(tw)}{2t},$$ for $w$? My first thoughts were that we would have to be careful about the domain of $f(w)$ so that the inverse was actually a function ...
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5answers
113 views

Graph of the function $x^y = y^x$, and $e$ (Euler's number).

Earlier, I was using the Desmos Graphing Calculator, and I wanted to remind myself of what the graph of the function $x^y = y^x$ looked like. If you have never seen what it looks like before, it is ...
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1answer
41 views

Prove graphically that the Lambert equation has exactly zero, one or two roots

I need some help on the below problem. Consider the Lambert equation: $xe^x = a$ for real values of x and a (a) Show graphically that the equation has exactly one root $ \xi(a) \ge 0 $ if $ ...
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1answer
63 views

Two kind of equations involving natural log and exponentiation

I know how to solve equations using Lambert's W function like $xe^x=k$ or $e^x+x=k$ But how can I solve this two kinds of equations involving natural log ? $e^x \ln(x)=k$ and $e^x+\ln(x)=k$ I ...
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1answer
76 views

Transcendental equations involving more than 2 terms

I now how to solve transcendental equations involving only two terms like: $xe^x=k$ $x=W(k)$ Where W(x) is the Lambert's Omega function. But how can I solve (for $x$) a more general case? Like: ...
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2answers
61 views

Logarithms and ratios.

This is the question: $$\log_b 64 = \frac{3}{b}$$ And have to find $b$. So I tried a bit and got this:$$\frac{b}{\log b} = \frac{\log 64}{3}$$ But have no idea what to do next. Thanks for your ...
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2answers
69 views

Closed form solution to $x\log_2(1+\frac{a}{x}) = b$ using Lambert W.

Is there an expression for the solution to \begin{equation} x\log_2(1+\frac{a}{x}) = b \end{equation} where $a$ and $b$ are constants, and $x$ is the variable? I am aware that there are no solutions ...
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1answer
31 views

Help solving non-trivial logarithmic inequality

I have the following equation: $$\dfrac{2\pi G\lambda M^4}{m^2}\ln\left(\dfrac{\phi_i}{\phi_e}\right)+2\pi G\left(\phi_i^2-\phi_e^2\right)\ge 65$$ which, for the purpose of this question, I'll ...
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1answer
48 views

How to solve the equation 5x=0.01^x [duplicate]

Hi I recently posted a this question earlier and got some excellent answers but to take it a little further I liked k170's answer however it contained a Lambert W Function in the answer and I was ...
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1answer
26 views

Simplify expression with lambert w-Function

I have an expression and i am almost sure what it equals: $ e^{-W_{-1}\left(-\frac{log\left(x\right)}{x}\right)} $ I only need a simplified version of this expression for $x\geq e$. I assume: ...
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1answer
181 views

Branch points of the Lambert W function

Let $W_{k}(z)$ be the kth branch of the Lambert W function. My question pertains to the branch point that $W_{0}(z)$ shares with $W_{-1}(z)$, and $W_{1}(z)$ at $z = - \frac{1}{e}$. By the inverse ...
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2answers
38 views

$\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$

How to prove $\int _{0}^{\infty }\! \left( {\it W} \left( -{{\rm e}^{-1 -\epsilon}} \right) +1+\epsilon \right) {{\rm e}^{-\epsilon}}{d\epsilon}={\rm e} - 1$ where W is the Lambert W function? Maple ...
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0answers
30 views

Closed form solution for DDEs?

I am solving the equation $X-A-B\exp(-Xy)-C\exp(-Xz)=0$ where $X, A, B$ and $C$ are 2x2matrices and $y$ and $z$ are scalars. What will be the closed form solution ...
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0answers
26 views

solving/approximating the transcendental inequality $c \le αx + β(b^x) + γx(b^x)$

I couldn't find a representation of $x$ using Lambert $W$ function and I doubt this is even possible. Assuming there is no clean solution and numerical methods must be used, is there a way to ...
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2answers
117 views

Finding solutions to $ x^x = 2x$

A friend claims it isn't possible to find a closed form for the smaller positive real solution of $x^x = 2x$. Numerically we have seen that $0.346...$ and $2$ are solutions, but are failing to do ...
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1answer
84 views

Finding inverse of $f(x) =\frac{\ln(x)}{x}$

How do you find the inverse of the following function $$f(x) = \frac{\ln (x)}{x}$$ What looked like a simple question made my head hurt during exam.
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62 views

Lambert W function, W(x), representation for entire domain

The Taylor series for the Lambert W function is $W_0(x)=\sum_{n=1}^\infty\frac{(-n)^{n-1}x^n}{n!},\left\{x\in\mathbb{R}:|x|<\frac{1}{e}\right\}$. Is there any exact closed form way to express ...
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1answer
111 views

Solving $a=\Big(1+\frac{b}{x}\Big)^x$ for $x$

How to solve this equation for $x$? $$a=\Bigg(1+\frac{b}{x}\Bigg)^x$$ It's not a task that I was asked to solve by someone. I just have to solve it because it's a part of my project. If it's ...
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1answer
57 views

Montonicity of Lambert W

Is Lambert $W(x)$ function, an increasing function from $0\rightarrow\infty$? How about in negative axis and complex plane? Note $W(x)$ is given by $$W(x)e^{W(x)}=x.$$ Charts could help understand.
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The Lambert function has two real branches for $x∈[−1/e,0)$: the principal branch $W_0$ and the branch $W_-1$

I am trying to understand Lambert W function. I am new to this special function. What is the actual meaning of the word two real branches of Lambert W function for any real $x$. How to find the branch ...
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1answer
72 views

Solving equation involving self-exponentiation

How do I solve the equation $\displaystyle x=ay^2(by)^{\frac 1y}$ for $y$, where $a$ and $b$ are constants? I've been trying to manipulate this into a form on which I can use the Lambert W function, ...
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0answers
39 views

How to check if some equation can be solved using Lambert $\operatorname{W}$ function.

I'm very interested in Lambert $\operatorname{W}$ function and I want to know how to check if some equation can be solved using this function. Example $1$: $$e^xx=a$$ For this equation it is obviously ...
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1answer
51 views

Lambert W function identity from differential equation

For constants $v,K$ and a function $C(t)$, can you prove that if : $$ \frac{dc}{dt} = - \frac{vc(t)}{K + c(t)},~\text{with } c(0) = c_0 $$ Then the solution: $$ \left[ K \ln c(t) + c(t) ...
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1answer
24 views

Lambert W function multiplication with scalar

Let $W$ be the Lambert W function, $Y$ be a real valued function and $x \in \mathbb{R} $. Given $ Ye^Y = x \iff Y = W(x) $ is it true that $Y = kW(\frac{1}{k}x)$ for non-zero $k \in \mathbb{R} $ ? ...
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95 views

Can $x^{x^x}=k$ be solved using the W function?

The lambert W function is defined to be the inverse of $f(x)=xe^x$, and the equation $x^x=k$ can be solved fairly easily using the function: $$x^x=k$$ $$\ln(x^x)=\ln(k)$$ $$x\ln(x)=\ln(k)$$ ...
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1answer
27 views

$y = ln(p+qe^x)/x$, solve $x$

$y = \ln(p+qe^x)/x$ $p$ and $q$ are constants. Express $x$ in terms of $y$. I believe I have to use Lambert W function, but I'm stumped. Thinking help is needed. Thank you very much!
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1answer
45 views

Solving $ f'(x) =-\log( f(x) +a ) $

Can the solution of $$ f'(x) = -\log( f(x) + a ) $$ with $f(0)=0$ and $a \in (0,1)$ be well approximated by the Lambert W function for $x>0$? It seems that morally this might be the case (by ...
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1answer
35 views

Differential equation $x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x)$

In a proof of the series expansion of the Lambert-W-function, I need that it is the only non-zero function satisfying: $$ x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x) $$ Is it true?
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1answer
38 views

How do I apply the product log function (W(x)) to this equation?

I have the following: $$3(e^{hv\over kT}-1)v^2 = e^{hv\over kT}\frac {hv^3}{kT}$$ Which is the numerator of the derivative of Planck's energy distribution formula when the derivative is set to $0$. ...
0
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2answers
49 views

Can someone show me how to solve Lambert functions, such as the one here?

I would like to understand what process (steps) are required to arrive at the answer of 43.559... as shown in the following equation. I have looked at Wikipedia and I have also looked at the MathWorld ...