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Topic: Cannot get inverse of y=x/logx to work
Replies: 5   Last Post: May 30, 2011 2:03 PM

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G. A. Edgar

Posts: 2,510
Registered: 12/8/04
Re: Cannot get inverse of y=x/logx to work
Posted: May 30, 2011 9:31 AM
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In article <XKFEp.5313$>, Brad
Cooper <> 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.
> (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.

G. A. Edgar

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