Lambert W Function: Solving $r=ue^u$

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In summary, the Lambert W function is a special function used to solve equations involving exponential or logarithmic terms. It has applications in various fields such as physics, biology, chemistry, engineering, and finance. The function cannot be expressed in terms of elementary functions and is typically calculated using numerical methods or software programs. However, it may have limitations in terms of finding real solutions and computational efficiency.
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Dustinsfl
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Based on Pickslides reference to the Lambert W function, I am now trying to solve this:

$$
r = \frac{q\left(1-\exp\left\{-\frac{u^2}{\varepsilon}\right\}\right)}{u(q-u)}
$$

So now I have this in the form $r = ue^u$ and I need to transform it to $u = w(r)$

I am not quite sure on how to do that though.
 
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  • #2
dwsmith said:
So now I have this in the form $r = ue^u$...

Hi dwsmith, :)

Did you transform the original equation into the form \(X=Ye^{Y}\) ? If so can we please see the result?

Kind Regards,
Sudharaka.
 

FAQ: Lambert W Function: Solving $r=ue^u$

What is the Lambert W function and what does it solve for?

The Lambert W function, named after mathematician Johann Heinrich Lambert, is a special function that solves the equation $xe^x=y$ for $x$. In other words, it finds the value of $x$ when $xe^x$ is equal to a given value $y$.

How is the Lambert W function used in science?

The Lambert W function is commonly used in various fields of science, such as physics, engineering, and finance, to solve equations that involve exponential or logarithmic terms. It can also be used to simplify complex mathematical expressions and to find the roots of certain equations.

What are the applications of the Lambert W function?

The applications of the Lambert W function include solving equations in physics, such as the heat equation and the Schrödinger equation, modeling population growth in biology, and calculating the yield of chemical reactions in chemistry. It is also used in various engineering and financial models.

How is the Lambert W function calculated?

The Lambert W function cannot be expressed in terms of elementary functions, so it is typically calculated using numerical methods or approximations. Some software programs, such as MATLAB and Mathematica, have built-in functions for calculating the Lambert W function.

Are there any limitations to the use of the Lambert W function?

While the Lambert W function is a powerful tool for solving certain equations, it does have some limitations. It may not have a real solution for certain values of $y$, and in some cases, it may have multiple real solutions. Additionally, it may not be computationally efficient for certain types of equations.

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