On 2016-06-03 07:03, Veit Elser wrote:
Two fun W (Lambert function) facts:
1) The Taylor series of W records the number of spanning trees of the complete graphs.
2) If 1/2+i y_n is the nth zero of the Riemann zeta function on the critical line, then asymptotically (large n)
y_n ~ 2 pi (n-11/8)/ W((n-11/8)/e)
-Veit
Less amazing, but useful: They appear in an inelegant but efficient series solution to Kepler's equation: http://www.tweedledum.com/rwg/pizza.html It's a bit disappointing that W doesn't solve Kepler's more elegantly. But suppose we posit that E(ε,M) := Kepler(ε,M) solves M = E - ε sin E . Does this dyadic function Kepler subsume Lambert W as a special case? If so, maybe we should embrace it. Failing that, I think "Dilbert Lambda"(y), which solves y = Λ exp Λ², is usually nicer than W.
On Jun 2, 2016, at 10:28 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Thanks. I vastly prefer the Lambert W function when it is not defined
with a branch cut but is allowed to extend to it full Lambertness on a Riemann surface.
Whether considering the Lambert function or the more simply defined
function of which it is the inverse function:
f(z) = z exp(z),
the aforesaid Riemann surface is just the subset of C^2 defined as
{(z,w) | z = w exp(w)}.
This is, in my opinion, the appropriate object of study.
—Dan
All we need is for you to tell us how it would look in Mathematica. --rwg