> : We all agree that f(z) = z^2 is analytic .. > > We have, however, for z^2 > > Real= sqrt((x^2-y^2)^2+(2x*y)^2)*cos((atan(2x*y/(x^2-y^2))+(0 or > pi)) > and > Imag= sqrt((x^2-y^2)^2+(2x*y)^2)*sin((atan(2x*y/(x^2-y^2))+(0 or > pi)) . > > Neither Real nor Imaginary part of z^2 is analytic (in real terms) in the > unit disk. > > If you like to see the plots request a fax transmission.
This is too stupid for words. From private correspondance I have learnt that Tleko has been told by several people that atan(Im(z)/Re(z)) is *not* equal to arg(z) in computer lingo, and is at best a very dubious formulation in mathematics. He insists, however, in doing his plots with atan(Im/Re), like above, instead of atan2(Im,Re) and gets excited like I know what over the fact that there is a phase jump for sign changes of x. Hopeless. For a quote from my correspondance:
[written by me, dak, and quoted by Tleko:] I have looked at the faxes. I could not believe my eyes. I could as well be writing to /dev/null. Please be so kind and *listen*. Just replace atan(y./x) in *all* of your programs by atan2(y,x). That is the mistake in *every* single program of yours, and I have written you already about 3 times *why* this is an mistake, and what you should do to amend it.
Replace atan(y./x) by atan2(y,x) in program 1. Replace atan(y./x) by atan2(y,x) in program 2. Replace atan(y./x) by atan2(y,x) in program 3. Replace atan(y./x) by atan2(y,x) in program 4.
The function atan cannot return anything outside the range -pi/2..pi/2, and therefore is *not* able to return the proper arg of x+iy which covers the range -pi..pi. It will switch to return values out of phase by pi when x gets negative, and all your plots show exactly this effect due to the inappropriate use of atan.
Once you replace atan(y./x) by atan2(y,x), it should be possible for you to be doing your own correct plots.
-- David Kastrup Institut fuer Neuroinformatik, Ruhr-Universitaet Bochum Email: firstname.lastname@example.org Telephon: +49-234-700-5570 ----------------------- [written by Tleko:]
Thank you for the message. atan2 is not an analytic function. Neither are z=x+iy or z=x-iy. You apparently have not read the earlier postings.
[Complete requote of treatise with mistakenly using atan(y/x) for arg(z) deleted]
Just want to warn you off investing any more time with this individual. Better argue with your wallpaper. At least it might become damp and peel off, which is some reaction.