11/14/2009, 09:14 AM
Hi.
I saw this:
http://eom.springer.de/s/s087230.htm
Apparently, it seems there is a formula that can extend a Taylor series to a whole cut complex plane, called a Mittag-Leffler star of the function. This is interesting: because then perhaps maybe we could apply Faulhaber's formula to this to yield a continuum sum that works even for functions which are not entire: it appears the reason the original Faulhaber's formula (Faulhaber-on-Taylor formula) was not working is because continuum sum is a "global" operation (unlike derivative), as I mentioned in a recent post to the thread "Continuum sum formula rescued?", that is, the operation not only depends on the behavior of the function being put into it near the sum indices, but also very far away, and since a Taylor series of finite convergence radius only looks locally like the function they expand, we can explain the failure of the Faulhaber formula when applied to Taylor expansions of non-entire functions. Yet if the Mittag-Leffler series, on the other hand, converges over a whole cut plane, then this problem should not arise, as the global behavior will be correct.
If this could be done, it might then enable the usage of Ansus' sum formula for tetration to extend tetration to any complex base and height.
But the problem is I can't test it, since the description on the page is incomplete: what does the notation \( c_{\nu}^{(n)} \) mean? If \( c_{\nu} \) is the \( \nu \)-th Taylor coefficient, then what's \( c_{\nu}^{(n)} \)? What are the magic numbers \( k_n \)? I can't access the references, because I don't have access to a university library.
I saw this:
http://eom.springer.de/s/s087230.htm
Apparently, it seems there is a formula that can extend a Taylor series to a whole cut complex plane, called a Mittag-Leffler star of the function. This is interesting: because then perhaps maybe we could apply Faulhaber's formula to this to yield a continuum sum that works even for functions which are not entire: it appears the reason the original Faulhaber's formula (Faulhaber-on-Taylor formula) was not working is because continuum sum is a "global" operation (unlike derivative), as I mentioned in a recent post to the thread "Continuum sum formula rescued?", that is, the operation not only depends on the behavior of the function being put into it near the sum indices, but also very far away, and since a Taylor series of finite convergence radius only looks locally like the function they expand, we can explain the failure of the Faulhaber formula when applied to Taylor expansions of non-entire functions. Yet if the Mittag-Leffler series, on the other hand, converges over a whole cut plane, then this problem should not arise, as the global behavior will be correct.
If this could be done, it might then enable the usage of Ansus' sum formula for tetration to extend tetration to any complex base and height.
But the problem is I can't test it, since the description on the page is incomplete: what does the notation \( c_{\nu}^{(n)} \) mean? If \( c_{\nu} \) is the \( \nu \)-th Taylor coefficient, then what's \( c_{\nu}^{(n)} \)? What are the magic numbers \( k_n \)? I can't access the references, because I don't have access to a university library.
