On Fri, Jun 3, 2016 at 4:20 PM, Bill Gosper <billgosper@gmail.com> wrote:
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 .
Solving x Sinh[x] == z, x==W[Y[z]], where (Y[x] - W[Y[x]]^2/Y[x])/2==x (and W:=LambertW) Define Y: In[413]:= Clear@Y; Y[z_?NumericQ] := y /. FindRoot[z == (y - ProductLog[y]^2/y)/2, {y, 2 z}] I.e., Y=InverseFunction[Function[y, 1/2 (y - ProductLog[y]^2/y)]] E.g., x=W[Y[π]] solves x Sinh x = π: In[414]:= # Sinh@# &@ProductLog[Y[Pi]] Out[414]= 3.141592653589793 In[415]:= # Sinh@# &@ProductLog[Y@3511] Out[415]= 3511. In[416]:= # Sinh@# &@ProductLog[Y@-1] Out[416]= -1.000000000000001 - 3.80109470280995*10^-16 I In[419]:= # Sinh@# &@ProductLog[Y[10^-9]] Out[419]= 9.999999999979317*10^-10 In[420]:= # Sinh@# &@ProductLog[Y[10^9]] Out[420]= 1.000000000000001*10^9 Y[x] is well approximated on [0,∞) by 2√(x²+a x/(x^2+b)) a ~ 683, b ~ 2222, but this has four unequal ripples and might be much improved. But can we make W out of Y? --RWG 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