LambertW

**galactus** · January 31st 2007, 05:21 AM

Hello all:

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

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

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:

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

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

**CaptainBlack** · January 31st 2007, 05:34 AM

Originally Posted by galactus

Hello all:

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

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

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:

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

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

**ThePerfectHacker** · January 31st 2007, 07:53 AM

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,
$ax+b=e^x$ for $a\not = 0$ .
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

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