What I don't understand is this

The closed form solution is ln(2)/W(ln(2)) = 1.559610469462369349970388768765002993285

Using the definition of the W,

n = W(n) eW(n)

The original problem:
xx = 2

Change to exponential form:
ex ln(x) = 2

Take ln both sides:
x ln(x) = ln(2)

Let y = 1/x:
-ln(y)/y = ln(2)

Let z = y/ln(2):
-ln(z/ln(2)) ln(2)/z = ln(2)

Divide both sides by -ln(2):
ln(z/ln(2))/z = -1

Move z over to the rhs:
-ln(z/ln(2)) = z

Exponentiate:
ln(2)/z = ez

Move the z to the rhs:
ln(2) = z ez

which, by definition of the W function, implies that z = W(ln(2)). And thus x = ln(2)/W(ln(2)).

Posted By: jafski, Jul 11, 14:31:40