In article <XKFEp.5313$aH5.firstname.lastname@example.org>, Brad Cooper <Brad.Cooper_17@bigpond.com> wrote:
> Hi, > > I cannot get the correct result for the inverse calculation of > y = x/log(x). > > > On The fourth page (document pp. 240) of... > > Partial Sums of Infinite Series, and How They Grow > R. P. Boas, Jr. > > http://mathdl.maa.org/images/upload_library/22/Ford/Boas.pdf > > (It is 22 pages of pdf and takes a while to load) > > The writer says... > > The successive approximations to the inverse of y = x/logx are > (if we write Ly for logy, L2y for log logy) > > x = y*(Ly + L2y) + L2y/Ly + y*L2y/(Ly)^2*(1 - 1/2*L2y) + > y*L2y/(Ly)^3*(1 - 3/2*L2y + 1/3*(L2y)^2) + > y*L2y/(Ly)^4*(1 - 3*L2y + 11/6*(L2y)^2 - 1/4*(L2y)^3) > > In the Computer Algebra Systsem MuPAD I wrote... > > Inv := Y*(LY + L2Y) + L2Y/LY + Y*L2Y/(LY)^2*(1 - 1/2*L2Y) + > Y*L2Y/(LY)^3*(1 - 3/2*L2Y + 1/3*(L2Y)^2) + > Y*L2Y/(LY)^4*(1 - 3*L2Y + 11/6*(L2Y)^2 - 1/4*(L2Y)^3); > > x := y -> subs(Inv, Y = y, LY = ln(y), L2Y = ln(ln(y))); > > > This creates the inverse log function x = x(y) as shown in the paper. > > > Now, 1.295855509/ln(1.295855509) = 5.0 (approx.) > > > But in MuPAD I get x(1.295855509) = -3909.235291, not very close to 5.0 :-) > > > I am interested to know if another CAS obtains a better result. > > Am I missing something more basic in terms of the Mathematics? > > Any help much appreciated. > > Cheers, > Brad
Interesting question. Consider the function x/log(x) ... it decreases for x in (0,e) and increases for x in (e,infinity). The minimum value y=e occurs when x=e. Now for the part of the graph where x>e, Boas has given an asymptotic expansion. For example, substitute y=5 into the Boas series to get x=11.6, substitute x=11.6 into x/log(x) to get 4.7, approximately right.
But you are asking about the OTHER part of the graph, where x<e . For that, the Boas series is nowhere near, as you noted. On that part of the graph, the asymptotics look like x ~= 1+1/y+(3/2)/y^2+(8/3)/y^3+(125/24)/y^4+(54/5)/y^5 Using your values, substitute y=5 in there, and get 1.293 as you wanted.
Others noted that the inverse is given in terms of the Lambert W function: y = -y*W(-1/y) ... well, these two parts of the graph are given as two branches of the W function. In Maple's notation, -y*LambertW(0,-1/y) is the branch smaller than e, and -y*LambertW(-1,-1/y) is the branch larger than e.