# 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $y=\ln(a)$ ...
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### 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 ...
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### 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 ...
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### 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.
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1 vote
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### 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 ...
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://...
1 vote
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### 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?
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### 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 ...
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### 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 ...
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### 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 $f(x) = x$: $$a^x+bx+c = 0$$...
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1 vote
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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 ...
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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 ...
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 ...
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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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