could someone explain to me how to prove that $z\mapsto (|z|^2-1)^2$ is not complex differentiable when $|z|>1$?

$\partial_z (|z|^2 - 1)^2 = \partial_z (z \bar z - 1)^2 = 2 \partial (z \bar z - 1) = 2 \bar z$.

@Jonas doesn't that only prove that it is not analytic?

What does complex differentiable mean for you?

The thing is that your $|z|$ does not care about directions.

if there is a $c\in \mathbb{C}$ so that $f(z)=f(z_0)+c(z-z_0)+r(z)$ where $\lim_{z->z_0}\frac{|r(z)|}{|z-z_0|}=0$

yeah, i get that. it just maps everything to the real line. i am just shakey about writing the proof of it; i'm new to complex analysis.

so i guess here the $c$ is $\partial_z f \mid_{z_0}$

Important to know: A function is holomorphic if it is complex differentiable.

Then it is also analytic, i.e., has infinitely many derivatives and converges to its Taylor series.

Or rather the series converges to the function.

(That is a theorem).

But, let us Blimey a bit about your limit. Say we wish to figure out what this is. Note that in $1$ it is equal to $0$, hence, the derivative is $$\lim_{h \to 0} \frac{|h|^2 - 1}{h}.$$

Say we now have $h$ on the line through $(1, 1)$ and $(1, 0)$. Then the limit is: $$\lim_{t \to 0} \frac{|1 + i t|^2 - 1}{1 + i t} = \lim_{t \to 0} \frac{t^2}{1 + i t} = 0.$$

While over the real axis we have: $$\lim_{t \to 0} \frac{t^2 - 1}{t} = ...$$

And now I hope I did not mess up rotally.

i see.

gahh i still cannot figure this out. the problem i'm working on is to construct a function $f$ with $f'(z)=1$ at every point $|z|\leq 1$ but at nowhere else in $\mathbb{C}$

Alright.

using $f=\left\{\begin{array}{lcl}0&:&|z|\leq 1 \\ (|z|^2 - 1)^2 &:&|z|> 1 \end{array}\right.$ gives me one that almost works, but $f'(z)=0$.

What if you add $z$?

... ah. yes. thank you.

(then divide)

@JasonBourne Where?

@JasonBourne Ahh. okay.

@JasonBourne Well, welcome to eternal september.

(Do not forget that is one of the reasons why lhf is possible).

@JasonBourne Naah, my point is without such kind of people who accept an trickily incorrect answer to the questions, there would be no lhf.

Fuck yeah, Chopin.

Have a good rest or beer and rest...

Which I will do: nighty night.

Good night.

@Jonas so, when i'm talking about why $f$ is not differentiable outside of $\overline{\Delta}(0,1)$, you used $\partial_zf$ (the formal $z$-derivative, right?). by what condition does $\partial_zf(z)=2\overline{z}$ imply that $f$ is not differentiable?

@JasonBourne the $d$'s cancel, it's just $x/y$, reciprocal of which is $y/x$, multiply by $d/d$. $\qed$

hey guys

two courses, intermediate algebra and college algebra

hi

@JasonBourne Well done - I think the only subject badge I have is number theory

@JasonBourne Neat :)

I'm not usually online at this time of day - couldn't sleep, so thought I would see if I could find a question to answer :)

@JasonBourne Hmm - not sure. Possibly a "feature" of the Javascript implementation of your current browser?

Darn - someone upvoted a later answer which was basically a copy of mine :(

here

@Eric Do you know that the symmetric group has order $n!$?

It is where he says that (12)a != (12)B implies that a != B, thus, there are atleast as many even as there are odd permutations

@OldJohn yes i do

I don't see how he can reach such a conclusion just by showing that a != B

you know that if $\alpha$ is even, then $(12)\alpha$ is odd (and vice versa)?

@JasonBourne Yes i understand what the implication proof of the proof is. I think that statement is pretty intuitive, it is just that line in the proof that is bothering me

@JasonBourne Darn again - the answer which copied mine has just been accepted :((

@OldJohn Sure, that part is easy, its essentially the same as taking an even 2k, getting an odd, 2k+1, and and even 2k+1+1 = 2(k+1)

@JasonBourne The book is Contemporary Abstract Algebra by Galligan

It sucks

OK - so, if you take a list of all the even permutations, then "multiply" by $(12)$, you get all the odd permutations - so there are equal numbers of each

The author goes on on, and his proofs are very wordy, and shys away from symbolic notation.

@Eric I would say that the proof he gives of that theorem is very clearly written

@JasonBourne Actually I disagree, unless the author does everything in one step, I think showing the calculus lation step by step is easier to follows

@OldJohn It is, this proof is just me.

@OldJohn Anyway what set do the permutations a and B belong to?

Are we just assuming that everything is in $S_n$?

@Eric yes

ok

yeah but that means that |{odd}| <= |{even}|

Yeah excatly

oh

Oh, wel that makes alot more sense

But i still dont understand his deal with the a!= B, stuff

@OldJohn yoyo!

I haven't seen you in a while!!!

@OldJohn I finished all my problems! All 40 of them!

@BenjaLim well done that man!!

Not been here much recently - been busy with other things

@JasonBourne Marcus' Number Fields.

@OldJohn I realised that; I posted on your wall a few days ago but got no reply so I guessed you were busy.

@JasonBourne Yes haven't you seen all my questions on ANT ?

@OldJohn And even more exciting is that now I am going to start a course on Algebraic Geometry!

@BenjaLim Ah - sorry! I have not been looking at FB either recently - trying to do more active things, rather than spending time online :)

@BenjaLim Horrors!

@OldJohn Deep and serious shit!

@JasonBourne @OldJohn It's a reading course using Kempf's book Algebraic Varieties

I actually always thought that would be interesting to look into - I even have a couple of beginners books on it, but not got past the first chapter or so

ah. @OldJohn You need to have a strong algebra background!

@BenjaLim Yeah that sounds like serious shit!

@OldJohn I think halfway in the book there is sheaf cohomology!!! When I opened that book I was like farkkkkkk

I enjoyed reading the first couple of chapters of Reid's Undergrad introduction - my algebra is good enough for that :)

@BenjaLim that is deep and mysterious stuff - tread very warily!!!

@OldJohn If you want to start with commutative algebra, I recommend the section on Dummit and Foote. It is blended together with algebraic geometry in such a way that it is very readable.

@OldJohn Inside, I'm trembling with fear! Because my supervisor said that he used Kempf's book as a first course in AG

@BenjaLim Thanks - I might do that one day. At the moment, I am working on my old favourite area: analysis :)

@OldJohn WHILE BEING A MASTERS STUDENT AT CAMBRIDGE!!!

@JasonBourne their rings have $1$.

@JasonBourne And let me tell you the number of times I have taken a look at D&F

@BenjaLim Impressive

@JasonBourne I have used that book as a reference, to learn material from it, I find the exposition very clear. Just yesterday I was reading on Groebner bases and that was clear for me too.

@OldJohn Hmmm, as an analyst do you have an example to show that $C[0,1]$ is not Noetherian?

@JasonBourne You seem to like books alot

I was thinking of looking at like ideals of piecewise functions.

@BenjaLim I can't even remember what Noetherian means!

@JasonBourne How many of those books have you actually used?

@OldJohn Of if every ascending chain of ideals eventually stabilizes.

@JasonBourne What can you say about Knapp's books?

Yeah - it came to me as you were typing :)

@OldJohn I'll have to think about it a bit more.

But I don't have any feel for the ideal structure in $C[0,1]$, unfortunately - never looked at it algebraically

@OldJohn yeah me too. I can tell you the analytic properties like yeah it's complete in the sup norm, not compact, etc

but algebraically somehow my mind becomes retarded :D

@BenjaLim mine too

@JasonBourne What about his Basic and Advanced Real Analysis books?

@JasonBourne Wasn't trying to be arrogant or anything. Just stating an opinion.

Seriously?.. crap

@BenjaLim @JasonBourne Time I tried getting back to sleep - it is 4 am here now

@JasonBourne So what would you recommend after rudin?

@OldJohn 9 points ahead now!! And real madrid coming up!

^baby rudin

@OldJohn I was thinking of functions like here: math.stackexchange.com/a/157007/38268

@BenjaLim yay!!

@JasonBourne I want an opinion, nothing more

it is up to me what i do with it

@Eric Is your analysis up to the level of Rudin?

Well i wouls say i am about 50% the level of rudin

Goodnight all

my prof uses alot of his problems for homework

@Eric Can you sketch a proof as to why a sequentially compact $X$ is necessarily compact?

and I would like to continue to study analysis without back tracking relearning everything

@Eric are you an undergrad at a university?

@BenjaLim we dont do topology in our course sadly

@BenjaLim yes

@Eric Chapter 2 in Rudin is like the most important chapter.

@Eric If you don't know and are not familiar with things like compactness

Well we dont use rudin

how did you guys cover results like $C[0,1]$ is complete in the sup norm (IIRC) and stuff?

@BenjaLim I basically go to the worst university ever man

the math program is absurdly bad

@Eric which country are you in?

University of North Florida

Right.

Basically, the only rigorous math courses a math major has to take are Algebra 1 (dont do rings) And Advanced Calculus 1 and 2, all other courses like complex analysis, number theory, vector analysis, are all taught WITOUT proofs

@JasonBourne I only delete comments that reveal personal details.

@Eric wow.

they dont even offer topology or algebra 2 or intro to measure theory

@JasonBourne Besides the things I say do not necessarily reflect the opinions of the math. dept at ANU.

@BenjaLim I know its bad. The, professors are pretty good but considering that on average 60% of all high school student drop out in my city, its a (big) stretch to expect kids right out of high school to dive head first into rudin

@JasonBourne I am all ears

@Eric I bought my copy for 10 AUD $\approx$ 10 USD

@JasonBourne On analysis you are definitely qualified, I read your honours thesis.

@JasonBourne Well then please suggest something.

@JasonBourne correct.

My plan is to read rudin and a follow up to rudin, (folland, knopp, etc) at the same time

bye guys

I already put one out there myself, but I didnt get much inreturn

ok in that post, by advanced calculus i mean A rigorous reteaching of Calculus1,2,3

@BenjaLim, you solved that noetherian problem?

by real analysis i mean, analysis, I mean going into measure theory and so on L^p spaces, and such

jason by calculus i mean a course in the computational methods of integral and differentiable calculus, without any proofs what so eevr

@JasonBourne Ok, but would real and complex analysis be too hard?

Well, some of my professors said they think that is a crappy book to learn from. That basically it is more like a reference book.

@BenjaLim, anyway, here's one possible way to do it. Take $m$ to be the ideal of all cts functions that vanish at 0. We show that it is not finitely generated. Suppose the contrary, and that $f_1,\cdots,f_n$ generates the ideal. The key is that there's always a continuous function $g$, that vanishes at 0 at a speed slower than $f_1,\cdots,f_n$, i.e. $\lim_{x \to 0^+} f_i/g = 0$ for all $i$. Once you know this, write $h_1f_1+ \cdots h_nf_n = g$, divide both sides by $g$ and take limit to 0.

@JasonBourne Well is it like baby rudin times 2? The same level?

Absurd at the point 0. (You can take $g$ that vanishes at 0 only and nowhere else so that division presents no problem). Construction of such $g$ is not difficult. @BenjaLim

ok

@JasonBourne If you were serious about analysis, what would you personally recommend?

Rudin?

Sorry, If you were recommending books to a student, that you felt was serious about analysis, what would you recommend?

Ok, thanks man

yes

Im trying to find another proof of quadratic reciprocity

By finding I mean looking for one to read

Not creating one lol

uhm

Some book by silverman somthing or other

Do you know where I can find an elementry proof other then Eisenstein's geometric proof?

Of quadratic reciprocity

I just need a proof of the conditions on when I can flip the legendre symbols, I already have proved the cases for $(\frac{-1}{p})$, $(\frac{2}{p})$

Ireland-Rosen give something like four proofs in successive chapters. I think the one using quadratic Guass sums is elementary.

There are many proofs of Quadratic reciprocity

The Gauss sum proof is something good to read.

I know wikipedia says there are hundreds

Im trying to find one with out gauss sums, or other stuff

just basic modular arithmetic

and some algebra

I think that is second to the pythagorean theorem.

Deepness of these two results differ by so much though.

What do you mean by deepness?

Whats the most general reciprocity law for higher powers?

Artin reciprocity law.

But Ethan, you would need to study a long time to get there ;)

Number theory.

You?

@Ethan Theorems 3.3 and 3.4 of Ivan Niven's The Theory of Numbers combine to give a relatively elementary proof of QR. It's mostly modular arithmetic and clever summations.

Banana is very subjective, and it has nothing to do with your interest to be honest.

@BenW. Is there someway I can find that proof online? I don't have the book

@Sanchez

How 'advanced' is artin reciprocity can it give the same if not more in depth statements about residues of higher powers then say quadratic reciprocity?

@Jason, lean towards analytic for now, but interested in both to be honest.