How to find value of Lambert W function ?

Question

W(ln2) = ?

Answer 1

Mathematica implements the Lambert W function as ProductLog [z]. There's an online evaluator here: http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog . If you must compute it yourself, then you're left with using some kind of iterative method, such as using Newton's method to find the root of the function xe^x - ln 2 (and if this is just to solve (1/2)^n = n, then you may save yourself a lot of work by just using Newton's method directly on it). The wikipedia page describes many such methods, so I suggest you read it first.

By the way, don't feel bad about using numerical methods to solve a problem. Despite the impressions you get from your elementary algebra class, _most_ problems we can come up with using only "elementary" operations cannot be solved algebraically, and which things can be solved algebraically is largely a function of which operations we consider important enough to bother constructing an inverse function for it and putting a button on the calculator. Even for functions that have inverse functions with a button on the calculator, when you go down into the circuitry to figure out how the calculator actually computes the value of the function, you will find that it is usually just an application of Newton's method or some other iterative root-finding algorithm. Occasionally, a taylor series is used instead, but the point is that it's always just a numerical approximation, so the only difference between using newton's method to find the root of 1/2^n - n and trying to find √2 using your calculator is that in the second case the invocation of newton's method is hidden in the calculator's circuitry. Well that, and that √2 is a much easier expression to manipulate algebraically than "the unique positive real root of x²-2," but the Lambert function does solve that problem in that you may now describe the solution using W(ln 2)/ln 2, and manipulate it algebraically using known properties of the W function (e.g. W(x)/x = e^(-W(x)) and other such identities).

Answer 2

In Maple both genuine branches are denoted with the help of LambertW(0,y) and LambertW(-a million,y). i imagine that has similarities in Mathematica. i'm no longer attentive to any basic courting between both.

Answer 3

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