02/13/2011, 10:27 AM

Oh, now with the picture I see what you mean.

To get a formula we can use the Lambert W function.

The Lambert W function is the inverse of the function M(x)=x*e^x

and we can express the self root x^(1/x) with help of M:

y = x^(1/x) = exp(-M(-ln(x)))

You can verify this with a bit of calculation.

Then we can obtain the inverse:

exp(-W(-ln(y))) = x

More exactly x can be two values, left and right from e, which correspond to the two branches of W:

So when you want to get the left value - as in your case - you choose x_L and get:

would be simply 3 again.

To get a formula we can use the Lambert W function.

The Lambert W function is the inverse of the function M(x)=x*e^x

and we can express the self root x^(1/x) with help of M:

y = x^(1/x) = exp(-M(-ln(x)))

You can verify this with a bit of calculation.

Then we can obtain the inverse:

exp(-W(-ln(y))) = x

More exactly x can be two values, left and right from e, which correspond to the two branches of W:

So when you want to get the left value - as in your case - you choose x_L and get:

would be simply 3 again.