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
