Maybe this was already adressed elsewhere, so if this is so, some kind reader may please link to that entry.

Playing again with the structures of Andy's slog I came to the following observation (so far with base e only, but I think it's trivial to extend).

Consider the power series for slog(z) as given by Andy's descriptions, and define slog0(z) by inserting zero as constant term instead of -1 as in slog(z). slog0(e^^h) gives now h+1 for the argument e^^h.

But moreover, now the series for slog0(z) can formally be inverted.

One can observe, that its coefficients are near that of the series for log(1+z), so let's define the inverse to the slog-function as tetration-function

taylorseries(T0) = serreverse(slog0) - taylorseries(log(1+x))

Then the coefficients of T0() decrease nicely, and we can compute e^^h (best for fractional h in the range -1<h<0 )

e^^h = T0(1+h) + log(2+h)

The series for T0() look much nicer than that of the slog(), but of course the coefficients are directly depending on the accuracy of the coefficients of the slog()-function, so I still used slog with, say, matrix-size of 96 or 128 for a handful of correct digits.

I've never worked with Jay D. Fox's extremely precise solutions for the slog-matrix so I cannot say anything how the coefficients of T0() would change. Would somebody like to check this?

Gottfried

Appendix: 32 Terms of T0(h) taken from slog0(z) with matrixsize of 64

Appendix 2:

Jay D. Fox has provided very accurate coefficients for the slog-function. Using the first 128 of that leading coefficients to recompute T0(h) I arrive at the remarkable solution for e^^pi where 20 digits match Jay's best estimate:

thread see at http://math.eretrandre.org/tetrationforu...php?tid=63 "Improving convergence of Andrew's slog"

post see at http://math.eretrandre.org/tetrationforu...920#pid920

The data of the taylor-series for slog with 700 terms are also in that thread.

Playing again with the structures of Andy's slog I came to the following observation (so far with base e only, but I think it's trivial to extend).

Consider the power series for slog(z) as given by Andy's descriptions, and define slog0(z) by inserting zero as constant term instead of -1 as in slog(z). slog0(e^^h) gives now h+1 for the argument e^^h.

But moreover, now the series for slog0(z) can formally be inverted.

One can observe, that its coefficients are near that of the series for log(1+z), so let's define the inverse to the slog-function as tetration-function

taylorseries(T0) = serreverse(slog0) - taylorseries(log(1+x))

Then the coefficients of T0() decrease nicely, and we can compute e^^h (best for fractional h in the range -1<h<0 )

e^^h = T0(1+h) + log(2+h)

The series for T0() look much nicer than that of the slog(), but of course the coefficients are directly depending on the accuracy of the coefficients of the slog()-function, so I still used slog with, say, matrix-size of 96 or 128 for a handful of correct digits.

I've never worked with Jay D. Fox's extremely precise solutions for the slog-matrix so I cannot say anything how the coefficients of T0() would change. Would somebody like to check this?

Gottfried

Appendix: 32 Terms of T0(h) taken from slog0(z) with matrixsize of 64

Code:

`T0(h) = 0`

+ 0.0917678575394 *h

+ 0.175505903737 *h^2

+ 0.0164995173026 *h^3

+ 0.0191448752458 *h^4

+ 0.00133512590560 *h^5

+ 0.00231496855708 *h^6

- 0.0000239484773943 *h^7

+ 0.000304699490128 *h^8

- 0.0000364374120911 *h^9

+ 0.0000455639411655 *h^10

- 0.0000114433561149 *h^11

+ 0.00000769463425171 *h^12

- 0.00000262533621718 *h^13

+ 0.00000145886258700 *h^14

- 0.000000604261454214 *h^15

+ 0.000000301260921078 *h^16

- 0.000000129633563113 *h^17

+ 0.0000000606527811916 *h^18

- 0.0000000293680769725 *h^19

+ 0.0000000147234175905 *h^20

- 0.00000000637812232351 *h^21

+ 0.00000000263672711471 *h^22

- 0.00000000137069796843 *h^23

+ 0.000000000858320261831 *h^24

- 0.000000000396925930057 *h^25

+ 7.48739318521E-11 *h^26

- 1.71265782834E-11 *h^27

+ 7.18443441580E-11 *h^28

- 5.88967800348E-11 *h^29

- 7.64569615630E-12 *h^30

+ 2.78278848579E-11 *h^31

+O(h^32)

Appendix 2:

Jay D. Fox has provided very accurate coefficients for the slog-function. Using the first 128 of that leading coefficients to recompute T0(h) I arrive at the remarkable solution for e^^pi where 20 digits match Jay's best estimate:

Code:

`37149801960.55698549 914478420500428635881 \\ using T0() with n=160 terms: e^^Pi = e^^(3+frac(Pi)) = e^e^e^[T0 (1+ frac(Pi) ) + log( 2 + frac(Pi)) ]`

37149801960.55698549 914478420500428635881 \\ Using T0() with n=128 terms

37149801960.55698549 872339920573987 \\ J.d.Fox to about 25 digits precision; difference at 20. dec digit

post see at http://math.eretrandre.org/tetrationforu...920#pid920

The data of the taylor-series for slog with 700 terms are also in that thread.

Gottfried Helms, Kassel