Infinite towers & solutions to Lambert W-function

[brangelito]

The solution to the equation:

Code:

x*c^x -1 = 0

Seems to be given by:

Code:

lim n->inf
(1/c)^^n

Put c=e, and you get the Lambert W-function.

Is this already known?

[bo198214]

The solution to the equation:

Code:

x*c^x = 1

Seems to be given by:

Code:

lim n->inf
(1/x)^^n

You mean 1/c instead of 1/x?!

The limit \( x=\lim_{n\to\infty} \frac{1}{c}[4]n \) satisfies the equation
\( \left(\frac{1}{c}\right)^x = x \) (if I add another \( \frac{1}{c} \) at the bottom of the infinite tower, the value must stay the same).
This equation is equivalent to your equation above:
\( 1=c^x x \)