Analytic functions and identity theorem

Matt N.

Mar 12, 2012 19:14

I think it's about time I learned the difference between analytic and holomorphic function.

My memory has stored them on the same neuron. : /

Ooh. Complex analytic functions are called holomorphic. And analytic means the Taylor series of the function converges on a neighbourhood of every point in the domain. Heh.

And what was that with Laurent series?

I'll ignore that for now.

Ello : )

t.b.

'Ello!

Matt N.

I was really surprised this morning when a comment of yours appeared after I edited an answer.

t.b.

@MattN The hard thing to prove is of course that a complex analytic function is the same thing as a function satisfying the Cauchy-Riemann equations.

Matt N.

Didn't expect to see that much of you this week : )

t.b.

@MattN yeah, it was pretty simultaneous :)

(I was spamming Brian's answers with comments this morning...)

Matt N.

Mar 12, 2012 19:25

I only saw one : )

Asaf Karagila

The seminar went fine.

t.b.

The only ones remaining are here.

Asaf Karagila

I only managed to do a part of what I planned. I'll continue next week.

What's up?

t.b.

@MattN that's what you get when you develop a meromorphic function in the annular region between two poles.

@AsafKaragila nice to hear. Did you prove that all sets have the BP?

Asaf Karagila

No. I only proved and stated the things I needed for the construction. I didn't even begin with the actual construction with the inaccessible.

Matt N.

Mar 12, 2012 19:28

@tb I see. That question is on my want-to-read list (read: favourites), postponed for when I have time.

@AsafKaragila Congratulations on kicking some Israeli ass!

Daniil

Is anyone here familiar with counter-free Buchi automata?

(or counter-finite DFA for that matter)

Matt N.

I'm still looking up what the Cauchy Riemann equations are.

Asaf Karagila

@MattN The determinant of the Jacobian.

t.b.

@AsafKaragila ???

Asaf Karagila

I think.

Matt N.

Mar 12, 2012 19:29

LOL : D

Asaf Karagila

I remember that it's like the difference between the partial derivatives (real part/imaginary part)...

t.b.

@MattN those expressing that the differential of a function $f: G \to \mathbb{C}$ is complex linear (when seen as a function $f: G \subset \mathbb{R}^2 \to \mathbb{R}^2$) as opposed to only real linear.

Asaf Karagila

Something which looks very much like the determinant of the differential or something like that.

t.b.

I don't know. $\frac{1}{2}\left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right)f = \overline{\partial}f = 0$ doesn't look particularly like a determinant to me (often people look at real and imaginary parts of this separately)

Matt N.

@tb I'm not sure what you are saying but according to Wikipedia $f : \mathbb{C} \to \mathbb{C}$ is differentiable at $x + iy$ iff $f$ satisfies these equations at $x + iy$.

Asaf Karagila

Mar 12, 2012 19:35

Well, the last time I saw the CR equations was almost three years ago! :-)

Excuse me for trying to be a part of the conversation! :-)

I'll be back shortly, anyway.

t.b.

@MattN yes, that's complex differentiable. But to define it you look at the real derivative and ask when it is a complex linear function, that is: whether it is of the form $\begin{pmatrix} a& -b \\ b & a\end{pmatrix}$.

Matt N.

@tb For that I'd have to look up meromorphic but I'll ignore that for now. Properly doing some complex analysis is scheduled for after handing in the BSc's project. If that is going to happen at all.

t.b.

(you want the differential to commute with the linear map given by multiplication by $i$).

Daniil

Is there something like Caley theorem but for monoids?

t.b.

@MattN A meromorphic function is essentially a quotient of two holomorphic maps. (since you're learning commutative algebra: an element of the field of fractions of the (local) ring of holomorphic functions)

Matt N.

Mar 12, 2012 19:40

@tb For $f: C \to C$ , $f(z) = z^2$ you define $\frac{\partial}{\partial z}$ where $z = x + iy$ by $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, is that what you're saying?

@tb Oh yes, that rings a bell. I have heard that before... somewhere. : )

Complex analysis is a slippery slope.

All I wanted to look up is "analytic" and now we're talking about Cauchy Riemann equations and whatnots. : )

t.b.

@MattN yes, I look at $f$ as a function of the real and imaginary part and get a function from a subset of $\mathbb{R}^2$ to $\mathbb{R}^2$. Then I differentiate it in the usual way and ask whether the Jacobian is a complex linear map.

Matt N.

Ok. Just wanted to make sure I understood.

t.b.

@Daniil something like this?

Matt N.

I have a question:

Daniil

@tb perfect, thanks!

Matt N.

Mar 12, 2012 19:47

What do analytic functions look like? In particular: why can't an analytic function not vanish on a bounded set?

@AsafKaragila Did Hardy remove his answer in response to your comment? (or was it a different question?)

t.b.

@MattN no, he deleted it because he thought he was on the right track...

(no joke)

Matt N.

I can't follow. On the right track = his answer was correct?

t.b.

@MattN look at the open mapping theorem and the identity theorem for holomorphic functions.

Matt N.

@tb Lovely, thank you.

t.b.

@MattN It was this question and when he deleted his answer he left a comment saying "Sorry about multiple edits; I think I've got it where I want it now."

Oh, @TheChaz is in the house! How was the wedding?

Matt N.

Mar 12, 2012 19:55

@tb Heh : )

The Chaz

Wonderful. I might need a week to recover from the festivities...

t.b.

Sounds like everything was like it should be... :)

The Chaz

Oh yeah. She said "yes", so that's good! A couple 4:00am nights has worn me out though... Let me know if this picture of me and my son is visible, would you?

Asaf Karagila

@MattN What Theo said.

The Chaz

(@t.b.)

t.b.

Mar 12, 2012 20:02

@TheChaz Yes, very nice! How old is he, about 4-5 years old?

The Chaz

Just turned 3 in February, actually :)

t.b.

Oh, really? I'm always very bad at estimating...

The Chaz

With his new haircut he looks a lot older

t.b.

What on earth do people learn in US undergrad institutions? I would say most of the math you learn here is far less than 200 years old.

Some of the ideas date back further, but the rigor needed is not something that should be classified as "modern notation only", I believe.

Matt N.

He says it's not 100% accurate : )

I'm not sure what I need the identity theorem for. If I have $f: \Omega \to \mathbb{R}$ analytic and for $B \subset \Omega \subset \mathbb{R}$ bounded, $f\mid_B = 0$ then I can have two cases:
If $B$ is open then I get a contradiction since $f(B) = \{0\}$ is closed.
If $B$ is closed then I take an open subset $O$ and do the same for $O$.

Asaf Karagila

Mar 12, 2012 20:12

Is this $\Omega$ the set of countable ordinals?

t.b.

@AsafKaragila it's an uppercase Omega, dummy!

Asaf Karagila

@tb Which is the notation Matt used in the question about maps from $\omega_1$ into $\mathbb R$.

Matt N.

@AsafKaragila Of course, did you not know that the countable ordinals are a subset of the reals?

Asaf Karagila

@MattN Actually, I did know that... you were the one who didn't know. :-)

Matt N.

@tb I think that's common notation for the set of countable ordinals. At least, I copied it off somewhere else : )

t.b.

Mar 12, 2012 20:14

@MattN uppercase?

Never seen that.

Matt N.

@tb Yes since lower case is the natural numbers.

t.b.

Every sane person on this planet uses $\omega_1$ for that...

Matt N.

@AsafKaragila I'm not sure whether I did or not. You are confusing : )

Asaf Karagila

I saw a lesser spotted eagle today on the way to Jerusalem. Not to mention the infinite number of kites and storks passing through Israel on their way to Europe.

t.b.

@MattN What exactly are you assuming about your $B$ here?

Matt N.

Mar 12, 2012 20:17

@tb That it's a bounded subset of an open subset of the reals.

t.b.

But what if $B$ is a finite set?

Asaf Karagila

Is it me or MSE and MO both stopped working?

Matt N.

@tb Then it's closed?

t.b.

@AsafKaragila It's probably you. Both work fine for me.

The Chaz

@AsafKaragila In what sense? They are up for me.

t.b.

Mar 12, 2012 20:18

@MattN but how does it contain an open subset?

Matt N.

But of course it's connected as well otherwise I can't apply the open mapping theorem! Sorry for not stating my assumptions properly.

t.b.

You need it to be connected and contain at least two points.

Matt N.

@AsafKaragila Same here. It must be you. : )

Asaf Karagila

$B$ is connected, but not simply connected. Its fundamental group is the free groups with two generators.

Ah, now it works again.

Matt N.

@tb Yes.

t.b.

Mar 12, 2012 20:21

But still you might not have an open subset a priori, so you need the identity theorem for that case.

Matt N.

I don't see how a non-empty, connected subset of $\mathbb{R}$ that contains more than one point can't have an open, non-empty subset T_T

Asaf Karagila

$\{0\}$?

t.b.

But we're talking about subsets of the complex numbers.

Asaf Karagila

$\mathbb R$?

Matt N.

@tb No, I am thinking about functions on the reals.

t.b.

Mar 12, 2012 20:23

@MattN Real holomorphic functions? They tend to be constant.

Matt N.

@tb No, I never said holomorphic! I said analytic. : )

t.b.

After half an hour of inquiring about complex analytic functions?

Matt N.

Yes. And I was only inquiring about complex functions because you gave me a lot of information about them that I needed to process and make sure I understood.

t.b.

But then you need the identity theorem because the open mapping theorem is blatantly false for real analytic functions.

Think of the sine for example.

Oh, they're giving Mr. abc a hard time

Matt N.

: )

t.b.

Mar 12, 2012 20:29

Oh, question about PDO's impending... :)

Hi Nimza

Matt N.

Hi Nimza : )

Nimza

Hey guys! Does anybody know about N-solvability of finite spherical subsets?

Hi @MattN )

t.b.

Context?

Nimza

Polynomials

Hi @tb :)

t.b.

What is a spherical subset of polynomials? But the short answer is most likely: no.

Nimza

Mar 12, 2012 20:34

A finite subset Omega of a unit sphere S^{n-1} is called N-solvable if any polynomial of degree N of a variable x = (x_1,...,x_n) can be represented as a sum of polynomials of degree N of scalar variables <omega_i, x>, where omega_i is in Omega (or of its projections on finite number of directions)

t.b.

No, as I suspected, I never heard of anything of the kind. Sorry.

Nimza

:(

t.b.

???

Matt N.

@tb Right, I'm distracted by a phone call. Of course $\sin ( (\frac{\pi}{4}, \frac{3\pi}{4}) ) = (\frac{\sqrt{2}}{2},1]$ isn't open.

@tb : )

I answered one of his questions the other day : )

t.b.

@MattN so he actually asked a question? I've never seen him doing that...

Matt N.

Mar 12, 2012 20:40

@tb Yes! He wanted a sequence $f_n$ with pointwise limit zero such that $f_n^\prime$ does not converge.

Though he wrote "$f_n^\prime$ diverges" which sounds kind of funny to my ears.

t.b.

@MattN he's not the only one around here knowing many mathematical words

Matt N.

Hello skullpatrol.

Skullpatrol

Hi Matt.

Whatz up?

Matt N.

Well not much since Wikipedia claims that "This is not true for real-differentiable functions" about the identity theorem. So I have two theorems that I can't apply.

And yourself?

Skullpatrol

Not much.

t.b.

Mar 12, 2012 20:53

@MattN yes, but you were talking about analytic functions. For smooth functions the standard counterexample is $f(x) = e^{-1/x^2}$ for $x \gt 0$ and $f(x) = 0$ for $x \leq 0$.

Fortuon Paendrag

Hello everyone!

Skullpatrol

Hello

Matt N.

Hi Fortuon.

Fortuon Paendrag

My exam, I think I messed up a question. Zorn zorn zorn. Why do I never remember him..

t.b.

@MattN a real analytic function can always be seen as a complex analytic function in some connected neighborhood of a compact interval

Matt N.

Mar 12, 2012 21:03

@tb Nice, thanks for the counterexample!

Skullpatrol

Does anyone here use MathJax 2.0?

Matt N.

@tb Is this obvious or is it a theorem I should know?

Never mind, I'll ignore that for now.

t.b.

@MattN it's a lemma :) (I don't dare to say it's obvious). For every point of the compact interval you have a positive radius of convergence by definition of analyticity and this allows you to interpret the function as a holomorphic function on a (complex) ball of that radius. Now use compactness.

Matt N.

@tb Yay, I can see that! Cool. : ) Thank you!

Skullpatrol

@robjohn Pardon the interuption, but do you use MathJax 2.0?

Asaf Karagila

Mar 12, 2012 21:17

Hm. The results of the teacher reviews came in today.

t.b.

Flunked AT?

Asaf Karagila

In the first class I got 4.2-4.9/5 and the second gave me 4.7-4.9

On all counts.

I'm impressed. That's a good review.

Matt N.

Congratulations!

t.b.

So are these the classes you attended or those where you were teaching?

Asaf Karagila

TA'ing.

Matt N.

Mar 12, 2012 21:18

Oh.

t.b.

That's really good. Congratulations!

Asaf Karagila

I wonder if these marks would have been the same after the exam, what with all that pigeonhole principle fiasco...

Matt N.

Probably not.

Now they think you stabbed them in the back.

Asaf Karagila

I'm not sure...

Matt N.

I am.

t.b.

Mar 12, 2012 21:20

Like students always do: blame it on the teachers, not on themselves...

Matt N.

: )

Asaf Karagila

Maybe some would have given me lower marks, or perhaps more people would have voted and would have given me bad marks.

I did get a few thanking emails after the results of the exam, and I did get a few people coming to thank me, also after the exam.

t.b.

Well, a prof I like once quipped: you could serve your students croissants and capuccino every day you lecture and half of them would complain that the croissants were cold and the capuccino not hot enough.

5

Matt N.

@AsafKaragila Seems much more personal where you are. Here at the cattle farm you don't even recognise people's faces (neither tutor nor classmates) let alone their email address, names or office number.

Skullpatrol

@tb LOL

Asaf Karagila

Mar 12, 2012 21:23

Well, I do my best to make this class personal. I enjoy teaching them and I enjoy teaching this subject.

Skullpatrol

@tb In my opinion, teachers who spoon feed their students are not really helping them in the long run.

Matt N.

I think I finally have it. I'm too distracted : (
If $f$ is analytic on an connected, open set $\Omega$ and zero on a bounded connected non-empty subset $B$ that contains at least two points then I can pick an open subset of $B$ on which $f$ agrees with the zero function (on $\Omega$) so $f$ *is* the zero function on $\Omega$.

Seems like too many constraints on $B$.

$f$ agrees on $O \subset B$ open with the zero function $z: \Omega \to \mathbb{R}$ is what this is saying.^

t.b.

@Skullpatrol that's probably true. On the other hand, it's very hard to see where to draw the line, especially if you're far more advanced and trained than your students. It takes some time and a lot of experience to remember all the pitfalls you're avoiding automatically, without thinking so to speak.

@MattN yep.

Matt N.

@tb Yay!!! : ) Thank you!

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