# Questions tagged [lambert-w]

For questions related to the Lambert W or product log function, the inverse of $f(z)=ze^z$.

676 questions
Filter by
Sorted by
Tagged with
43 views

### Finding zeroes for function $f(t) = e^{k(t-1)} -t$ for $k> 0$ analytically

I tried using Lambert W function the following way $$e^{k(t-1)} -t=0$$ $$e^{k(t-1)}=t$$ $$-ke^{-k} = -kte^{-kt}$$ $$W(-ke^{-k}) = W(-kte^{-kt})$$ $$-k = -kt \implies t = 1$$ but this only gives me one ...
94 views

### How to find the solution of $x(e^x)+\ln(x)+c = 0$, where $c$ is constant? [closed]

For $x(e^x)+\ln(x)+c = 0$, where $c$ is constant. I believe the solution is solved using Lambert $W$-function and derivatives, but I can't figured it out, I can't find any ideas can someone tell me ...
1 vote
47 views

37 views

### Can we use the Lambert W solution $y=-3W(K_2x^{-4/3})$ instead of $y =-3W(\frac 1 3\sqrt[3]{-\frac{K_1}{x^4}})$ if we choose an appropriate constant?

During the process of solving the separable differential equation $4y - x(y-3)y' = 0$, our solution acquires a constant when we go from $\frac 4 x = \frac{y-3}{y}y'$ to $\ln x + C_1 = y - 3 \ln y$. We ...
• 698
1 vote
104 views

### $x^y = y^x$ what is $y$?

So today I came across an answer on MathStackExchange. I read it and found it well explained. But I stuck on a step where the user write "Solve using the properties of $W$ function". I don't ...
• 1,812
1 vote
65 views

### Is there any general formula that describes the sum and difference of Lambert W Function?

I know $\ln(a)+\ln(b)=\ln(ab)$ and $\ln(a)-\ln(b)=\ln(\frac{a}{b})$ or $\ln(a^{b}) = b\ln(a)$ and other rules for the ln (Natural Logarithm). But are there any sum or difference laws for the lambert ...
53 views

### Solving for ${x}^{{x}^{3}}=5$

I am trying to solve $${x}^{{x}^{3}}=5$$ I took logarithms on both sides, and got $${x}^{3}\cdot \ln \left( x \right) -\ln \left( 5 \right) =0$$ Then I took the help of Maple, when I solve for x ...
78 views

25 views

### Taylor expansion of the alternative branch of the lambert W function

I know that the Taylor expansion of the principal branch of the Lambert W function is, $$W_0(z)=\sum\limits_{n=0}^\infty \frac{(-n)^{n-1}}{n!}z^n$$ But what is the Taylor expansion for $W_{-1}$?
• 31
1 vote
### Inverse of $x\exp(-x)$
What would be the inverse of the function $y=x\exp(-x)$ when $1\leq x\leq \infty$ and $0\leq y\leq 1/e$. I tried to solve this with Lambert W function and the solution came out to be $-W(-y)$. But in ...