Would You Agree With This Statement?
mike3
I'd agree with this statement:
"Holomorphic functions are the central object of study in 'complex
analysis'. Without them, i.e. when one moves to the more general class
of C->C functions which are not holomorphic (real/imag parts do not
satisfy the Cauchy-Riemann equations) but still 'more well behaved'
than general functions C->C (namely smooth partials), the resulting
'complex' analysis *degenerates* into 'real analysis' and 'vector
analysis'."
I'm pretty darned sure you'd agree with this statement, too, so much
I'd bet money on it. Gone is the complex derivative, the "essence" or
"heart" of complex analysis, so complex analysis is gone and what we
have is something that more or less can be considered equivalent to
vector fields of R^2. Do you? (I'm betting around a million dollars on
it right now! :) No, just kidding :) )
But this is not so interesting as what I'm wondering about what
happens when we approach this degenerate condition -- how does our
complex analysis _break down_ as we get "closer" to there? What I mean
by "approach" this degeneracy is this: Consider some sequence of
functions, each of which resemble some nonholomorphic function in the
more general class above, but they themselves are all holomorphic. A
more "rigorous" definition is this:
A sequence of functions f_n(z) from C->C, defined everywhere, is said
approach another function g: C->C, also defined everywhere, if
lim_{n->inf} f_n(z) = g(z)
for all points z in C. (We could extend this to a continuous
collection of functions by letting n be a real parameter and maybe
have f_n(z) continuous, differentiable, even smooth in n.)
So let all f_n(z) for all finite n be holomorphic, and let f be non-
holomorphic but with smooth partials. My favorite is the "pillow
function" pl(z) = (sin(Re(z)) + cos(Im(z))) + i(sin(Re(z)) - cos(Im
(z))). (Graph the real/imag parts to understand the name.) Now, what
happens to the complex derivative of f_n(z) as n is increased? I would
expect it becomes more and more unstable, or explodes and takes on
ever more wacky values, or something like that? What sort of behaviors
would we expect? But then consider the "vector-analysis"-like
"directional derivatives" defined by
dirdiff_u f(z) = lim_{h->0} (f(z + hu) - f(z))/(hu), h a real number
where u is a complex number representing the "direction". (Note this
is not exactly the same as a "vector analysis" directional derivative,
for example there is that factor of u in the denominator which is
impossible in regular vector analysis as vector division is not
necessarily defined.) It will equal the complex derivative if f has
one, regardless of the choice of u. But it can also exist even for
nonholomorphic f that do not have complex derivatives. (The "true"
"vector analysis" derivative is the directional derivative in the
direction Re(u)*i + Im(u)*j of the R^2 vector field F(v) = Re(f(z))*i
+ Im(f(z))*j and v = Re(z)*i + Im(z)*j. (Note here i and j are the
unit vectors forming the basis for R^2: i = (1, 0), j = (0, 1)) not
imaginary units) So what happens in my given scenario to _these_
derivatives, for a fixed u? And how, if at all, does this constrain
the destabilization of the complex derivative? Of course this all
assumes such a series of functions can even exist, which I'm not sure
of.
David C. Ullrich
as "pointwise convergence".
Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
holomorphic then the derivatives of f_n cannot remain
uniformly bounded on compact sets. The f_n themselves
cannot remain uniformly bounded on compact sets
either; this is easy to see from Montel's theorem.
(If (f_n) is uniformly bounded on compact sets then some
subsequence converges uniformly on compact sets to a
holomorphic function; since f_n -> f pointwise this
shows f is holomorphic. Now, if the derivatives are
bounded on compact sets and the functions converge
pointwise then it follows that the functions are
bounded on compact sets.)
On Mon, 1 Dec 2008 01:48:40 -0800 (PST), mike3 <mike...@yahoo.com>
wrote:
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
David Bernier
> Hi.
>
> I'd agree with this statement:
>
> "Holomorphic functions are the central object of study in 'complex
> analysis'. Without them, i.e. when one moves to the more general class
> of C->C functions which are not holomorphic (real/imag parts do not
> satisfy the Cauchy-Riemann equations) but still 'more well behaved'
> than general functions C->C (namely smooth partials), the resulting
> 'complex' analysis *degenerates* into 'real analysis' and 'vector
> analysis'."
If the domain is all of C, then functions with a finite number
of singularities in every bounded subset of C are also
interesting.
If D is a connected, simply connected subset of C, and
f is holomorphic in D, then Re(f) and Im(f) are both
harmonic functions, i.e.
Laplacian( Re(f)) == 0 in D and
Laplacian( Im(f)) == 0 in D.
The Laplacian operator appears in the "Heat equation",
(with boundary conditions). If the temperature
at some point doesn't change over time, then the
temperature function satisfies the Laplace equation,
except on the boundary where the boundary conditions
take hold. With unchanging temperature, we have
a steady-state solution. These are "very special"
functions (the harmonic ones).
For 1-D, the analogue of the Laplacian equation
is d^2 u/dx^2 == 0. So u(x) = ax + b,
with (say) a, b real for a real-valued u. So
with 1-D, harmonic function theory is simple.
Everything changes in 2-D.
Suppose we look for a meaning of "almost harmonic"
function for 2-D. This would apply to C^2
(twice continuously differentiable
real-valued functions on R^2). I don't know of one
that I would consider reasonable. There is also
in complex the "harmonic conjugate", i.e.
given f which is real harmonic, find g such that
f + ig is analytic (except singularities).
From what I remember, with fairly weak conditions,
any two harmonic conjugates of f differ by a constant.
Of course, there's a lot more that I don't know.
I seem to remember that there's a notion of capacity for
compact subsets of C which is of interest to
complex analysts...
David Bernier
mike3
> Your "approaches" as defined below is simply known
> as "pointwise convergence".
>
> Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
> holomorphic then the derivatives of f_n cannot remain
> uniformly bounded on compact sets. The f_n themselves
> cannot remain uniformly bounded on compact sets
> either; this is easy to see from Montel's theorem.
>
Hmm. So what would, say, a graph, of such a hypothetical
function sequence, converging to the pillow function, look like?
And would the derivatives "explode more" (get bigger/more wild
values) in the places the functions more closely resemble the
function?
Also, what about the question about agreeing with my
statement I put in quotes (actually it's not entirely mine, I
took the bit "Holomorphic functions are the central object of
study in 'complex analysis'." from the Wikipedia and added
some more stuff after it.)
> (If (f_n) is uniformly bounded on compact sets then some
> subsequence converges uniformly on compact sets to a
> holomorphic function; since f_n -> f pointwise this
> shows f is holomorphic. Now, if the derivatives are
> bounded on compact sets and the functions converge
> pointwise then it follows that the functions are
> bounded on compact sets.)
>
> On Mon, 1 Dec 2008 01:48:40 -0800 (PST), mike3 <mike4...@yahoo.com>
Robert Israel
Also of interest are subharmonic functions and superharmonic
functions.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
mike3
> mike3 wrote:
> > Hi.
>
> > I'd agree with this statement:
>
> > "Holomorphic functions are the central object of study in 'complex
> > analysis'. Without them, i.e. when one moves to the more general class
> > of C->C functions which are not holomorphic (real/imag parts do not
> > satisfy the Cauchy-Riemann equations) but still 'more well behaved'
> > than general functions C->C (namely smooth partials), the resulting
> > 'complex' analysis *degenerates* into 'real analysis' and 'vector
> > analysis'."
>
> [...]
>
> If the domain is all of C, then functions with a finite number
> of singularities in every bounded subset of C are also
> interesting.
>
> If D is a connected, simply connected subset of C, and
> f is holomorphic in D, then Re(f) and Im(f) are both
> harmonic functions, i.e.
> Laplacian( Re(f)) == 0 in D and
> Laplacian( Im(f)) == 0 in D.
>
But here f is _not_ holomorphic as it violates the Cauchy-Riemann
equations. E.g. the pillow function I mentioned, or the absolute
value function, or the real/imaginary part functions, etc. So then as
I mentioned, the resulting "complex" analysis degenerates to vector
analysis of what amounts to vector fields on R^2.
David Bernier
pl(x + iy) = sin(x) + cos(y) + i*(sin(x) - cos(y))
del(Re(pl))/del x (0, 0) = cos(0) = 1
del(Im(pl))/del y (0, 0) = sin(0) = 0 .
"lim_{n->inf} f_n(z) = g(z)
for all points z in C."
I interpret this as saying that the f_n converge point-wise to g.
This is not a very strong requirement, as can be seen
in real analysis on [0, 1] with f_n(x) = x^n, so that
lim_{n -> oo} (x^n ) = either 0 if x is in [0,1), or else 1 if x=1.
Another case you mention is n being a real parameter.
> expect it becomes more and more unstable, or explodes and takes on
> ever more wacky values, or something like that? What sort of behaviors
> would we expect? But then consider the "vector-analysis"-like
With the f_n being holomorphic, and converging to (say)
pl point-wise, I don't know what to expect. But what I
get is that pl doesn't satisfy the C.R. equations at
z = 0.
David Bernier
David C. Ullrich
wrote:
>On Dec 1, 8:05 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> Your "approaches" as defined below is simply known
>> as "pointwise convergence".
>>
>> Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
>> holomorphic then the derivatives of f_n cannot remain
>> uniformly bounded on compact sets. The f_n themselves
>> cannot remain uniformly bounded on compact sets
>> either; this is easy to see from Montel's theorem.
>>
>
>Hmm. So what would, say, a graph, of such a hypothetical
>function sequence, converging to the pillow function, look like?
I don't know that there is such a sequence (I doubt it; I
suspect that any pointwise limit of analytic functions
must be complex-differentiable at a lot of points).
mike3
> On Mon, 1 Dec 2008 11:53:48 -0800 (PST), mike3 <mike4...@yahoo.com>
> wrote:
>
> >On Dec 1, 8:05 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >> Your "approaches" as defined below is simply known
> >> as "pointwise convergence".
>
> >> Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
> >> holomorphic then the derivatives of f_n cannot remain
> >> uniformly bounded on compact sets. The f_n themselves
> >> cannot remain uniformly bounded on compact sets
> >> either; this is easy to see from Montel's theorem.
>
> >Hmm. So what would, say, a graph, of such a hypothetical
> >function sequence, converging to the pillow function, look like?
>
> I don't know that there is such a sequence (I doubt it; I
> suspect that any pointwise limit of analytic functions
> must be complex-differentiable at a lot of points).
>
What is "a lot of points", anyway? What criterion are you using to
determine how "many" points there are, anyway?
>
>
> >And would the derivatives "explode more" (get bigger/more wild
> >values) in the places the functions more closely resemble the
> >function?
>
> >Also, what about the question about agreeing with my
> >statement I put in quotes (actually it's not entirely mine, I
> >took the bit "Holomorphic functions are the central object of
> >study in 'complex analysis'." from the Wikipedia and added
> >some more stuff after it.)
>
And any comment on this?
<snip>
mike3
what's "del"? The partial derivative ("deeh") symbol? Because I'm
sure you don't mean the nabla operator?
> "lim_{n->inf} f_n(z) = g(z)
> for all points z in C."
>
> I interpret this as saying that the f_n converge point-wise to g.
> This is not a very strong requirement, as can be seen
> in real analysis on [0, 1] with f_n(x) = x^n, so that
> lim_{n -> oo} (x^n ) = either 0 if x is in [0,1), or else 1 if x=1.
>
What about if we had some sort of monotonicity requirement?
Say, diff_n(z) = | f_n(z) - g(z) | is monotone in n. Or is that too
much?
> Another case you mention is n being a real parameter.
>
> > expect it becomes more and more unstable, or explodes and takes on
> > ever more wacky values, or something like that? What sort of behaviors
> > would we expect? But then consider the "vector-analysis"-like
>
> With the f_n being holomorphic, and converging to (say)
> pl point-wise, I don't know what to expect. But what I
> get is that pl doesn't satisfy the C.R. equations at
> z = 0.
>
That's right, it's not supposed to, so I'm trying to see what happens
as a function
gets "closer and closer" to one that doesn't satisfy those equations,
how the
complex analysis starts to fail and degenerate.
Robert Israel
> On Dec 2, 5:16=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
> > On Mon, 1 Dec 2008 11:53:48 -0800 (PST), mike3 <mike4...@yahoo.com>
> > wrote:
> >
> > >On Dec 1, 8:05=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
> > >> Your "approaches" as defined below is simply known
> > >> as "pointwise convergence".
> >
> > >> Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
> > >> holomorphic then the derivatives of f_n cannot remain
> > >> uniformly bounded on compact sets. The f_n themselves
> > >> cannot remain uniformly bounded on compact sets
> > >> either; this is easy to see from Montel's theorem.
> >
> > >Hmm. So what would, say, a graph, of such a hypothetical
> > >function sequence, converging to the pillow function, look like?
> >
> > I don't know that there is such a sequence (I doubt it; I
> > suspect that any pointwise limit of analytic functions
> > must be complex-differentiable at a lot of points).
> >
>
> What is "a lot of points", anyway? What criterion are you using to
> determine how "many" points there are, anyway?
If a sequence of holomorphic functions converges pointwise on some
open set D, then the limit is holomorphic on a dense open subset of D.
The proof involves the Baire Category Theorem and Vitali's Convergence Theorem.
Shouldn't this be in your book, David?
Mariano Suárez-Alvarez
Not really. Before that degeneration occurs, there
is lots of things that are very much "complex".
The theory of sub- and superharmonic functions,
of pluri(sub)harmonic functions, of quasiholomrphic mappings,
and some purely topological variants like quasiconformal
mappings, retain *lots* of properties one usually
attaches to holomorphic functions.
-- m
mike3
But do these things apply to, say, the pillow function, the absolute
value function, the conjugation function, or the real/imaginary part
functions?
galathaea
yeah
those are the types of things you see there
for instance
if f is holomorphic
|f|^p is subharmonic forAll p > 0
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
David C. Ullrich
<isr...@math.MyUniversitysInitials.ca> wrote:
>mike3 <mike...@yahoo.com> writes:
>
>> On Dec 2, 5:16=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> > On Mon, 1 Dec 2008 11:53:48 -0800 (PST), mike3 <mike4...@yahoo.com>
>> > wrote:
>> >
>> > >On Dec 1, 8:05=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> > >> Your "approaches" as defined below is simply known
>> > >> as "pointwise convergence".
>> >
>> > >> Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
>> > >> holomorphic then the derivatives of f_n cannot remain
>> > >> uniformly bounded on compact sets. The f_n themselves
>> > >> cannot remain uniformly bounded on compact sets
>> > >> either; this is easy to see from Montel's theorem.
>> >
>> > >Hmm. So what would, say, a graph, of such a hypothetical
>> > >function sequence, converging to the pillow function, look like?
>> >
>> > I don't know that there is such a sequence (I doubt it; I
>> > suspect that any pointwise limit of analytic functions
>> > must be complex-differentiable at a lot of points).
>> >
>>
>> What is "a lot of points", anyway? What criterion are you using to
>> determine how "many" points there are, anyway?
>
>If a sequence of holomorphic functions converges pointwise on some
>open set D, then the limit is holomorphic on a dense open subset of D.
That's amazing, like you're chanelling Ullrich. (I didn't want to
state anything explcitly because I didn't have time to think about
it or to try to look it up...)
>The proof involves the Baire Category Theorem and Vitali's Convergence Theorem.
>
>Shouldn't this be in your book, David?
Heh. Yes, it should. Unfortunately, if the book contained everything
it should it would be five times longer, and the editor was concerned
about the page count as it is. There are many other omissions that
bother me a lot more (for example, nothing to do with the fact that
log|f| is subharmonic, using that word or not, eg no Jensen's formula.
Only a hint at the existence of Hadamard's theorem on factoring
entire functions, not even the statment of the result. Etc.)
The problem is the style the book is written in. I say in the
Introduction that a more accurate title would be "Complex
Explained in Excruciating Detail". Better yet might be
"Complex With a Big Emphasis on Explaining Why Things
Really Work, How it is that Various Tricks are Really
Methods, Etc." Among the people who like the book,
the pedagogic style seems to be what they like. But it
takes more pages than a dry theorem, proof, theorem,
oroof exposition.
David C. Ullrich
wrote:
>[...]
>
>That's right, it's not supposed to, so I'm trying to see what happens
>as a function
>gets "closer and closer" to one that doesn't satisfy those equations,
>how the
>complex analysis starts to fail and degenerate.
One standard topic that might interest you from this point
of view is the study of the "d-bar" operator. Define
d-bar f = df/dx + i df/dy.
f is holomoprhic if and only if d-bar f = 0, so one notion of
"close to holomorphic" would be "d-bar f is small".
Here "small" could mean various things. Things like
Cauchy's Theorem have analogous statements for
non-holomorphic functions, including terms involving
d-bar f.
(See for example Chapter n (n = 26?) of my book
"Complex Made Simple" for a few simple applications,
or see the proof of Mergelyan's Theorem in Rudin
"Real and Complex Analysis" for a big application...)
>> David Bernier
>>
>> > "directional derivatives" defined by
>>
>> > dirdiff_u f(z) = lim_{h->0} (f(z + hu) - f(z))/(hu), h a real number
>>
>> > where u is a complex number representing the "direction". (Note this
>> > is not exactly the same as a "vector analysis" directional derivative,
>> > for example there is that factor of u in the denominator which is
>> > impossible in regular vector analysis as vector division is not
>> > necessarily defined.) It will equal the complex derivative if f has
>> > one, regardless of the choice of u. But it can also exist even for
>> > nonholomorphic f that do not have complex derivatives. (The "true"
>> > "vector analysis" derivative is the directional derivative in the
>> > direction Re(u)*i + Im(u)*j of the R^2 vector field F(v) = Re(f(z))*i
>> > + Im(f(z))*j and v = Re(z)*i + Im(z)*j. (Note here i and j are the
>> > unit vectors forming the basis for R^2: i = (1, 0), j = (0, 1)) not
>> > imaginary units) So what happens in my given scenario to _these_
>> > derivatives, for a fixed u? And how, if at all, does this constrain
>> > the destabilization of the complex derivative? Of course this all
>> > assumes such a series of functions can even exist, which I'm not sure
>> > of.
David C. Ullrich
David C. Ullrich
It's easier to prove something when you know it's true:
Say (f_n) is a pointwise convergent sequence of functions
holomorphic in D. Let E_N be the set of points
where sup |f_n| <= N. Then E_N is closed and the
union of the E_N is D, so some E_N contains a disk.
Say a disk is "good" if (f_n) is uniformly bounded on said disk.
The above argument, specialized to some disk in D, shows that
D' is dense in D, where D' is the union of all the good disks
in D. The limit is holomorphic in D'.
(The other day I was being stupid, not realizing I was
done at this point, wanting to get uniform convergence
on compact subsets of D' and not realizing that that's
trivial: If K is a compact subset of D' then K is covered
by finitely many good disks, hence (f_n) is uniformly
bounded on K.)
mike3
> On Wed, 03 Dec 2008 06:15:39 -0600, David C. Ullrich
>
>
>
> <dullr...@sprynet.com> wrote:
> >On Tue, 02 Dec 2008 14:08:02 -0600, Robert Israel
> ><isr...@math.MyUniversitysInitials.ca> wrote:
>
>
> >>> On Dec 2, 5:16=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >>> > On Mon, 1 Dec 2008 11:53:48 -0800 (PST), mike3 <mike4...@yahoo.com>
> >>> > wrote:
>
> >>> > >On Dec 1, 8:05=A0am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >>> > >> Your "approaches" as defined below is simply known
> >>> > >> as "pointwise convergence".
>
> >>> > >> Yes, if f_n is holomorphic, f_n -> f pointwise and f is not
> >>> > >> holomorphic then the derivatives of f_n cannot remain
> >>> > >> uniformly bounded on compact sets. The f_n themselves
> >>> > >> cannot remain uniformly bounded on compact sets
> >>> > >> either; this is easy to see from Montel's theorem.
>
> >>> > >Hmm. So what would, say, a graph, of such a hypothetical
> >>> > >function sequence, converging to the pillow function, look like?
>
> >>> > I don't know that there is such a sequence (I doubt it; I
> >>> > suspect that any pointwise limit of analytic functions
>
> >>> What is "a lot of points", anyway? What criterion are you using to
> >>> determine how "many" points there are, anyway?
>
> >>If a sequence of holomorphic functions converges pointwise on some
> >>open set D, then the limit is holomorphic on a dense open subset of D.
>
> >That's amazing, like you're chanelling Ullrich. (I didn't want to
> >state anything explcitly because I didn't have time to think about
> >it or to try to look it up...)
>
> It's easier to prove something when you know it's true:
>
> Say (f_n) is a pointwise convergent sequence of functions
> holomorphic in D. Let E_N be the set of points
> where sup |f_n| <= N. Then E_N is closed and the
> union of the E_N is D, so some E_N contains a disk.
>
> Say a disk is "good" if (f_n) is uniformly bounded on said disk.
> The above argument, specialized to some disk in D, shows that
> D' is dense in D, where D' is the union of all the good disks
> in D. The limit is holomorphic in D'.
>
> (The other day I was being stupid, not realizing I was
> done at this point, wanting to get uniform convergence
> on compact subsets of D' and not realizing that that's
> trivial: If K is a compact subset of D' then K is covered
> by finitely many good disks, hence (f_n) is uniformly
> bounded on K.)
>
So then is there any way to express the degeneration of the
complex analysis at all, and to watch it break down?
David C. Ullrich
wrote:
>On Dec 3, 8:17 am, David C. Ullrich <dullr...@sprynet.com> wrote:
I have no idea what that question means.
mike3
> On Tue, 23 Dec 2008 00:57:42 -0800 (PST), mike3 <mike4...@yahoo.com>
To see what happens as you get closer and closer to where it
no longer applies and degenerates to vector analysis and real
analysis. E.g. a nonconstant function where the real/imag. parts
vary independently of each other, such as sin(x) + i cos(y),
where x and y are the real/imag. parts respectively of the complex
number put in. But apparently you can't do that, it seems...
David C. Ullrich
wrote:
>On Dec 23, 5:16 am, David C. Ullrich <dullr...@sprynet.com> wrote:
There are any number of things you could be actually asking
here. I already gave the best answer I could think of, weeks
ago, right here in this thread.
>But apparently you can't do that, it seems...
Good strategy, guy. Not happy about the lack of answers to
incoherent questions, try to intimidate people into reading
your mind. Good luck with that, I'm certain it's going to
be very effective.
Ioannis
I don't think he means that in a demeaning way ("...you can't do that,"). My
parser detects a syntax error. He probably means "But apparently _one_ cannot do
that", in the genitive sense, as "one, me included", like "it cannot be done".
You seem to have interpreted it as a personal attack because of the "you".
If that's not the case my parser has a bug.
> David C. Ullrich
--
Ioannis
David C. Ullrich
wrote:
Hmm, you could be right. Glad you pointed that out.
>You seem to have interpreted it as a personal attack because of the "you".
>
>If that's not the case my parser has a bug.
>
>> David C. Ullrich
David C. Ullrich
amy666
its been a long time since i last used those.
if i recall correctly , subharmonics are functions where the closed contour is =< 0
and superharmonic are those where the closed contour gives >= 0.
( thus both superharmoic and subharmonic = holomorphic )
is that correct or am i confused ?
> --
> Robert Israel
> isr...@math.MyUniversitysInitials.ca
> Department of Mathematics
> http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver,
> BC, Canada
regards
tommy1729
amy666
> On Dec 1, 8:05 am, David C. Ullrich
> <dullr...@sprynet.com> wrote:
> > Your "approaches" as defined below is simply known
> > as "pointwise convergence".
> >
> > Yes, if f_n is holomorphic, f_n -> f pointwise and
> f is not
> > holomorphic then the derivatives of f_n cannot
> remain
> > uniformly bounded on compact sets. The f_n
> themselves
> > cannot remain uniformly bounded on compact sets
> > either; this is easy to see from Montel's theorem.
> >
>
> Hmm. So what would, say, a graph, of such a
> hypothetical
> function sequence, converging to the pillow function,
> look like?
> And would the derivatives "explode more" (get
> bigger/more wild
> values) in the places the functions more closely
> resemble the
> function?
no it would explode less.
call me a fan-boy , but this relates to continu iterations.
for entire functions a closed contour integral = 0.
let G(x,y) be the y'th iterate of T(x) and G(x,oo) = F(x)
let G(x,y) be Coo for fixed y or fixed x.
let F(x) be holomorphic and let G(x,y) approach 0* for rising y ( * the 0 of " closed contour integral = 0 )
and 0** for rising y too. ( ** the 0 of the riemann cauchy equation )
something like that will do.
regards
tommy1729
David C. Ullrich
wrote:
>mike3 wrote :
>
>> On Dec 1, 8:05 am, David C. Ullrich
>> <dullr...@sprynet.com> wrote:
>> > Your "approaches" as defined below is simply known
>> > as "pointwise convergence".
>> >
>> > Yes, if f_n is holomorphic, f_n -> f pointwise and
>> f is not
>> > holomorphic then the derivatives of f_n cannot
>> remain
>> > uniformly bounded on compact sets. The f_n
>> themselves
>> > cannot remain uniformly bounded on compact sets
>> > either; this is easy to see from Montel's theorem.
>> >
>>
>> Hmm. So what would, say, a graph, of such a
>> hypothetical
>> function sequence, converging to the pillow function,
>> look like?
>> And would the derivatives "explode more" (get
>> bigger/more wild
>> values) in the places the functions more closely
>> resemble the
>> function?
>
>no it would explode less.
You can't even read. There _is_ no sequence of
analytic functions converging to that function
(I conjectued this was so, Israel gave something
that would be either a proof or a hint depending
on the reader, and I wrote it out as a complete
proof, right here in this thread).
Since there is no such sequence, talking about how
its derivatives behave doesn't make much sense.
David C. Ullrich
wrote:
You're confused.
Why bother asking?
>
>> --
>> Robert Israel
>> isr...@math.MyUniversitysInitials.ca
>> Department of Mathematics
>> http://www.math.ubc.ca/~israel
>> University of British Columbia Vancouver,
>> BC, Canada
>
>regards
>
>tommy1729
David C. Ullrich
Robert Israel
> Robert wrote :
> > Also of interest are subharmonic functions and
> > superharmonic
> > functions.
>
> its been a long time since i last used those.
>
> if i recall correctly , subharmonics are functions where the closed contour
> is =< 0
>
> and superharmonic are those where the closed contour gives >= 0.
>
> ( thus both superharmoic and subharmonic = holomorphic )
>
> is that correct or am i confused ?
Rather than be confused, why can't you look it up?
For example: <http://en.wikipedia.org/wiki/Subharmonic_function>
and <http://eom.springer.de/S/s090900.htm>
amy666
> amy666 <tomm...@hotmail.com> writes:
>
> > Robert wrote :
>
> > > Also of interest are subharmonic functions and
> > > superharmonic
> > > functions.
> >
> > its been a long time since i last used those.
> >
> > if i recall correctly , subharmonics are functions
> where the closed contour
> > is =< 0
> >
> > and superharmonic are those where the closed
> contour gives >= 0.
> >
> > ( thus both superharmoic and subharmonic =
> holomorphic )
> >
> > is that correct or am i confused ?
>
> Rather than be confused, why can't you look it up?
free information on the internet is not always reliable.
ive already wasted time being confused about statements that were actually wrong on the internet.
and then i posted a 'disproof' of a ' proof ' whereas it was just a mistake on the internet.
>
> For example:
> <http://en.wikipedia.org/wiki/Subharmonic_function>
> and <http://eom.springer.de/S/s090900.htm>
> --
> Robert Israel
> isr...@math.MyUniversitysInitials.ca
> Department of Mathematics
> http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver,
> BC, Canada
regards
tommy1729
amy666
assuming at oo values reach limit = 0.
forgot that.
> >
> >is that correct or am i confused ?
>
> You're confused.
>
> Why bother asking?
why bother answering me ?
you always blaim me for not willing to learn.
but your replies are like ;
you cant read
your wrong as always
etc
but you dont teach or explain anything.
even if i am willing to learn , you dont post math , you just insult me.
i find it hard to believe your a teacher ...
why should you bother to post and NOT learn me anything , and why should i bother reading your insults without the math ?
besides i asked Robert Israel.
>
> >
> >> --
> >> Robert Israel
> >> isr...@math.MyUniversitysInitials.ca
> >> Department of Mathematics
> >> http://www.math.ubc.ca/~israel
> >> University of British Columbia
> Vancouver,
> >> BC, Canada
> >
> >regards
> >
> >tommy1729
>
> David C. Ullrich
>
> "Understanding Godel isn't about following his formal
> proof.
> That would make a mockery of everything Godel was up
> to."
> (John Jones, "My talk about Godel to the post-grads."
> in sci.logic.)
tommy1729
David C. Ullrich
wrote:
No, that has nothing to do with it. What you said
is simply wrong. You're confusing one thing with
another.
>> >
>> >is that correct or am i confused ?
>>
>> You're confused.
>>
>> Why bother asking?
>
>why bother answering me ?
Just for fun.
>
>you always blaim me for not willing to learn.
>
>
>but your replies are like ;
>
>you cant read
>
>your wrong as always
>
>etc
>
>but you dont teach or explain anything.
I explain lots of things to lots of people.
>even if i am willing to learn , you dont post math , you just insult me.
>
>i find it hard to believe your a teacher ...
>
>why should you bother to post and NOT learn me anything , and
>why should i bother reading your insults without the math ?
Who said you should?
David C. Ullrich
wrote:
>Robert wrote :
>
>> amy666 <tomm...@hotmail.com> writes:
>>
>> > Robert wrote :
>>
>> > > Also of interest are subharmonic functions and
>> > > superharmonic
>> > > functions.
>> >
>> > its been a long time since i last used those.
>> >
>> > if i recall correctly , subharmonics are functions
>> where the closed contour
>> > is =< 0
>> >
>> > and superharmonic are those where the closed
>> contour gives >= 0.
>> >
>> > ( thus both superharmoic and subharmonic =
>> holomorphic )
>> >
>> > is that correct or am i confused ?
>>
>> Rather than be confused, why can't you look it up?
>
>free information on the internet is not always reliable.
What's an example of an error in basic math on Wikipedia?
Or on the Springer site that he cited?
>ive already wasted time being confused about statements that were actually wrong on the internet.
>
>and then i posted a 'disproof' of a ' proof ' whereas it was just a mistake on the internet.
>
>
>>
>> For example:
>> <http://en.wikipedia.org/wiki/Subharmonic_function>
>> and <http://eom.springer.de/S/s090900.htm>
>> --
>> Robert Israel
>> isr...@math.MyUniversitysInitials.ca
>> Department of Mathematics
>> http://www.math.ubc.ca/~israel
>> University of British Columbia Vancouver,
>> BC, Canada
>
>regards
>
>tommy1729
David C. Ullrich
Jesse F. Hughes
> Robert wrote :
>>
>> Rather than be confused, why can't you look it up?
>
> free information on the internet is not always reliable.
Right. So rather than depend on free information on the internet, you
seek answers on Usenet.
It's good that you no longer depend on free internet from the
internet.
--
"Puts his arm around you, fiddles with your hair. You know, and he says, come
on, you know, just because you like a bit of a kiss and a cuddle with another
man doesn't make you gay. Which, you know, I've thought a lot about. But I
think it does. I think it does." --- The Office (interviews)
** Posted from http://www.teranews.com **
amy666
yeah , i was wrong ...
but then what is the name of what i described ?
regards
tommy1729
David C. Ullrich
wrote:
If you ask again after not being obnoxious for one
month I'll tell you.
Or you could just look it up. I mean you don't even
have to search for it, you've been _given_ links...
amy666
which links ?
this is the first reply to that question and i dont see links , how could there be earlier links ?
David C. Ullrich
wrote:
You can't be bothered to learn to use Google or Wikipedia.
When someone goes to the trouble of looking up something
_for_ you and telling you where to find it you don't even
notice. Now weeks later you want me to go through the
thread for you finding the reply where those links were
given?
Good luck with that.
spudnik
for cropcircles, other than "Doug and Dave with 2by9s,"
the obviously fake Druids.
> You can't be bothered to learn to use Google or Wikipedia.
> >> >> >> That would make a mockery of everything Godel
amy666
i think your hallucinating.
really.
sure ive seen links , but not to that.
David Bernier
Is this what you described?
(a) functions where the closed contour is =< 0
and
(b) those where the closed contour gives >= 0
A contour integral gives a complex number, such as 5 - 2pi*i .
There is no natural ordering "<" for the complex numbers.
Are you interested in contour integrals which are real-valued?
David Bernier
Ref.:
[amy:]
its been a long time since i last used those.
if i recall correctly , subharmonics are functions where the closed contour is =< 0
and superharmonic are those where the closed contour gives >= 0.
( thus both superharmoic and subharmonic = holomorphic )
is that correct or am i confused ?
[amy, over.]