[math-fun] continued fractions yielding pi
On 2/21/14, Warren D Smith <warren.wds@gmail.com> wrote:
Well, two natural questions arise:
Q1. is there any known series expansion for pi (or something closely related) which, like the usual series for e, converges factorially rather than merely geometrically?
If said series arose from an analytic function's Taylor series, then this would necessarily be an "entire" function, ruling out stuff like arcsine, arctan, elliptic functions, elliptic integrals, log, and polylogs, but permitting, e.g, the reciprocal of the Gamma function, and erf and erfc, and new functions.
--various continued fractions for erf and erfc are available here: http://functions.wolfram.com/GammaBetaErf/Erfc/10/01/ http://functions.wolfram.com/GammaBetaErf/Erf/10/01/ These would allow computing pi using a sum of two continued fraction evaluations to compute integral_0^infinity exp(-t^2) dt = sqrt(pi)/2. For example, let F1(z) = (sqrt(pi)/2)*erfc(z) F2(z) = (sqrt(pi)/2)*erf(z) so that for any z>0 we have sqrt(pi)/2 = F1(z) + F2(z) and could choose the value of z to maximize convergence speed or whatever. Then we have such continued fractions as F1(z) = exp(-z^2) * 1/(2z+) 2/(2z+) 4/(2z+) 6/(2z+) 8/(2z+...) F2(z) = exp(-z^2) * z/(1-x+) (2x/(3-x+)) (4x/(5-2x+)) (6x/(7-2x+)) (8x/((9-2x+...) where x=2*z*z. Similarly one could compute pi in a different way via a sum of continued fractions by computing certain gamma function values as a sum of two incomplete gamma functions to get the whole Euler-gamma-integral.
I think F1 below (for erfc) has crummy convergence. The numerators in the cf terms are increasing linearly, while the denominators are constant. The recurrence for successive convergents' numerators & denominators will overemphasize the older term over the newer term, which retards the decay of the oscillation around the limit. Gosper can likely say more. Rich ------- Quoting Warren D Smith <warren.wds@gmail.com>:
On 2/21/14, Warren D Smith <warren.wds@gmail.com> wrote:
Well, two natural questions arise:
Q1. is there any known series expansion for pi (or something closely related) which, like the usual series for e, converges factorially rather than merely geometrically?
If said series arose from an analytic function's Taylor series, then this would necessarily be an "entire" function, ruling out stuff like arcsine, arctan, elliptic functions, elliptic integrals, log, and polylogs, but permitting, e.g, the reciprocal of the Gamma function, and erf and erfc, and new functions.
--various continued fractions for erf and erfc are available here: http://functions.wolfram.com/GammaBetaErf/Erfc/10/01/ http://functions.wolfram.com/GammaBetaErf/Erf/10/01/
These would allow computing pi using a sum of two continued fraction evaluations to compute integral_0^infinity exp(-t^2) dt = sqrt(pi)/2.
For example, let F1(z) = (sqrt(pi)/2)*erfc(z) F2(z) = (sqrt(pi)/2)*erf(z) so that for any z>0 we have sqrt(pi)/2 = F1(z) + F2(z) and could choose the value of z to maximize convergence speed or whatever.
Then we have such continued fractions as F1(z) = exp(-z^2) * 1/(2z+) 2/(2z+) 4/(2z+) 6/(2z+) 8/(2z+...)
F2(z) = exp(-z^2) * z/(1-x+) (2x/(3-x+)) (4x/(5-2x+)) (6x/(7-2x+)) (8x/((9-2x+...) where x=2*z*z.
Similarly one could compute pi in a different way via a sum of continued fractions by computing certain gamma function values as a sum of two incomplete gamma functions to get the whole Euler-gamma-integral.
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