(08/27/2009, 02:51 PM)bo198214 Wrote: I dont think one can not understand this without having taken a proper look at the pictures.I'm working on pictures of the isexp at the moment, will get back to this maybe early next week at the earliest.

Quote:Fortunately what you (also previously) describe does not only apply to the intuitive slog but I remember the pictures that Dmitrii made about his Cauchy slog.I'm not particularly familiar with Dmitrii's "Cauchy slog", though looking at the images I'm not really seeing a noticeable difference between that and the intuitive slog.

(They are scattered over the forum. Particularly this image depicts I think L2, as the left border. It depicts cslog(G) where G is the region bounded by L1 and exp(L1), though L1 extends here also to the lower fixed point. This region is particularly also used by Kneser in his construction which though is not of computational nature.)

As for Kneser's construction, I gave an initial look and was quickly overwhelmed; it will take me some time to properly decipher it, so I can't really comment on it yet.

Quote:So remembering these pictures I can follow you at least a bit.Well, here's where I think that the intuitive slog must be uniquely defensible: as I've mentioned before, any cyclic shift from the islog on the real axis (which would be necessary if two slogs are not the same) would necessarily destroy the exact approximation of the regular slog at either the upper or lower fixed points, if not both.

If I believe what you write I am asking myself where the specific *intuitive* slog is relevant. I guess you could do that construction with every slog and always get as a limit the regular slog. Because you would continue every slog that is defined on G to the whole plane except to the primary fixed points and cuts and somehow it seems as these coarse properties suffice for your construction instead of it relying on the fine grained structure directly on G.

Thus, there is no other slog which properly approximates the regular slogs at both the upper and lower fixed points, and which takes on real values without singularities on its principal branch. This can be seen by a simple thought experiment about a Fourier series representing the cyclic wobble function: necessarily it cannot be smooth as we travel infinitely far in both imaginary directions. Perhaps in one imaginary direction, but not the other.

As an example, consider . Notice that admits a proper Fourier series representation for our purposes: a function that is cyclic on the real axis, being equal to 0 at the integers.

This wobble function will be very small on the real axis, though it will admittedly take on non-real values. But it's small enough that perhaps we don't at first notice it as we work numerically.

As we move towards the upper fixed, the islog will go towards , and the thus the qslog will differ from the islog by smaller and smaller amounts, such that the regular slog is properly approximated at the upper fixed point.

However, as we move towards the lower fixed point, there comes a point where the cyclic shift becomes unstable, and the regular slog is not well approximated for the lower fixed point, even despite having not introduced new singularities.

Note that in order to pick a cyclic function which does not allow complex shifts for the real axis, it must be symmetric, and now the regular slog behavior is destroyed at both fixed points.

To my knowledge, one cannot construct a Fourier series that is smooth in both imaginary directions. Only one direction can be smooth, and only if one allows complex results for real inputs. Otherwise, neither direction is smooth.

If you know otherwise, then my argument would seem to fall apart.

Edit: Replaced with , to make clear my point about the periodicity.

~ Jay Daniel Fox