[Poll: Recommended Analysis textbook] University mathematics - First year 2008 Watch

Poll: Which analysis book would you recommend to a fresher? and why?
Mary Hart's Guide to Analysis (5)
26.32%
R.P. Burn's book Numbers and Functions (5)
26.32%
Fundementals of Mathematical Analysis by Rod Haggarty (1)
5.26%
A First Course in Mathematical Analysis by J.C. Burkill (7)
36.84%
Other - Please state (1)
5.26%
fusionskd
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#1
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I'm to start university this year (hopefully Imperial) and I was wondering if it would be worth going over a few of the topics during my summer holidays, that I'm to cover during my first year at uni...and if so, what kind of topics should I look at and where can I find good resources/textbooks in aid of this?

Thank you
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JohnnySPal
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In all honesty, I turned up to Warwick and felt at no disadvantage for having done next-to-no work over the summer. I don't know about Imperial, but we even had a module called Foundations that was an introduction to what type of stuff we'll be doing.

If you have to do some Analysis in the first year it'd probably be best to do that, if anything. You know how they say university style maths is completely different to what you'll have done before? Well, it seems like the more basic Analysis stuff is almost designed to get you into that way of thinking. If you turn up to uni having an idea of what's going on and even with a vague idea of how to go about proving things rigorously, you'll at be a great advantage!
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fusionskd
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(Original post by JohnnySPal)
In all honesty, I turned up to Warwick and felt at no disadvantage for having done next-to-no work over the summer. I don't know about Imperial, but we even had a module called Foundations that was an introduction to what type of stuff we'll be doing.

If you have to do some Analysis in the first year it'd probably be best to do that, if anything. You know how they say university style maths is completely different to what you'll have done before? Well, it seems like the more basic Analysis stuff is almost designed to get you into that way of thinking. If you turn up to uni having an idea of what's going on and even with a vague idea of how to go about proving things rigorously, you'll at be a great advantage!
Thanks for your help! Can I ask, what is Analysis about, and what is it used for?
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Arrogant Git
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Analysis is basically a very rigorous, formal way of looking at limits of sequences and series, as well as calculus. For example, how to deal, formally with infinite sums.
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JohnnySPal
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Analysis is the branch of maths that (eventually) rigrously builds up differentiation and integration and justifies all the various things you can do with it. It does this by (respectively) building up the notion of a limit - differentiating is finding the limit of the change in y over the change in x at each point on a curve, i.e. the gradient at each point - and by rigorously considering the summation of series.

The more basic analysis you're more likely to meet next year is sequences and their limits, e.g. Considering sequences such as a_n = 1/n; and series, e.g. Considering \sum_{n=1}^{\infty} \frac{1}{n}.

Now you'll probably wonder why it takes so much maths to say "well, the sequence 1/n tends to 0 as n goes off to infinity" but, hmm... It's hard to succinctly explain. One says that a sequence a_n converges to a limit a if (and only if) \forall \epsilon > 0 \text{ } \exists N \in \mathbb{N} such that |a_n - a|<\epsilon \text{ } \forall n>N. Might look like crap, but if you think about it very hard you'll see it makes sense: The above says that, by taking a term far enough in the sequence, you can get as close as you like (i.e. within an arbitrary epsilon) to the limit a, and more importantly by taking any subsequent term in the sequence you'll never get any further away.

(Personally, I understood the above random greek once I drew an arbitrary convergent sequence on a graph and drew horizontal "bands" around the value y=a, where a is our limit. Ahh, and fyi, the upside down A means "for all" and the backwards E means "there exists". So the statement reads "for all epsilon greater than 0 there exists...").

Go read some analysis books if that's interested you for soem reason. Or, ya know, go use the search function and look for proper replies to similar questions to yours :p:

You'll also prove that the sum above diverges (i.e. \sum_{n=1}^{k} \frac{1}{n} gets arbitrarily large as you increase k), but interestingly \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}, i.e. the alternating sum, is actually equal to log2.
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fusionskd
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#6
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(Original post by JohnnySPal)
Analysis is the branch of maths that (eventually) rigrously builds up differentiation and integration and justifies all the various things you can do with it. It does this by first building up the notion of a limit and by rigorously considering the summation of series.

The more basic analysis you're more likely to meet next year is sequences and their limits, e.g. Considering sequences such as a_n = 1/n; and series, e.g. Considering \sum_{n=1}^{\infty} \frac{1}{n}.

Now you'll probably wonder why it takes so much maths to say "well, the sequence 1/n tends to 0 as n goes off to infinity" but, hmm... It's hard to succinctly explain. One says that a sequence a_n converges to a limit a if (and only if) \forall \epsilon > 0 \text{ } \exists N \in \mathbb{N} such that |a_n - a|<\epsilon \text{ } \forall n>N. Might look like crap, but if you think about it very hard you'll see it makes sense: The above says that, by taking a term far enough in the sequence, you can get as close as you like to the limit a, and more importantly by taking any further term in the sequence you'll never get any further away.

(Personally, I understood the above random greek once I drew an arbitrary convergent sequence on a graph and drew horizontal "bands" around the value y=a, where a is our limit. Ahh, and fyi, the upside down A means "for all" and the backwards E means "there exists". So the statement reads "for all epsilon greater than 0 there exists...").

Go read some analysis books if that's interested you for soem reason. Or, ya know, go use the search function and look for proper replies to similar questions to yours :p:

You'll also prove that the sum above diverges (i.e. \sum_{n=1}^{k} \frac{1}{n} gets arbitrarily large as you increase k), but interestingly \sum_{n=1}^{\infty} \frac{1}{n^2} is actually equal to something like ln2 (can't remember what exactly!).
Thanx! +rep

Do you know where I can get some analysis books?
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EierVonSatan
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MA131 messes with your brain :p:
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JohnnySPal
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#8
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Amazon. Do Imperial send you a reading list through?

If not, and you want a book, I can recommend Mary Hart's Guide 2 Analysis and R.P. Burn's book Numbers and Functions. Both cover most of the basic analysis you'll meet throughout university.
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JohnnySPal
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#9
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(Original post by EierVonSatan)
MA131 messes with your brain :p:
The number of times I wished I could go back to it in the third year though :p: once you get it it's SO easy, but until then it will be the bane of your existence!
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qgujxj39
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#10
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(Original post by JohnnySPal)
You'll also prove that the sum above diverges (i.e. \sum_{n=1}^{k} \frac{1}{n} gets arbitrarily large as you increase k), but interestingly \sum_{n=1}^{\infty} \frac{1}{n^2} is actually equal to something like ln2 (can't remember what exactly!).
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi ^2}{6}, if I recall correctly.

Not that I know any analysis, I just read it somewhere.
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Creole
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#11
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(Original post by JohnnySPal)
Amazon. Do Imperial send you a reading list through?

If not, and you want a book, I can recommend Mary Hart's Guide 2 Analysis and R.P. Burn's book Numbers and Functions. Both cover most of the basic analysis you'll meet throughout university.
Winner.
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JohnnySPal
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(Original post by tommm)
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi ^2}{6}, if I recall correctly.

Not that I know any analysis, I just read it somewhere.
Yeah, i just editted my post. it's actually the alternating sum 1-(1/2)+(1/3)-(1/4)+... that converges to log2. I shoulda realised 1/n^2 converged to that - I had to prove it in an exam I did last year! (that was one GIT of a paper I tell you)
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EierVonSatan
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#13
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(Original post by tommm)
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi ^2}{6}, if I recall correctly.

Not that I know any analysis, I just read it somewhere.
yuppers, it was discovered as an attempt to define pi i think
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qgujxj39
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#14
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(Original post by EierVonSatan)
yuppers, it was discovered as an attempt to define pi i think
It was just Euler showing off his skillz.
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fusionskd
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#15
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(Original post by JohnnySPal)
Amazon. Do Imperial send you a reading list through?

If not, and you want a book, I can recommend Mary Hart's Guide 2 Analysis and R.P. Burn's book Numbers and Functions. Both cover most of the basic analysis you'll meet throughout university.
Just to confirm, these are the books:

http://www.amazon.co.uk/Guide-Analys...5353400&sr=8-1
http://www.amazon.co.uk/Numbers-Func...5353536&sr=1-1

From the reviews, the books look pretty good, I'm probably going to buy them!
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El Matematico
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(Original post by fusionskd)
Just to confirm, these are the books:

http://www.amazon.co.uk/Guide-Analys...5353400&sr=8-1
http://www.amazon.co.uk/Numbers-Func...5353536&sr=1-1

From the reviews, the books look pretty good, I'm probably going to buy them!
I dont have the Guide to Analysis book, but I do have Numbers and Functions by R.P.Burn (thats the right one in the link) and I would recommend it. Another that I quite like is Foundations of Mathematical Analysis by Rod Haggarty, but I do think that Numbers and Functions is a good book.
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fusionskd
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(Original post by El Matematico)
I dont have the Guide to Analysis book, but I do have Numbers and Functions by R.P.Burn (thats the right one in the link) and I would recommend it. Another that I quite like is Foundations of Mathematical Analysis by Rod Haggarty, but I do think that Numbers and Functions is a good book.
Thanks! Do you mean 'Fundamentals of Mathematical Analysis' by Rod Haggarty because I didn't find Foundations.

This one: http://www.amazon.co.uk/Fundamentals...5364210&sr=8-1
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fusionskd
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#18
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I also found this book:

A First Course in Mathematical Analysis by J. C. Burkill
http://www.amazon.co.uk/First-Course...5364210&sr=8-1

Not sure if anyone has heard of it, whats your take on it?
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eulerwaswrong
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(Original post by fusionskd)
Thanks for your help! Can I ask, what is Analysis about, and what is it used for?
you want to go to imperial yet dont know what analysis is?

sorry if that sounded rude - but id get reading
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eulerwaswrong
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(Original post by JohnnySPal)
Amazon. Do Imperial send you a reading list through?

If not, and you want a book, I can recommend Mary Hart's Guide 2 Analysis and R.P. Burn's book Numbers and Functions. Both cover most of the basic analysis you'll meet throughout university.
as said above - very good book
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