problem on analytic functions

Forum for the GRE subject test in mathematics.
brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

problem on analytic functions

Postby brain » Wed Dec 08, 2010 7:28 pm

http://upload-pictures.info/share-21F3_4D00122E.html

The curve here intersects itself, so it's not simple. What should we do?

prong
Posts: 24
Joined: Thu Sep 24, 2009 12:17 am

Re: problem on analytic functions

Postby prong » Wed Dec 08, 2010 11:05 pm

What is its winding number around z_0?

le6tan
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Joined: Sat Apr 10, 2010 8:30 am

Re: problem on analytic functions

Postby le6tan » Thu Dec 09, 2010 3:46 am

winding number is 2, so the answer should be 4*i*pi ...

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Thu Dec 09, 2010 6:31 am

le6tan wrote:winding number is 2, so the answer should be 4*i*pi ...

That's right. Which theorem takes into account the winding number?

trevaskis
Posts: 22
Joined: Sun Nov 07, 2010 1:33 pm

Re: problem on analytic functions

Postby trevaskis » Thu Dec 09, 2010 6:39 am

Cauchy's Theorem (one of them anyway)

http://en.wikipedia.org/wiki/Residue_theorem

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Thu Dec 09, 2010 8:39 am

trevaskis wrote:Cauchy's Theorem (one of them anyway)

http://en.wikipedia.org/wiki/Residue_theorem


Ok, thank you. Now it's clear.

trevaskis
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Re: problem on analytic functions

Postby trevaskis » Thu Dec 09, 2010 8:59 am

Princeton review covers this pretty briefly.

brain
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Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Thu Dec 09, 2010 9:19 am

trevaskis wrote:Princeton review covers this pretty briefly.


You are right. Even the chinese remainder theorem is not present there.

prong
Posts: 24
Joined: Thu Sep 24, 2009 12:17 am

Re: problem on analytic functions

Postby prong » Fri Dec 10, 2010 12:25 am

You should really know the Chinese Remainder Theorem anyway, if you're going to go to grad school.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Fri Dec 10, 2010 5:45 pm

prong wrote:You should really know the Chinese Remainder Theorem anyway, if you're going to go to grad school.


If it were so important then why it was not mentioned in the "Cracking ..." book? I think you put too much stress on this theorem.

cbreeden
Posts: 25
Joined: Sun Jul 18, 2010 8:37 am

Re: problem on analytic functions

Postby cbreeden » Fri Dec 10, 2010 7:25 pm

Breaking News! Everything a math major should learn in their four years as an undergraduate is now accessible in a single book!

mdornbos
Posts: 27
Joined: Sun Sep 05, 2010 11:11 am

Re: problem on analytic functions

Postby mdornbos » Sat Dec 11, 2010 2:36 am

I think the princeton review book is very good for a START to the studying for the math gre. But nothing beats getting out old books, notes, and tests from the classes you took. I do agree, however, that there are things that should be added in maybe in place of others.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 12:40 pm

cbreeden wrote:Breaking News! Everything a math major should learn in their four years as an undergraduate is now accessible in a single book!


The "Cracking..." book is meant for preparation, not to fill your gaps. If a theorem is not included there, it is not neccesary for the test, especially the Chinese one.

PieceOfPi
Posts: 25
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Re: problem on analytic functions

Postby PieceOfPi » Sat Dec 11, 2010 1:01 pm

brain wrote:
cbreeden wrote:Breaking News! Everything a math major should learn in their four years as an undergraduate is now accessible in a single book!


The "Cracking..." book is meant for preparation, not to fill your gaps. If a theorem is not included there, it is not neccesary for the test, especially the Chinese one.


Oh you will be very surprised in the actual exam...

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 1:30 pm

PieceOfPi wrote:
brain wrote:
cbreeden wrote:Breaking News! Everything a math major should learn in their four years as an undergraduate is now accessible in a single book!


The "Cracking..." book is meant for preparation, not to fill your gaps. If a theorem is not included there, it is not neccesary for the test, especially the Chinese one.


Oh you will be very surprised in the actual exam...


Can you give me a few examples that would surprise me or you are just making quesses?

PieceOfPi
Posts: 25
Joined: Mon Aug 02, 2010 3:16 pm

Re: problem on analytic functions

Postby PieceOfPi » Sat Dec 11, 2010 1:41 pm

brain wrote:
Can you give me a few examples that would surprise me or you are just making quesses?


Absolutely! For example, the "Cracking..." book does not mention anything about sequences of functions (which I wonder why, because this is really the heart of analysis), but I have seen questions that required to know this. I have also seen questions from either actual exam or older exams on classification of surfaces, finding Jordan blocks, special kinds of linear transformations (e.g. orthogonal, unitary, self-adjoint), linear interpolation, partial differential equations, triple integrals, and my friend told me some questions required him to remember Stoke's and/or Gauss's theorems. I believe most of these concepts were not covered in the "Cracking..." book.

I agree with whoever said that "Cracking..." book is a good starting point for preparing for this test. On the other hand, you certainly need to go a bit deeper once you are finished with that book. I actually have not seen anything on Chinese Remainder Theorem yet, but it is an important fact from number theory / abstract algebra, so I wouldn't be surprised if you see something about CRT on the exam.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 1:59 pm

PieceOfPi wrote:
brain wrote:
Can you give me a few examples that would surprise me or you are just making quesses?


Absolutely! For example, the "Cracking..." book does not mention anything about sequences of functions (which I wonder why, because this is really the heart of analysis), but I have seen questions that required to know this. I have also seen questions from either actual exam or older exams on classification of surfaces, finding Jordan blocks, special kinds of linear transformations (e.g. orthogonal, unitary, self-adjoint), linear interpolation, partial differential equations, triple integrals, and my friend told me some questions required him to remember Stoke's and/or Gauss's theorems. I believe most of these concepts were not covered in the "Cracking..." book.

I agree with whoever said that "Cracking..." book is a good starting point for preparing for this test. On the other hand, you certainly need to go a bit deeper once you are finished with that book. I actually have not seen anything on Chinese Remainder Theorem yet, but it is an important fact from number theory / abstract algebra, so I wouldn't be surprised if you see something about CRT on the exam.


Look, don't use my post to express your personal opinion and spread statements that might be truely false. If you know questions from the exam that fall beyond the book coverage, post them. This site is for helping people take the test, it is not a means for deception. I am not gonna let anyone to use my posts for deluding people.

alex
Posts: 37
Joined: Wed Oct 20, 2010 12:54 am

Re: problem on analytic functions

Postby alex » Sat Dec 11, 2010 2:29 pm

it's true, sequences of functions questions are in the freely available practice exams. stokes theorem was with out a doubt on the test in november. i don't remember particular questions, and even if i did, posting them gives others an unfair advantage.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 2:36 pm

alex wrote:it's true, sequences of functions questions are in the freely available practice exams. stokes theorem was with out a doubt on the test in november. i don't remember particular questions, and even if i did, posting them gives others an unfair advantage.


Without particular questions, OUT OF MY POST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
This post is on analytic functions, not your hallucinations of what is fair and what is not.
To all: stick to the theme or get out.

PieceOfPi
Posts: 25
Joined: Mon Aug 02, 2010 3:16 pm

Re: problem on analytic functions

Postby PieceOfPi » Sat Dec 11, 2010 2:38 pm


brain
Posts: 28
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Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 3:18 pm

PieceOfPi wrote:http://www.ets.org/Media/Tests/GRE/pdf/gre_0809_math_practice_book.pdf

See #64.


To some extent you are right, no function sequances are in the book but the bigger problem is that uniform convergence is not defined also. However, I still don't see any questions that can't go without the Chinese theorem.

alex
Posts: 37
Joined: Wed Oct 20, 2010 12:54 am

Re: problem on analytic functions

Postby alex » Sat Dec 11, 2010 3:20 pm

chinese remainder theorem is unrelated to the topic of this post. please stay on topic.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 3:25 pm

alex wrote:chinese remainder theorem is unrelated to the topic of this post. please stay on topic.


Read the whole post, Prong claimed great importance of that theorem. I let Prong defend its claim. With particular questions!

prong
Posts: 24
Joined: Thu Sep 24, 2009 12:17 am

Re: problem on analytic functions

Postby prong » Sat Dec 11, 2010 3:39 pm

The importance of the CRT is difficult to understate, but my intention was not to imply that it appears specifically on any particular questions.

It's just a basic fact that you should know. Not knowing it is like not knowing that the indefinite integral of x^2 is (x^3)/3. It's like not knowing that every ideal is the kernel of a ring homomorphism, and vice versa, and that if f: A->B is a homomorphism, A/ker f is isomorphic to B.

One shouldn't rely too much on practice books. There is often more than one way to solve a problem.

alex
Posts: 37
Joined: Wed Oct 20, 2010 12:54 am

Re: problem on analytic functions

Postby alex » Sat Dec 11, 2010 3:46 pm

brain wrote:This post is on analytic functions, not your hallucinations of what is fair and what is not.
To all: stick to the theme or get out.


i just think you're being pretty unreasonable is all. CRT was brought up, by you, as an example of an important theorem excluded in the princeton book, not something that was strictly on the topic of analytic functions. It was brought up as part of an ongoing discussion. Everything that I said in my earlier post was also a natural product of the discussion and you told me to get out if I wasn't here to talk about analytic functions. You need to get a grip. We're all here just trying to help each other out.

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 3:48 pm

prong wrote:The importance of the CRT is difficult to understate, but my intention was not to imply that it appears specifically on any particular questions.

It's just a basic fact that you should know. Not knowing it is like not knowing that the indefinite integral of x^2 is (x^3)/3. It's like not knowing that every ideal is the kernel of a ring homomorphism, and vice versa, and that if f: A->B is a homomorphism, A/ker f is isomorphic to B.

One shouldn't rely too much on practice books. There is often more than one way to solve a problem.


If you can't prove your words with particular questions, keep your comparisons and suggestions out of my post!

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

Re: problem on analytic functions

Postby brain » Sat Dec 11, 2010 3:56 pm

alex wrote:
brain wrote:This post is on analytic functions, not your hallucinations of what is fair and what is not.
To all: stick to the theme or get out.


i just think you're being pretty unreasonable is all. CRT was brought up, by you, as an example of an important theorem excluded in the princeton book, not something that was strictly on the topic of analytic functions. It was brought up as part of an ongoing discussion. Everything that I said in my earlier post was also a natural product of the discussion and you told me to get out if I wasn't here to talk about analytic functions. You need to get a grip. We're all here just trying to help each other out.


How do you help others as you say? By writing worthless stuff in post? I said the theorem was not in the book as a remark to the book. I didn't say the theorem was underlying for the test. The most natural thing is when you don't have questions helpful for the exam, stop spamming my post!

aaaaa
Posts: 5
Joined: Sat Dec 11, 2010 6:20 pm

Re: problem on analytic functions

Postby aaaaa » Sat Dec 11, 2010 6:35 pm

Dude, quit your whining. You got your question answered, so who cares if people make a few more posts that may be relevant and helpful to some people. If you're so worried about tangents being explored, go PM an admin to close the thread.

Every Math GRE has a question about solving some modular equations. Usually it looks like "ax + by = c (mod m), dx + ey = f (mod m). Find x, y (mod m)." In this case, it's just a little linear algebra. But it's not a stretch to imagine them saying "Let x = 3 (mod 11), x = 4 (mod 17), find x (mod 187)." For this, you should know CRT.

Here's an example that showed up on a real test: suppose a complete graph K_n on n vertices has e edges. Find n (I don't remember the exact value they gave for e). Of course, they asked it in more of a convoluted way, but it's the type of thing that would show up in any basic discrete math, combinatorics, or graph theory course that you should know. The number of edges is sum of i from i=1 to n-1 which is n(n-1)/2. This is a fairly straightforward exercise, and I'm pretty sure it's not mentioned in the PR Math GRE book. You can't expect that book to teach you everything you need to know... yeah it has a few nice tips, but there's a reason people talk about your undergraduate preparation. If you ever take the test, you'll realize that the PR book's practice problems are completely different from the problems on the real test.

enork
Posts: 33
Joined: Fri Sep 18, 2009 3:16 am

Re: problem on analytic functions

Postby enork » Mon Dec 13, 2010 2:40 pm

I think there is a big distinction between "things you should know as a math major" and "things you should know to do well on the math GRE". It's been a few years since I took the test, but if I recall, almost none of what I learned in college was at all applicable.

prong
Posts: 24
Joined: Thu Sep 24, 2009 12:17 am

Re: problem on analytic functions

Postby prong » Tue Dec 14, 2010 12:25 am

aaaaa wrote:Here's an example that showed up on a real test: suppose a complete graph K_n on n vertices has e edges. Find n (I don't remember the exact value they gave for e). Of course, they asked it in more of a convoluted way, but it's the type of thing that would show up in any basic discrete math, combinatorics, or graph theory course that you should know. The number of edges is sum of i from i=1 to n-1 which is n(n-1)/2.


I think a better way to do this is to view it as n choose 2. You have n vertices, and the edges are just unordered pairs of 2 vertices. That is, making an edge just means choosing two distinct vertices. Therefore there are n choose 2 = n!/(2!(n-2)!) = n(n-1)/2 of them. (Or you can just know that n choose 2 is n(n-1)/2.)

Sorry that this isn't about analytic functions, brain :).

aaaaa
Posts: 5
Joined: Sat Dec 11, 2010 6:20 pm

Re: problem on analytic functions

Postby aaaaa » Tue Dec 14, 2010 3:06 am

Yeah people have told me I should do n choose 2 on this type of thing before, but for some reason, my intuition usually gives me ideas other than binomial coefficients, I guess I'm not as comfortable with them as I am with other techniques. I guess the way I think of it seems more algorithmic: you have to draw n-1 edges from vertex 1, n-2 from vertex 2, and so on...




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