[seqfan] Re: Is this constant already in the OEIS?

[Brad Klee]

Hi Felix,

This could be a hard question to answer. According to the Mathematica
Integration Oracle, the calculation appears to be wrong ( by an order of
magnitude ).

SymInt = Integrate[x^(-p) Cos[x], {x, 0, 3 Pi/2}][[1]];
N[(#/10000)] -> SetPrecision[SymInt /. p -> (#/10000), 20] & /@
  Range[3080, 3090] // Column

There's probably a reference we could find to check this symbollic form,
then the only issue is precision of evaluation for special functions.

At Wolfram they are really mad about all these Old German special
functions: Hypergeometric, Jacobian Elliptical, Weierstrass Elliptical, you
name it.

I've tested Jacobian Sn, Cn, and Dn as well as certain values of
Weierstrass P, related to the timelike geodesics in Schwarzschild spacetime
geometry. I got this first principles algorithm to crank out ten
approximations of these functions in no time. We can't say what's happening
in the kernel, but I'm pretty sure that those people at Wolfram Research
have figured out the "nome". I can't figure out what a "nome" is doing in
an equation; I thought gnomes perferred to stay in lawns and gardens???

Long story short, a fair selection of Mathematica special functions are
passing my independent Q.A. tests, developed out of physics. If quality is
uniform, Hypergeometric should be just as good. Compare with an article
that has no citations, no explanation of methods, Etc...

Anyone else have an opinion, who's the more accurate?

Here's the turn around. What about other numerical constants? For example,
quantization conditions. Erdos was interested in this subject:

http://jmahaffy.sdsu.edu/courses/s12/math342B/lectures/sources/elliptic_spiral_Earth.pdf

Figure 9 has m = 0.628415

http://oeis.org/search?q=6%2C2%2C8%2C4%2C1%2C5&sort=&language=english&go=Search

The timelike geodesics of the Schwarzschild spacetime geometry can be
quantized when proper time is projected out. Wolfram has a picture of this
about halfway through the following:

http://blog.stephenwolfram.com/2016/02/black-hole-tech/

I've already started tabulating "Magic Numbers" for trajectory closure, but
have yet to determine a high precision scheme for getting the numerical
values exactly correct.

Something along these lines could be another interesting direction for
OEIS, as long as someone can figure out the question of how to make the
root finder converge reliably.

Thanks,

Brad


On Sun, Sep 25, 2016 at 3:00 PM, Felix Fröhlich <felix.froe at gmail.com>
wrote:

> Dear sequence fans
>
> I went through some papers from Mathematics of Computation and found R. P.
> Boas and V. C. Klema, A constant in the theory of trigonometric series,
> Math. Comp. 18 (1964), 674-674,
> http://dx.doi.org/10.1090/S0025-5718-1964-0176283-9. According to the
> paper, the certain digits of the constant at the time of publication were
> 0.3048 (more digits may have been computed since then). Is that constant
> already in the OEIS? I guess it may be A255504. Can someone confirm this?
>
> Best regards
> Felix Fröhlich
>
> --
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>