Re: a conjecture !

On Oct 8, 12:00 pm, amy666 <tommy1...@xxxxxxxxxxx> wrote:
amy666 a écrit :
sigh !

no we dont assume holomorphic from the start !

we conjecture holomorphic follows !

Then it would be good 1) to learn what it means 2) to
express yourself
more clearly 3) to learn a few theorems (like
Cauchy's) 4) to sigh a
little cless...

i know what it means !!

you guys are confused by it , not me !

Temper, temper.
The problem with what you write, Tommy, is that use often write
phrases that make no sense to those who actually know the mathematical
technical meanings of what you write, and so it becomes really
difficult to tell what you are really on about. We have to guess, and
only you can tell if they are accurate guesses. With that in mind,
lets go through your list, which I will paste in here since it wasn't
in what you replied to:

a function f(z) is entire apart from potential removable singularities if

1) it converges for all of z

This one, for example, makes no sense as stated. Sequences converge,
series converge, you can talk about f(z) converging to a limit as z
goes to a. This last is the only one that has any hope of being what
you mean, when put into the form, f(z) coverges to some limit
(depending on z, of course) as z goes to a for every complex a.
Is that what you meant? Or did you mean something to the effect that
f(z) is defined as a power series, and that series converges for every
complex z. A complex-valued function of a complex variable is, after
all, just a set of ordered pairs of complex numbers and can be
provided in a very large variety of ways. If you meant that the power
series converges everywhere, then there is nothing to discuss, as then
the function is already known to be entire.

2) it has no poles

I have not seen the concept of pole discussed outside of the context
of a function that holomorphic at a lot of places. The pole is then
an isolated singularity, the center of some circle of radius say, r,
inside of which, except for the center, it is holomorphic. It is
unclear to me what a pole is for the most general complex-valued
function of a complex variable. What do you mean by it? Without that
information, no one can proceed.

3) it has no essential singularities

A comment similar to the one for condition 2). What do YOU mean by an
essential singularity?

4) it is Coo

This one is clear enough. Obviously that is a necessary condition,
and you understand that it is not a sufficient condition .

5) its first derivate is nowhere oo

I assume you mean derivative. If a function is Coo, then 5) is
already excluded, as it would not even be differentiable at such a
point. I have to assume that you mean that it is Coo outside of some
isolated "potential removable singularities." Since there is no such
thing as a potential removable singularity, as a previous poster
noted, I assume that is just a bit of tangled English and you really
meant to allow "some removable singularities." There is still a
problem, as it is unclear what a removable singularity might be in
the context of a general complex-valued function of a complex
variable. How does this sound as a definition of removable
singularity? A removable singularity is a point a in the complex
plane where f(a) is defined, and the limit as z approaches a of f(z)
is b, where b is not equal to f(a). You should then specify what you
mean by "some" removable singularities. A finite set? A countable
set? What?

6) it is Goo , analogue to Coo where G is gaussian curvature

How do you define the Gaussian curvature of your function? I have
only seen Gaussian curvature in the context of surfaces in the plane
which really doesn't work here. It is always a real number in the
definitions I have seen. However, I am a babe in the woods as far as
differential geometry is concerned so I am willing to concede that my
ignorance of such things is do to my ignorance and not to yours. But
I won't concede it unless you present a definition or a reference to a
place where it can be found.

That being said, your version of Gaussian curvature would have to give
you a complex number associated with every complex number which is to
say a function g(z) which is also a complex-valued function of a
complex variable.

7) the equation f(z) = A has a solution for all A apart from perhaps a single exception

So you have seen Picard's little theorem and make its conclusion about
entire functions a part of your precondition for their existence.
Nothing wrong with that.

8) f'(z) has the same 7 properties above

Nothing wrong with this one.

9) G_1 f(z) has the same 8 properties as above

I assume you mean that G_1 f(z) is your version of the Gaussian
curvature of f at z. In other words, it is what I called g(z).


I hope that, after having read my detailed comments about the
conditions of your conjecture you are beginning to realize the
inadequacy of your presentation. It is impossible to discuss your
conjecture because, as stated, it doesn't make sense. There is
nothing to work with because nobody can figure out what you mean. You
have given some evidence of actually being interested in mathematics
which is why I am giving you this help. I don't think that you are a
troll. I hope you can appreciate how frustrating and annoying it can
be to deal with language such as yours which is so loose as to be
impossible to turn into anything meaningful without further
clarification and which looks like it might be impossible to fix up
anyway even with greater clarificationl. In other words, if you try
to fix up the problems I have discussed, you find that your thinking
was flawed and you will have to go back to the drawing board.


Regards,
Achava


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