Who can solve ln x = 1/x? I can get an approximate solution to any degree of accuracy, but is there an exact solution?

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- Dec 31st 2008, 01:49 PMbillyatesNeed solution to mixed equation.
Who can solve ln x = 1/x? I can get an approximate solution to any degree of accuracy, but is there an exact solution?

- Dec 31st 2008, 02:15 PMgalactus
Let's just use Newton's method. It's always a good standby.

Let's make an initial guess of 1.5:

Iterate:

Keep going until you reach the desired accuracy.

The solution is 1.76291539065....

As far as an exact solution, I do not think so.

is pretty close of an approximation. - Dec 31st 2008, 03:39 PMmr fantastic
The exact solution can be written in terms of the Lambert W-function:

.

Therefore where W is the Lambert W-function.

W(1) is the Omega constant: http://en.wikipedia.org/wiki/Omega_constant.