MaplePrimes - answers and comments on Question, asymptotics of Lambert W

via Wrightomega

Axel Vogt — Mon, 30 May 2011 18:06:56 Z

LambertW(x);
convert(%, Wrightomega);
subs(ln(x) = t, %);
asympt(%, t,2);
subs(t=ln(x), %);

                                  ln(ln(x))       1
              ln(x) - ln(ln(x)) + --------- + O(------)
                                    ln(x)            2
                                                ln(x)

asymptotics

Robert Israel — Tue, 31 May 2011 10:32:51 Z

asympt(LambertW(0,x),x) didn't work in any version of Maple that I tried. However, you could do this:

> asympt(LambertW(0,exp(t)),t);

> eval(%, t = ln(x));

freeze and thaw

Alec Mihailovs — Tue, 31 May 2011 11:08:34 Z

This could be also written as

t:=freeze(ln(x)):
thaw(MultiSeries:-asympt(LambertW(exp(t)),t,3));

                                                       2
                                ln(ln(x))     ln(ln(x))
            ln(x) - ln(ln(x)) + --------- + O(----------)
                                  ln(x)              2
                                                ln(x)

It's not shorter or better - just looks different.

The difference with Axel Vogt's reply is in the O-term. MultiSeries:-asympt has the correct form of it, and the top level asympt doesn't.

Alec