Help Obtaining Numerical Approximation of Lambert W Solution

Question

I am studying a particular generating function $$\frac{2e^x}{e^{2x}+1+2x}$$ and I thought I would try to solve the equation $$e^{2x}+1+2x=0$$ to determine for what value of $x$ if any the function would be undefined. Because $$e^{2x}=-1-2x \Rightarrow e^{-(-2)x}=-2\left(x-\left(\frac{-1}{2}\right)\right)$$ Then the real solution is $$x=\frac{-1}{2}-\frac{1}{2}W\left(\frac{1}{e}\right)$$ As I have never used the lambert function I am trying to understand it. I know that it satisfies $$x=W(x)e^{W(x)}$$. How would I obtain a rough numerical approximation if I, say, didn't have a computer handy, or my cellphone to calculate? How can one "see" a solution without the aid of technology?

Answer 1

If you search for Lambert function in Wikipedia (http://en.wikipedia.org/wiki/Lambert_W_function), you could find interesting approximations for large values of the argument such as $$W(x)\approx L_1-L_2+\frac{L_2}{L_1}+\frac{L_2(L_2-2)}{2L_1^2}+\frac{L_2(2L_2^2-9L_2+6)}{6L_1^3}+\cdots$$ where $L_1=\log(x)$ and $L_2=\log(L_1)$ (this is for the pricipal branch).

Let us try for $x=1234$; the approximation gives $W(1234)\approx 5.42680$ while the exact solution should be $\approx 5.42669$.

For the general numerical solution, Newton and better Halley methods (as shown by Corless in the same page) are very efficient.