Thanks for the hint. It reminded me of my first encounter with fractional calculus many years ago when I was reading,
Calculus http://www.jsoftware.com/books/pdf/calculus.pdf I looked at the Wikipedia page but I cannot see, at least not yet, how it could help me to define an extension of ^: with a complex function (even if it is holomorphic) left argument and a real (floating) right argument. On Thu, May 17, 2018 at 10:03 PM, Roger Hui <[email protected]> wrote: > > I still remember an old post by Mark D. Niemiec on that subject: > > > > [Jforum] Power in the Rational Domain > > http://www.jsoftware.com/pipermail/general/2003-June/014722.html > > You may want to look at "Fractional Calculus", > https://en.wikipedia.org/wiki/Fractional_calculus , which make sense of > things like that 1.25-th derivative and the 2.33-rd integral. > > > > On Thu, May 17, 2018 at 4:50 PM, Jose Mario Quintana < > [email protected]> wrote: > > > Brian, > > > > No, I am afraid I was not trying to address your question. I was just > > suggesting that a counterpart of the usual saying "the approximation > breaks > > down at that certain point" (dividing the real line) is "the > approximation > > breaks down at a certain line" (dividing the complex plane). > > > > Bob, > > > > I had seen before that video and I liked a lot. Whereas the domain > > coloring grid (of isomagnitud and isophase lines) is a grid of the > > transformation of the domain (the image), in that video what is shown is > > the transformation a square grid (if I am not mistaken). One very > appealing > > feature of the video, at least for me, was the smooth transition from the > > grid of the domain into the transformed grid corresponding to the > image. I > > can only guess the details how this was done. In my opinion, ideally, it > > should be done by fractional powers in the sense of ^: (e.g., u^:1 -: > > (u^:0.5)@:(u^:0.5) so to speak). I still remember an old post by Mark D. > > Niemiec on that subject: > > > > [Jforum] Power in the Rational Domain > > http://www.jsoftware.com/pipermail/general/2003-June/014722.html > > > > I am not sure if this approach would always work for analytical > (continued) > > functions. (I regret not having taken a complex variable course as an > > optative in college.) > > > > > > On Wed, May 16, 2018 at 1:13 PM, robert therriault < > [email protected]> > > wrote: > > > > > Pretty amazing visualization of what the zeta function is by Grant > > > Sanderson. > > > > > > https://www.youtube.com/watch?v=sD0NjbwqlYw > > > > > > His videos are mathematically grounded and incredibly communicative in > > the > > > ways that they use animation and graphs. > > > > > > Cheers, bob > > > > > > > On May 16, 2018, at 9:45 AM, Brian Schott <[email protected]> > > > wrote: > > > > > > > > Pepe, > > > > > > > > I was able to compare the "domain"-induced viewrgb with the wikipedia > > > > version and I see the difference you noted. > > > > Was that example an attempt at addressing my question about where > lines > > > of > > > > finite length can be drawn for this case? > > > > I ask that because you used the phrase "at that line". > > > > > > > > > > > > On Tue, May 15, 2018 at 6:50 PM, Jose Mario Quintana < > > > > [email protected]> wrote: > > > > > > > >> I am glad to hear that it runs on JQt/Linux. It also seems to run > on > > > JHS > > > >> (at least it works with a Kindle Paperwhite 3 running on JHS/Linux > > > >> (BusyBox) using a custom J interpreter). > > > >> > > > >> You might find the following clumsy verb useful, > > > >> > > > >> domain=. |.@|:@({.@[ + ] *~ j./&i.&>/@+.@(1j1 + ] %~ -~/@[))&>/ > > > >> > > > >> It produces the vertices of a square grid corresponding to (the > lower > > > and > > > >> upper points of) a given complex interval and resolution; for > example, > > > >> > > > >> In particular, > > > >> > > > >> viewrgb 12 ccEnhPh zetahat"0 domain _20j_30 20j30 ; 0.1 > > > >> > > > >> reproduces, to some extent, the first graph on the Wikipedia page > for > > > the > > > >> Riemann zeta function. The leftmost section of the graph produced > by > > J > > > >> looks suspicious and might indicate that Ewart's default > approximation > > > >> (zetahat) breaks down at that point (or rather, at that line). > > > >> > > > >> > > > >> > > > >> > > > > ------------------------------------------------------------ > ---------- > > > > For information about J forums see http://www.jsoftware.com/ > forums.htm > > > > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > > > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
