Mittag-Leffler series for generating continuum sum? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/14/2009, 09:18 PM (This post was last modified: 11/19/2009, 09:31 PM by mike3.) I also stumbled upon this very interesting paper: http://arxiv.org/pdf/hep-th/9206074 It mentions methods that sum power series in the Mittag-Leffler star. One formula it gives, is this: given a principal branch of an analytic function, represented by its power series $f(z) = \sum_{n=0}^{\infty} a_n z^n$ at z = 0, $f(z) = \int_{0}^{\infty} \exp(-\exp(t)) \sum_{n=0}^{\infty} a_n \frac{(tz)^n}{\mu(n)} dt$ with $\mu(n) = \int_{0}^{\infty} \exp(-\exp(t)) t^n dt$. But it the formulas don't seem to work when tested numerically. Try it with the reciprocal series for $f(z) = \frac{1}{1 - z}$, which they mention in the paper, i.e. $f(z) = \sum_{n=0}^{\infty} z^n$. Then try evaluating using these formulas at $z = -2$, which outside the convergence radius for the series, but inside the Mittag-Leffler star. It seems to give huge values. Of course (and I highly suspect this is the case), I've missed something here... what might it be? ADDENDUM: I see now, this does work... it's just that the terms "hump up" for reasonably large z-values before they get smaller and the thing converges... it needed 768 terms to converge to a few places for z = -2, but got ~0.3333 like I'd expect for $\frac{1}{1 - -2} = \frac{1}{3}$. I think it gets better after that point since once you're over the "hump" the terms get small fairly quick, I suppose 1024 terms would get much more accuracy but calculating $\mu(n)$ takes a long time. « Next Oldest | Next Newest »

 Messages In This Thread Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 09:14 AM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/14/2009, 04:54 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 08:24 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 09:18 PM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/14/2009, 10:05 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/14/2009, 10:24 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/17/2009, 03:21 AM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/17/2009, 12:10 PM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/17/2009, 06:54 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/17/2009, 08:44 PM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/17/2009, 09:20 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/17/2009, 11:22 PM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/18/2009, 08:05 AM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/18/2009, 09:55 AM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/18/2009, 10:43 AM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/18/2009, 11:01 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/28/2009, 10:36 PM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 11/28/2009, 10:56 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 11/29/2009, 03:48 AM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 12/11/2009, 11:16 AM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 12/11/2009, 11:45 AM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 12/12/2009, 01:47 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 12/12/2009, 08:18 PM RE: Mittag-Leffler series for generating continuum sum? - by bo198214 - 12/12/2009, 09:53 PM RE: Mittag-Leffler series for generating continuum sum? - by mike3 - 12/13/2009, 02:48 AM Mittag-Leffler product ? Ramanujan's master theorem - by tommy1729 - 12/13/2009, 05:41 PM RE: Mittag-Leffler product ? Ramanujan's master theorem - by mike3 - 12/13/2009, 08:19 PM RE: Mittag-Leffler product ? Ramanujan's master theorem - by tommy1729 - 12/13/2009, 11:15 PM RE: Mittag-Leffler product ? Ramanujan's master theorem - by mike3 - 12/14/2009, 04:40 AM

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