jaydfox Wrote:Following up on the conjecture, my initial estimate puts closer to 0.9163, give or take. It would have been very cool if it had been true.

For reference, truncated to double precision (about 15 decimal places), is:

0.7856963885801098+0.916302621081289*I

This was calculated as follows:

1. I started with a 1200-term accelerated solution for Andrew's slog, base e, solved at 3548 bits of precision, then truncated to 2048 bits (of which about 1928 bits are valid, give or take).

2. I found the "residue" by subtracting the power series for , where is the primary fixed point, 0.3181315+1.3372357*I.

3. I recentered the residue from z=0 to z=1 (using PARI, but can be done with a Pascal matrix as well).

4. I added (to the residue) the power series for , thus recovering the slog, recentered with a minimum possible loss of precision.

5. I reverted the (just derived) power series of slog at z=1 to get the sexp at z=0. I only kept the first 600 terms for further evaluations. They become noticeably inaccurate after about 250 terms (i.e., they stopped converging on the power series of ), though the inaccuracy of each additional term is less than the additional precision provided, up to about 600 terms, give or take, depending on the distance from the origin at which you evaluate the power series. By this, I mean particularly that sexp(1) and sexp(-1) continue to converge on e and 0, respectively. Take that with a grain of salt. I wouldn't try to use more than about the first 250-400 terms thus derived for high-precision calculations, depending on your needs.

6. Finally, I evaluated the power series at i, giving an answer that is probably not accurate to more than 25 digits, though probably accurate to at least 15 digits, but I have no easy method of validating that claim at the moment.

Edit: For reference, I have attached a SAGE object, which represents the power series I derived as described above, truncated to 256 bits of precision (it's probably only good for about 80-100 bits anyway). Simply load the series into a variable, e.g., sexp, and then you can evaluate as simply as typing sexp(1) and pressing enter:

Code:

`Flt256 = RealField(256)`

C256 = ComplexField(256)

I = C256.0

sexp = load('sexp_600_pseries.sobj')

sexp(-1)

sexp(0)

sexp(1)-Flt256(e)

sexp(I)

As for results, you should get, respectively, 0, 1, 0, and the complex value that I described above.

~ Jay Daniel Fox