# LambertW

• Jan 31st 2007, 04:21 AM
galactus
LambertW
Hello all:

I seen a problem on another site where someone was wanting to know the inverse.

$\displaystyle y=4+x+5e^{x-3}$

As Pka said, the derivative is always positive, so it has an inverse, but can not be found by elementary means.

I ran it through Maple and it kicked back:

$\displaystyle x=y-4-LambertW(\frac{5e^{y}}{e^{7}})$

What is Lambert?. I am unfamiliar. Does anyone know how this would be done without technology?. I thought it would something interesting to look into. I did find an interesting page on Wiki
• Jan 31st 2007, 04:34 AM
CaptainBlack
Quote:

Originally Posted by galactus
Hello all:

I seen a problem on another site where someone was wanting to know the inverse.

$\displaystyle y=4+x+5e^{x-3}$

As Plato said, the derivative is always positive, so it has an inverse, but can not be found by elementary means.

I ran it through Maple and it kicked back:

$\displaystyle x=y-4-LambertW(\frac{5e^{y}}{e^{7}})$

What is Lambert?. I am unfamiliar. Does anyone know how this would be done without technology?. I thought it would something interesting to look into. I did find an interesting page on Wiki

The wikipedia pge is a good overview, and computational schemes.

For real arguments > -1/e Newton-Raphson is quite a good method og evaluating it.

RonL
• Jan 31st 2007, 06:53 AM
ThePerfectHacker
In my engineering class we were studing a hanging cable. The problem is that the equations that describe it are hyperbolic functions and hence cannot be solved through "normally". However, I have been able to reduce the problem to solving,
$\displaystyle ax+b=e^x$ for $\displaystyle a\not = 0$.
In that case the linear-exponential equations can always be solved (I imagine). The full solution is here. http://www.mathhelpforum.com/math-he...e-problem.html