In article <email@example.com>, Tleko <firstname.lastname@example.org> wrote: @In article <email@example.com>, Ilias Kastanas wrote: @> @>@:>tleko wrote: @>@:> Indeed we have z=z* . @>@:> We write z=r.exp(i.@) @>@:> z*=r.exp(i.(-@)) @>@:> to obtain z-z*=r.((cos@+i.sin@)-(cos(-@)+i.sin(-@-pi))) = 0 . @ @>@:>)@@In the article <firstname.lastname@example.org> Ilias Kastanas wrote: @>@:>)@@Is it possible not to realize this can only hold for y=o, i.e. for @>@:>)@@real z, and not in general? @>@:> @>@:> Yes it is. 2.r.i.sin@ = 2.i.y is valid for any real y @>@:> not only for y=0.
@> No it is not. x+iy = x-iy holds for y=0, not for "any real y"... @ @> Of course you deleted this, and you are pretending to "answer" by @> stating that r sin@ = y. Denial goes a far way. @ @> How about a straight answer? Are you claiming x+iy = x-iy holds @> for all real y? YES or NO?
@ I did NOT write x+iy=x-iy you did.
You wrote z = z* and z - z* = 0 a number of times.
It so happens that z = x + iy and z* = x - iy. If this comes as a surprise look it up: e.g. Hewitt & Stromberg, "Real and Abstract Analysis", p. 48, or G.H. Hardy, "Pure Mathematics", p. 82.
In those same pages, by the way, you will discover that Re(z) = x and, wonder of wonders, Im(z) = y.
Equality can be so inconvenient... You claimed z = z*, and thus x + iy = z = z* = x - iy. Do you insist on it, yes or no?
@ I wrote in the article <email@example.com>: @ z=r.exp(i.@) , z*=r.exp(i.(-@))
Say, r = sqrt(2), @ = pi/4, z = 1 + i, z* = 1 - i. What sort of hallucination can possibly suggest that 1 + i = 1 - i ?
@ to obtain z-z*=r.((cos@+i.sin@)-(cos(-@)+i.sin(-@-pi)))=0 @ @ < where Imag(z*)=r.sin(-atan(y/x)-pi)=r.sin(atan(y/x)) >.
This is wrong, and will stay wrong no matter how many times you repeat it, or how strenuously you ignore the errors in it, errors that a number of people pointed out.
@ It shows that x+iy and x-iy are not analytic functions in the unit @ disk.
This is wrong, and will stay wrong no matter how many times you repeat it, or how strenuously you ignore the numerous proofs showing that z is analytic and z* is not.
@ I will be glad to send you the plots and the 7th volume of 'Critical @ Comments ..' catalogued in the Library of Congress and several @ university libraries in the United States and abroad.
Thank you, but I cannot think of a more idle exercise. Plots are irrelevant to analyticity of z or z*, to say the least. And lengthy volumes are unnecessary for matters short and simple. If the 7th volume or its cohorts make nonsensical claims they are nonsense, whether cata- logued in the Library of Congress or not. If they consist of dozens of people telling you you are wrong and you ignoring them or thanking them for their contribution, it is hardly news.