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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0
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2answers
37 views

Finding the inverse of the function $f(k, x) = k^{x}x.$

Recently, I have been looking at the function $f(x) = e^{x}x,$ where its inverse is the Lambert W function. I was intrigued by the fact that it is rather hard to calculate its solution, in comparison ...
11
votes
2answers
241 views

Proof of strictly increasing nature of $y(x)=x^{x^{x^{\ldots}}}$ on $[1,e^{\frac{1}{e}})$?

The title is fairly self explanatory: I have been trying to rigorously prove that $y(x)=x^{x^{x^{\ldots}}}$ is a strictly increasing function over the interval $[1,e^{\frac{1}{e}})$ for a while now, ...
2
votes
1answer
51 views

Solve $\exp(x)(5-x)=5$ by hand

Is there a way to solve this equation by hand? $\exp(x)(5-x)=5$ Solutions: $x_1=0$ $x_2= 4.96511$
6
votes
0answers
74 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
4
votes
1answer
50 views

Methods for Finding Exact Solution For $e^{2x}+p(2x)$

I know there are ways using the Lambert W function, and have had answers to simpler examples, for example $$e^{2x}+1+2x=0\Rightarrow e^{2x}=-2x-1$$ has the solution ...
7
votes
2answers
540 views

Is there ANY possible way to solve this equation?

So I came up with this equation and it just seems like I can't solve it AT ALL for '$a$' $$a*b^a = c$$ EDIT: By the way, I'm only taking $b^a$, not both $b$ and $a$, just in case anyone was ...
3
votes
0answers
34 views

Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ...
0
votes
0answers
20 views

Where did I go wrong in this simplification involving the Lambert W Function?

I have been working on this problem for about a day and thought I finally found a way to simplify it, unfortunately when I plug my simplification back into maxima I get that they are not equal by what ...
0
votes
1answer
21 views

Big Theta of this modification of the secondary branch of the Lambert W function

I am looking to find the big-$\Theta$ of $-W_{-1}(-\frac{a}{n})$ in terms of elementary functions where $a$ is a constant. Looking around and I find that this should be $O(\log(n))$ and with maxima I ...
1
vote
2answers
45 views

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: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
3
votes
0answers
27 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 ...
1
vote
1answer
38 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 ...
1
vote
1answer
38 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}} ...
0
votes
3answers
70 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, ...
3
votes
1answer
35 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 ...
7
votes
2answers
160 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 ...
1
vote
1answer
147 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. ...
0
votes
0answers
12 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 ...
6
votes
3answers
130 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*} ...
0
votes
1answer
33 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 ...
1
vote
2answers
126 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
5
votes
2answers
245 views

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ ...
0
votes
1answer
49 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.
4
votes
1answer
95 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
3
votes
2answers
53 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 ...
1
vote
0answers
47 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 ...
1
vote
3answers
284 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), ...
0
votes
2answers
152 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.
1
vote
1answer
110 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. ...
2
votes
2answers
55 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 ...
4
votes
5answers
120 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 ...
1
vote
1answer
42 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 $ ...
0
votes
1answer
65 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 ...
1
vote
1answer
81 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: ...
3
votes
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 ...
6
votes
2answers
70 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 ...
0
votes
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 ...
0
votes
1answer
57 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 ...
0
votes
1answer
31 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: ...
4
votes
1answer
192 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 ...
2
votes
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 ...
0
votes
0answers
31 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 ...
0
votes
0answers
27 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 ...
5
votes
2answers
125 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 ...
2
votes
1answer
85 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.
1
vote
0answers
65 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 ...
1
vote
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 ...
1
vote
1answer
61 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.
0
votes
0answers
64 views

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 ...
3
votes
1answer
76 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, ...