Bad complex functions
Han de Bruijn
is not the same as a "good" complex function of _one_ complex variable.
An example. Let f(z) = z^2 = (x^2 - y^2) + i.(2.x.y) ; replace herein
(2.x.y) by (3.x.y) , giving g(z) = z^2 + i.Re(z).Im(z) . We suppose
g(z) is a kind of function not very worthwile to consider; maybe call
it a "bad" function for that reason.
Corollary. There are many more bad functions than good functions, i.e.
complex valued functions of two real variables are much more likely to
occur than complex valued functions of one complex variable. Functions
of the form f(z) are far more restrictive than their counterparts of
the form f(x,y) , where f is still complex. I think this is part of
an EXPLANATION why the "good" Complex stuff can be Made Simple indeed.
Han de Bruijn
Rotwang
> Conjecture. A complex valued function of two real variables in general
> is not the same as a "good" complex function of _one_ complex variable.
>
> An example. Let f(z) = z^2 = (x^2 - y^2) + i.(2.x.y) ; replace herein
> (2.x.y) by (3.x.y) , giving g(z) = z^2 + i.Re(z).Im(z) . We suppose
> g(z) is a kind of function not very worthwile to consider; maybe call
> it a "bad" function for that reason.
From the above example, it appears that by "good" complex function you
mean a function which is holomorphic. If so then your conjecture is
obviously true, in that it's easy to construct examples of functions
of a complex variable which are not holomorphic, indeed you have given
one above. Note that, according to the modern use of the word
"function", there isn't really any distinction between a function of a
single complex variable and a function of two real variables; since
each complex number z can be uniquely written as x + i*y with x, y
real, given a function f(x,y) of two real variables we can simply
define f(z) = f(x,y).
(Intuitively one often thinks of holomorphic functions as those that
depend only on z and not on x and y individually, or alternatively as
those that depend on z but not its complex conjugate zbar - taken
literally this is nonsense, but if you look in Appendix 7 of Complex
Made Simple you'll see that there is a sense in which this idea can be
made precise).
> Corollary. There are many more bad functions than good functions, i.e.
> complex valued functions of two real variables are much more likely to
> occur than complex valued functions of one complex variable.
If by "many more" you mean that the set of "good" (i.e. holomorphic)
functions has greater cardinality than the set of "bad" functions,
then this depends on how general you allow your bad functions to be.
If you demand that they are continuous then I /think/ that the two
sets have the same cardinality, for example. As for "more likely to
occur", this is meaningless unless you specify how your functions are
being chosen. Different kinds of functions will arise in different
contexts. It's certainly the case that holomorphic functions occur
often enough in e.g. physics that complex analysis turns out to have
many real-world applications.
> Functions
> of the form f(z) are far more restrictive than their counterparts of
> the form f(x,y) , where f is still complex. I think this is part of
> an EXPLANATION why the "good" Complex stuff can be Made Simple indeed.
Yes, of course it's true that if you relax the condition that the
functions you're dealing with are holomorphic then many simplifying
results no longer apply. But that doesn't make it any less remarkable
that such a simple condition, that of being complex-differentiable on
an open set, should have so much power over how a function can behave;
consider that the analogous property of real-valued functions, that of
being differentiable on an open set, is not nearly as restrictive -
there are differentiable functions which are not continuously
differentiable, infinitely differentiable functions which are not
analytic, and so on. So the fact that holomorphic functions have such
nice properties is really surprising, to me at least.
Han de Bruijn
> On 2 Mar, 10:51, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>
>>Conjecture. A complex valued function of two real variables in general
>>is not the same as a "good" complex function of _one_ complex variable.
>>
>>An example. Let f(z) = z^2 = (x^2 - y^2) + i.(2.x.y) ; replace herein
>>(2.x.y) by (3.x.y) , giving g(z) = z^2 + i.Re(z).Im(z) . We suppose
>>g(z) is a kind of function not very worthwile to consider; maybe call
>>it a "bad" function for that reason.
>
> From the above example, it appears that by "good" complex function you
> mean a function which is holomorphic.
Not exactly, The function needs not to be differentiable per se. Though
I don't know (yet) if it adds much if I say this. Sure differentiable,
as such, is MUCH more of another restriction. Especially if it is taken
in the complex sense: from whatever direction h you're approaching z in
[f(z+h) - f(z)]/h , the result is always the _same_ f'(z) . Compare the
infinity of directions in the complex plane with only two directions on
the _real_ axis and it becomes obvious that _complex_ differentiation is
much more demanding than the alike for real valued functions. It's not a
coincidence that holomorphic functions correspond to potential functions
in the plane; if f = u + i.v , then u,v are the solutions of Laplace
equations: d^2u/dx^2 + d^2u/dy^2 = 0 ; d^2v/dx^2 + d^2v/dy^2 = 0 .
If I remember well ..
Sure. But I want to know where the surprises come from.
Han de Bruijn
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
>Rotwang wrote:
>
>> On 2 Mar, 10:51, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>>
>>>Conjecture. A complex valued function of two real variables in general
>>>is not the same as a "good" complex function of _one_ complex variable.
>>>
>>>An example. Let f(z) = z^2 = (x^2 - y^2) + i.(2.x.y) ; replace herein
>>>(2.x.y) by (3.x.y) , giving g(z) = z^2 + i.Re(z).Im(z) . We suppose
>>>g(z) is a kind of function not very worthwile to consider; maybe call
>>>it a "bad" function for that reason.
>>
>> From the above example, it appears that by "good" complex function you
>> mean a function which is holomorphic.
>
>Not exactly, The function needs not to be differentiable per se. Though
>I don't know (yet) if it adds much if I say this. Sure differentiable,
>as such, is MUCH more of another restriction. Especially if it is taken
>in the complex sense: from whatever direction h you're approaching z in
>[f(z+h) - f(z)]/h , the result is always the _same_ f'(z) .
Complex-differentiable is the same as holomorphic. This is what
he and I assumed you meant by "good" - if you meant something
else you need to give at least a hint what you _do_ mean before
your question means anything.
Below you say something about the disctinction between
complex valued f(x,y) and complex-valued f(z). There
is literally no difference whatever between the two -
any function of x and y is a function of z, and conversely.
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.)
Hero
> Han wrote:
> > Rotwang wrote:
>>> Han wrote:
>
> >>>Conjecture. A complex valued function of two real variables in general
> >>>is not the same as a "good" complex function of _one_ complex variable.
>
>
> >> From the above example, it appears that by "good" complex function you
> >> mean a function which is holomorphic.
>
>
> Complex-differentiable is the same as holomorphic. This is what
> he and I assumed you meant by "good" - if you meant something
> else you need to give at least a hint what you _do_ mean before
> your question means anything.
>
> Below you say something about the disctinction between
> complex valued f(x,y) and complex-valued f(z). There
> is literally no difference whatever between the two -
> any function of x and y is a function of z, and conversely.
>
When You construct a function by joining constants, +, - ,* , / and z
into a term,
like f( z ) =( 3, 1 ) * z - pi / ( z * z ),
then this is of course a function of x and y, when given z = ( x, y ).
Now the other way round:construct a function by joining constants, +,
- ,* , / and x and y into a term,
like f ( x, y ) = ( x * x - y + y , 2 * x * y ) + 3 * ( x , 0 ).
How can You write this as a function of just "z", of an unseparated
pair ( x , y ), without using x or y on their own?
And this is connected of whether or not a function is holomorphic.
Actually holo means something like being "one", and morph refers to
"form".
With friendly greetings
Hero
David C. Ullrich
I said _function_, not "function constructed from constants, +, etc".
Any function of x and y is a function of z, since x = Re(z) and
y = Im(z). For example x * x - y is a function of z, because
x * x - y = Re(z) * Re(z) = Im(z).
>And this is connected of whether or not a function is holomorphic.
It certainly is. Which is exactly why people assumed he meant
holomorphic when he said "good". So far he's said no, that's
not what he meant, and given us no clue what he did mean.
In fact "function of z" and "function of x,y" mean the same
thing. But the things that you are saying I cannot write as
functions of z are not holomorphic, and that's not just a
coincidence: The reason for the connection is that all
the things you're allowing to build functions (addition,
etc) are holomorphic functions of two variables.
>Actually holo means something like being "one", and morph refers to
>"form".
>
>With friendly greetings
>Hero
>
David C. Ullrich
Han de Bruijn
The trivial way to do it is, of course, via x = Re(z) and y = Im(z) .
> And this is connected of whether or not a function is holomorphic.
> Actually holo means something like being "one", and morph refers to
> "form".
Han de Bruijn
Han de Bruijn
True. I'm still in search of a definition for covering the intuition.
But perhaps holomorphic just fits the bill ..
> In fact "function of z" and "function of x,y" mean the same
> thing. But the things that you are saying I cannot write as
> functions of z are not holomorphic, and that's not just a
> coincidence: The reason for the connection is that all
> the things you're allowing to build functions (addition,
> etc) are holomorphic functions of two variables.
>
>>Actually holo means something like being "one", and morph refers to
>>"form".
Han de Bruijn
Han de Bruijn
> Rotwang wrote:
>
>> On 2 Mar, 10:51, Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
>>
>>> Conjecture. A complex valued function of two real variables in general
>>> is not the same as a "good" complex function of _one_ complex variable.
>>>
>>> An example. Let f(z) = z^2 = (x^2 - y^2) + i.(2.x.y) ; replace herein
>>> (2.x.y) by (3.x.y) , giving g(z) = z^2 + i.Re(z).Im(z) . We suppose
>>> g(z) is a kind of function not very worthwile to consider; maybe call
>>> it a "bad" function for that reason.
>>
>> From the above example, it appears that by "good" complex function you
>> mean a function which is holomorphic.
>
> Not exactly, The function needs not to be differentiable per se. Though
> I don't know (yet) if it adds much if I say this. Sure differentiable,
> as such, is MUCH more of another restriction. Especially if it is taken
> in the complex sense: from whatever direction h you're approaching z in
> [f(z+h) - f(z)]/h , the result is always the _same_ f'(z) . Compare the
> infinity of directions in the complex plane with only two directions on
> the _real_ axis and it becomes obvious that _complex_ differentiation is
> much more demanding than the alike for real valued functions. It's not a
> coincidence that holomorphic functions correspond to potential functions
> in the plane; if f = u + i.v , then u,v are the solutions of Laplace
> equations: d^2u/dx^2 + d^2u/dy^2 = 0 ; d^2v/dx^2 + d^2v/dy^2 = 0 .
> If I remember well ..
Complex differentiation is the same as existence of the Cauchy-Riemann
(CR) equations, right? I've been close with the CR equations for quite
some time in the past. In (Computational) Fluid Dynamics, they stand
for a very special kind of fluid flow, namely Ideal Flow. An Ideal Flow
is a flow which is incompressible (common) AND irrotational (uncommon).
As follows, and the CR equations are easily recognized herein.
Incompressible: du/dx + dv/dy = 0
Irrotational : dv/dx - du/dy = 0 (d = partial)
http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft
http://hdebruijn.soo.dto.tudelft.nl/www/programs/plaatjes/slide01.htm
One would think that what engineers really want is a REAL flow and not
an Ideal Flow. But, unexpectedly perhaps, that's NOT what they really
want. With the apparatus I've been working on, what the engineers want
is the flow field, calculated in such a way that temperature stresses
cannot be worse in reality than they are in the calculations. We call
such calculations "conservative". An insightful moment of thinking has
revealed that not a realistic flow simulation but rather an Ideal Flow
simulation has the desired properties.
http://hdebruijn.soo.dto.tudelft.nl/www/grondig/veiliger.htm
It is known from theoretical fluid dynamics that the kinetic energy of
a flow field is minimal for Ideal Flow. It follows that the mean of the
temperature gradients, and therefore the mean temperature stresses, are
maximally, hence conservatively, calculated with Ideal Flow.
We conclude that, in this case, what's good and ideally in mathematics,
is also good and ideally for engineers !
Han de Bruijn
David C. Ullrich
There's no way anyone can tell what intuition you're talking
about. But you said that (x^2 - y^2) + i.(2.x.y) is good
and (x^2 - y^2) + i.(3.x.y) is bad - nobody has any idea
why that would be except that the first is holomorphic
and the second is not.
>> In fact "function of z" and "function of x,y" mean the same
>> thing. But the things that you are saying I cannot write as
>> functions of z are not holomorphic, and that's not just a
>> coincidence: The reason for the connection is that all
>> the things you're allowing to build functions (addition,
>> etc) are holomorphic functions of two variables.
>>
>>>Actually holo means something like being "one", and morph refers to
>>>"form".
>
>Han de Bruijn
David C. Ullrich
David C. Ullrich
Complex _differentiability_ (in an open set) is equivalent to the
_validity_ of the CR equations in that open set, yes.
You haven't read Chapter 0 yet, I gather...
David C. Ullrich
Han de Bruijn
Sure. You're always the pal who formulates better than I do.
> You haven't read Chapter 0 yet, I gather...
I have landed in the Chapter where triangles are employed for proving a
special case of Cauchy's integral theorem. Found a similar proof in an-
other book, though, that works with rectangles. Here are the triangles:
http://facweb.stvincent.edu/academics/mathematics/CIT/CIT.htm
But, what objections do you have against this one (and is it somewhere
in your book as well ?):
http://mathworld.wolfram.com/CauchyIntegralTheorem.html
paulde...@att.net
...
> Complex _differentiability_ (in an open set) is equivalent to the
> _validity_ of the CR equations in that open set, yes.
When I was at grad school many many years ago, this was not a widely
known fact among my peers. Most grad students assumed that you needed
continuity of the partial derivatives to guarantee differentiability.
But you don't need that -- the Looman-Menchoff theorem.
David Ullrich knows this for sure, and this is a tangent (no
mathematical meaning to tangent intended), but perhaps interesting for
some readers.
Paul Epstein
Hero
Regarding functions more general, like Your
Im : R x R --> R: z --> y,
is leading away from the speciality of C, which is a special
R x R - vectorspace over R.
Just like the set of matrices
( a - b
b a )
with the common operations is a somehow special structure
in the space of 2 x 2 matrices.
Using only termdefined functions, and the terms only build from
a set of elements and variables in these sets and from the operations
of a mathematical structure, highlights the differences between
a common R x R -vectorspace, C with an extra multiplication and
thirdly R x R with a metric.
Introducing the seperating functions Re and Im, or introducing
the common metric, or introducing the conjugate paves the way
for defining the dot-product with the now given terms.
> >And this is connected of whether or not a function is holomorphic.
>
> It certainly is. Which is exactly why people assumed he meant
> holomorphic when he said "good". So far he's said no, that's
> not what he meant, and given us no clue what he did mean.
> In fact "function of z" and "function of x,y" mean the same
> thing. But the things that you are saying I cannot write as
> functions of z are not holomorphic, and that's not just a
> coincidence: The reason for the connection is that all
> the things you're allowing to build functions (addition,
> etc) are holomorphic functions of two variables.
So let's now see, what C is in geometrical representation,
as Han is interested in fluid flow.
z = ( x, y ), w = ( u, v ) can represent points in a plane in
cartesian coordinates. z can act as a translation, when acting
by the operation of addition onto another element w.
And it can act as a rotation onto another element by
multiplying the two with this C-special multiplication.
These simple movements in a plane can all be generated by reflections.
Reflections are absent in pure C. Introduce the definition of
the conjugate and there we already have one possible reflection.
That's by the way the reason, we have in C only one
orientation of rotation, left is preferred by mathematicians.
So there is another C, which is right-turning and
term-defined fuctions, which replace z by conjugate z
are called anti-holomorphic.
With friendly greetings
Hero
Dave L. Renfro
> When I was at grad school many many years ago, this was
> not a widely known fact among my peers. Most grad students
> assumed that you needed continuity of the partial derivatives
> to guarantee differentiability. But you don't need that -- the
> Looman-Menchoff theorem.
>
> David Ullrich knows this for sure, and this is a tangent
> (no mathematical meaning to tangent intended), but perhaps
> interesting for some readers.
Even more tangentially (in both ways, and the pun is intended),
but perhaps less interesting and more arcane, is the following
I posted in sci.math on 29 December 2004:
****************************************************************
KRamsay <kram...@aol.com>
[sci.math Dec 28 2004 5:35:03:000PM]
http://groups.google.com/group/sci.math/msg/920f939c44ab1844
wrote (in part):
> The most relevant theorem in the book appears to be the
> Looman-Menchoff theorem, which Carlo Teubner already
> alluded to: if Omega is an open set, f is continuous and
> has both partial derivatives on Omega, and f satisfies the
> Cauchy-Riemann equations on Omega, then f is holomorphic
> on Omega. At the expense of a little bit more delicate proof,
> this slightly strengthens the easier result that also assumes
> the partial derivatives of f are continuous. And now we see
> that the continuity of f can't be removed.
In fact, Looman and Menchoff [= Men'shov] actually showed
that we only need the partials to exist finitely on a
co-countable set. Sindalovskii [3] observed (top of p. 360)
that it follows from Robert M. Fesq (MR 30 #4896) and Paul
J. Cohen (MR 21 #3004) [yes, the Cohen who proved the axiom
of choice and continuum hypothesis independence results]
that we can strengthen this even further. Instead of the
partials existing finitely on all but a countable set,
it is enough to assume that the partials exist finitely
on all but a subset of a countable union of closed sets
each having finite Hausdorff 1-measure, or equivalently,
as long as the partials exist everywhere except on a set
that can be covered by an F_sigma set with sigma-finite
Hausdorff 1-measure. Incidentally, this is a strictly
stronger notion of smallness than being simultaneously
meager (i.e. first category) in R^2 and having sigma-finite
Hausdorff 1-measure -- see Remark 1 in Renfro [3], where
in (a) I neglected to include the hypothesis that the
set is also meager.
By the way, I know next to nothing about the theory
behind these results. I just happen to be interested
in various hierarchies of "small set notions". When
I come across a paper that gives a non-trivial use of
some notion of smallness of interest to me, I often
write a note to myself about it or file a copy of the
paper away somewhere. So, when I saw these posts about
Cauchy-Riemann weakenings, I remembered having a paper
on this topic that involved a Hausdorff measure version
of F_sigma measure zero sets, and hence, I knew it would
be in my notebooks that deal with subsets of F_sigma
measure zero sets.
[1] Shaun A. R. Disney, Jack D. Gray, and Sidney A. Morris,
"Is a function that satisfies the Cauchy-Riemann
equations necessarily analytic?", The Australian
Mathematical Society Gazette 2(3) (1975), 67-81.
[2] Jack D. Gray and Sidney A. Morris, "When is a function
that satisfies the Cauchy-Riemann equations analytic?",
The American Mathematical Monthly 85 (1978), 246-256.
[3] Dave L. Renfro, 1 May 2000 sci.math post "HISTORICAL
ESSAY ON F_SIGMA LEBESGUE NULL SETS".
http://groups.google.com/group/sci.math/msg/00473c4fb594d3d7
[4] G. Kh. Sindalovskii, "Cauchy-Riemann conditions
in a class of functions with summable modulus, and
certain boundary properties of analytic functions",
Math. USSR Sbornik 56 (1987), 359-377.
****************************************************************
Dave L. Renfro
David C. Ullrich
<Han.de...@DTO.TUDelft.NL> wrote:
Yes, you can also do it with rectangles. I'm not sure what the
"though" means - there's no claim anywhere that the way
I do things is the only way to do them.
>Here are the triangles:
>
>http://facweb.stvincent.edu/academics/mathematics/CIT/CIT.htm
Yes, that appears to be the same proof as in CMS. Again, it's not
at all clear to me what your point is.
>But, what objections do you have against this one (and is it somewhere
>in your book as well ?):
>
>http://mathworld.wolfram.com/CauchyIntegralTheorem.html
Comment before looking: The fact that something is not in
the book doesn't imply that I have any "objection" to it.
A book containing everything that I don't object to would
be a little long.
Second comment before looking: Mathworld? Yech.
Mathworld is full of errors.
Ok, I looked. First, the main theorem that's proved is only
a very special case of CT, for simply connected regions.
Theorem 4.10 in CMS is much more general.
Second, the proof there uses Green's Theorem. You can
find comments on proving CT via GT in various places
in CMS: _Proving_ GT in the generality required to
make the proof on Mathworld correct is much harder
than you think. In particular it's certainly _not_ done
in the typical advanced calculus course - in the typical
advanced calculus course either a much less general
version of GT is proved, with severe restrictions
on the shape of the contour, or a more general
version is stated but the proof is full of unjustified
"obvious" assertions about the geometry. The proofs
in CMS (and any other decent text at that level) are
actual _proofs_.
Then at the bottom of the page we find the "yech"
I expected to find: A more general version of CT
is stated. There are three problems there:
(i) the more general version is only stated, not proved.
(ii) the more general version is still much less general
than the (standard) general version of CT you find in
CMS (as well as in most graduate texts on complex
analysis).
(iii) the more general version stated in equation
(14) on the page is wrong. Simply nonsense.
Incorrect. Mathworld meant to be stating the
standard result that appears as Theorem 4.13
in CMS. But there's a crucial hypothesis missing
in the version on Mathworld.
And the missing hypothesis is not just a technicality
(like for example if something had to be differentiable
and Mathworld forgot to say that, _that_ might count
as just a technicality).
If you choose two curves C_1 and C_2 at random
then equation (14) is simply not true.
Except for that I have no objection to what's
on Mathworld.
David C. Ullrich
David C. Ullrich
Yes, I knew that (I think Narasimhan's book on one complex variable
is where I saw this years ago). And yes, it's certainly an interesting
comment.
But in case you're curious, in the context of this thread that's not
the result I was referring to above - I was referring to the result
_assuming_ the partials are continuous. In the context of a
discussion with Han functions _are_ continuous, by definition.
I felt I was already being picky enough adding the words
"in an open set"...
As long as we've wandered from all-functions-are-continuous
land into math land, imo the _right_ way to think of the
CR equations is as in Chapter 0 of my book. You know
that there's no condition on the partials _at a single point_
that's equivalent to complex differentiability _at that point_.
But there _is_ a pointwise result - I think it's the right
way to look at it _because_ it holds even at a single point
and also because imo it clarifies what the CR equations
really mean:
Let's take "the point" to be the origin to simplify typing.
Recall that if f maps a neighborhood of the origin in R^2
to R^2 then f is _differentiable_ ("Frechet differentiable",
or "real differentiable" in the book) at the orgin if there
exists a linear map T: R^2 -> R^2 such that
f(x,y) = f(0,0) + T(x,y) + E(x,y),
where the error term E(x,y) satisifes
E(x.y) / ||(x,y)|| -> 0 as (x,y) -> (0,0).
Here when I say T is linear I mean R-linear.
It's a fact that f : C -> C is complex-differentiable
at a point if and only if it's real-differentiable
at that point and the "real derivative" T is
_complex_-linear.
Now the 2x2 matrix A representing the "real
derivative" T is composed of partials of the
real and imaginary parts of f, and the
condition that T be complex-linear turns
out to be just the CR equations. So:
f is complex-differentiable at a point if and
only if it is real-differentiable at that point
and satisfies the CR-equations at that point.
(So that's what the CR equations mean to me:
that the real derivative is complex-linear.)
>Paul Epstein
amy666
a while ago , someone asked if i could give an example of bad math online.
( during a debate about learning from the internet )
i think ullrich gives a good example here.
regards
tommy1729