Tetra-series

[Gottfried]

A new result, which I just posted in sci.math; I'll improve the formatting later (a bit lazy...)

Gottfried

Code:

Am 30.06.2008 17:00 schrieb alainverghote@gmail.com:
>> >> Gottfried
> >
> > Well, inverse function of tf(x) using lambertW()
> > is  -Lambert(-1/2exp(1/2e^x - 1)) +1/2e^x - 1
> > series near 0 , x +x^2 +4/3x^3 +29/12x^4 +51/10x^5 ...
> >
> >
Yepp, I got the same series - good!

But now - iterations and especially the sum of iterations
of this functions in the Lambert-representation should be
intractable, so I don't assume this can be helpful to get
more insight in the source of the inconsisteny-problem;

remember:
by the formal application of the matrix-approach
      asn(x)+asp(x)-x = 0  // expected
or
      asn(x,f(x)) = x - asp(x,f(x)) // expected
or
      asp(x,f°(-1)(x)) = x - asp(x,f(x)) // expected

is not true, at least for the function f(x) = exp(x)-1

Well - it was a try...

--------------------------------------

Meanwhile I refined my computation-process, so I've
now even the function fz(x) with the condition

  e^x - 1 = fz(x) + fz(fz(x)) + fz(fz(fz(x))) + ..
          = sum{h=1..inf} fz°h(x)

I got
fz(x) = 2*(x/4)/1! + 6*(x/4)^2/2! + 10*(x/4)^3/3! - 46*(x/4)^4/4! - 554*(x/4)^5/5!
       - 1690*(x/4)^6/6! + 27882*(x/4)^7/7! + 505986*(x/4)^8/8! + 2529590*(x/4)^9/9!
       - 61918794*(x/4)^10/10! - 1726391798*(x/4)^11/11! - 14268435022*(x/4)^12/12!
       + 352044609814*(x/4)^13/13! + O(x^14)

where the integer parts of the coefficients are
[2]
[6]
[10]
[-46]
[-554]
[-1690]
[27882]
[505986]
[2529590]
[-61918794]
[-1726391798]
[-14268435022]
[352044609814]
which have to be divided py powers of 4 and by factorials to give the
coefficients of the function.

The float representation of this function is
fz(x) =   0.500000000000*x^1 + 0.187500000000*x^2 + 0.0260416666667*x^3 - 0.00748697916667*x^4
         - 0.00450846354167*x^5 - 0.000573052300347*x^6 + 0.000337655203683*x^7 + 0.000191486449469*x^8
         + 0.0000265917660278*x^9 - 0.0000162726971838*x^10 - 0.0000103115449999*x^11
         - 0.00000177549511523*x^12 + 0.000000842437121499*x^13 + 0.000000632647393830*x^14
         + O(x^15)

Can we give a range for x where this converges ?
The quotients of subsequent coefficients give the following sequence

0.375000000000,0.138888888889,-0.287500000000,
0.602173913043,0.127105896510,-0.589222316145,
0.567106466538,0.138870223462,-0.611944959460,
0.633671534805,0.172185168687,-0.474480112208,
0.750972835462,0.212965249501,-0.333234112499,
0.928624638217,0.256835516535,-0.218432174828,
1.22327362079,0.303250445581,-0.126531683586,
1.82969444504,0.353026162019,-0.0509774937397,
3.94499797418,0.407780103462,0.0133041813614,-13.1183083394,
0.470026310618,0.0698970628404,-2.15481130004,
0.543690120454,...

-----------------------------------------------------------

Using 32 coefficients for the function and 60 iterates for the sum
I could approximate e^1 -1 relatively well. I got

    sum(h=1,60,fz°h(1.0)) - ( exp(1)-1 ) =  -3.24385306514 E-13

where the quality of approximation increased when terms of the
function and numbers of iterates are increased.

Fun... :-)

Gottfried Helms