Lindelof summation?
in [Math]
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From: mike3 on 12 Nov 2009 13:33 Hi. I saw this: http://en.wikipedia.org/wiki/Divergent_series#Lindel.C3.B6f_summation But it seems dubious: try it with the divergent Mercator series for log (3), it doesn't converge. What's really going on here? There seems to be an error in the Wikipedia. Also, I saw this: http://en.wikipedia.org/wiki/Mittag-Leffler_star How can one compute a Mittag-Leffler expansion for a general power series?
From: Gottfried Helms on 12 Nov 2009 14:58 Am 12.11.2009 19:33 schrieb mike3: > Hi. > > I saw this: > > http://en.wikipedia.org/wiki/Divergent_series#Lindel.C3.B6f_summation > > But it seems dubious: try it with the divergent Mercator series for log > (3), it doesn't converge. What's really going on here? There seems to > be an error in the Wikipedia. > Did you look at http://eom.springer.de/L/l058990.htm Sorry, can't say more at moment. Gottfried Helms > Also, I saw this: > http://en.wikipedia.org/wiki/Mittag-Leffler_star > > How can one compute a Mittag-Leffler expansion for a general power > series?
From: mike3 on 12 Nov 2009 20:46 On Nov 12, 12:58 pm, Gottfried Helms <he...(a)uni-kassel.de> wrote: > Am 12.11.2009 19:33 schrieb mike3: > > > Hi. > > > I saw this: > > >http://en.wikipedia.org/wiki/Divergent_series#Lindel.C3.B6f_summation > > > But it seems dubious: try it with the divergent Mercator series for log > > (3), it doesn't converge. What's really going on here? There seems to > > be an error in the Wikipedia. > > Did you look at > > http://eom.springer.de/L/l058990.htm > > Sorry, can't say more at moment. > > Gottfried Helms > I just looked at it, and the formula still seems to fail for the simple Mercator series. Yet it says that if we have a power series for a principal branch at the origin (as we do for the test case, log(1 + z)), then we should be able to Lindelof-sum it on the whole Mittag-Leffler star (in the case of log(1 + z) this coincides with having the branch cut at the negative real axis), but this doesn't seem to be right, the attempt to compute log (3) (i.e. set z = 2) fails. So either it applies only to a really limited and peculiar set of functions (that doesn't even include something as simple as log!), or I'm missing something here (which would be interesting, considering this describes essentially the same thing as WP, unless WP got it from here and this is wrong...).
From: mike3 on 15 Nov 2009 14:14 On Nov 12, 6:46 pm, mike3 <mike4...(a)yahoo.com> wrote: > On Nov 12, 12:58 pm, Gottfried Helms <he...(a)uni-kassel.de> wrote: > > > > > Am 12.11.2009 19:33 schrieb mike3: > > > > Hi. > > > > I saw this: > > > >http://en.wikipedia.org/wiki/Divergent_series#Lindel.C3.B6f_summation > > > > But it seems dubious: try it with the divergent Mercator series for log > > > (3), it doesn't converge. What's really going on here? There seems to > > > be an error in the Wikipedia. > > > Did you look at > > > http://eom.springer.de/L/l058990.htm > > > Sorry, can't say more at moment. > > > Gottfried Helms > > I just looked at it, and the formula still seems to fail for the > simple > Mercator series. Yet it says that if we have a power series for a > principal > branch at the origin (as we do for the test case, log(1 + z)), then we > should > be able toLindelof-sum it on the whole Mittag-Leffler star (in the > case of > log(1 + z) this coincides with having the branch cut at the negative > real > axis), but this doesn't seem to be right, the attempt to compute log > (3) (i.e. > set z = 2) fails. So either it applies only to a really limited and > peculiar > set of functions (that doesn't even include something as simple as > log!), or > I'm missing something here (which would be interesting, considering > this > describes essentially the same thing as WP, unless WP got it from here > and this > is wrong...). Does anyone else have any answers though as to why it doesn't seem to work?
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