What is the Product Log and how can it be expressed as a closed-form expression?

[LucasGB]

What Product Log Means??

Hi,

I have the following equation:

B = (k*a*Exp[-t*(B/N)*T/beta] - N)/(a*Exp[-t*(B/N)*T/beta])

Using Mathematica, I tried to solve it for B. Mathematica returned the following solution:

B = k-(N*beta*ProductLog[(E^((k*t*T)/(N*beta))*t*T)/(a*beta)])/(t*T)

What that means? What is ProductLog? How can I have a closed-form expression of that?

Thanks a lot,

Estêvão

 

[Mark44]

This might be helpful: http://reference.wolfram.com/mathematica/ref/ProductLog.html

 

[HallsofIvy]

"Product log" is also known as the "Lambert W function", the inverse function to $f(x)= xe^x$.

 

[LucasGB]

Thank you, guys, but I have to tell you: I'm really a begginer in math issues... Therefore, Wolfram Reference's definition and HallsofIvy's definition are still not sufficiently clear to me. Would you give some explanation of "Product Log for Dummies" kind? For example, Wolfram's text simply says that:

"ProductLog[z]
gives the principal solution for w in z=w*e^w"

But what that means? What "w" means? If I have something like:

ProductLog[(E^((k*t*T)/(N*beta))*t*T)/(a*beta)]

how can I transform that in a closed-form expression?

Thanks again,

Estêvão