Thread: Proofs... and Complex Analysis

Hey guys,

I know a lot of you love proofs and for some reason (which is hugely frustrating to me), you all seem to know how to prove statements with ease. I'm currently doing my 3rd (and final) undergraduate year in Mathematics and Geography, but for the life of me I have no clue how to prove statements in Maths! This semester I'm taking two courses in Mathematics, namely Applied Analysis and Complex Analysis. So far I've found AA pretty easy, but CA has been a huge struggle since proofs started entering the fray. I've worked my butt off but still don't seem to be able to prove things. Luckily I'm not the only one who's having trouble which has been reflected by a class average of 40% for the first test, with only 5 of us passing it (I got a "good" mark of 66%).

Is there a course that most people take, before they reach the analysis courses, that shows you how to prove statements or is it something that we as students are supposed to just pick up as we study from year to year? Surely there are a set of tried and trusted techniques that allow people to formulate a solution to a proof easily? I'm just really starting to stress out because our CA course is becoming more and more proof orientated as we go along and I'm able to answer fewer and fewer questions in our tutorials given to us each week.

I managed to get firsts for all my Maths courses in years 1 and 2, but this year it feels like I've hit a brick wall. I'm not a clever guy, I just work really hard at things until I understand them, but going over proofs, especially when they contain epsilons and deltas just baffles my mind completely.

Does anyone have any words of wisdom that would help me out here. Could you recommend any texts that deal with proof solving methods (if any)? Also, could anyone recommend any good texts related to an undergraduate introductory course into Complex Analysis (I have Visual Complex Analysis by Tristan Needham, but a lot of his examples, no matter how visual, do not shed much light on the matter for me)?

Thanks guys, cheers.

Unfortunately, I can't present you with a silver bullet. I can definitely understand your frustration. Complex Analysis is hard for me to begin with (I'm taking a grad level course in it right now), and I'm pretty good with proofs in an algebra setting. It can be challenging subject matter, depending on the breadth of the course.

I learned proofs by taking a (required) class titled "Introduction to mathematical reasoning." There are textbooks with this title and I would highly recommend picking one up. I don't have a particular one in mind for you. Perhaps checking out some reviews on Amazon will help. Here's one such book. The idea in such a course is to present you with some of the classical problems in math that required rigor to prove like showing that there are infinitely many primes or Fermat's Little Theorem (I really loved that one). A course like this will make you comfortable with the most common ideas of proofs like finite sets, infinite sets and their cardinality, proof by induction, proof by contrapositive, proof by contradiction and many, many others.

Regarding epsilons and deltas: I think you really have to wrap around the fact that first you pick an epsilon, and then you can find a delta which makes the statement work. A textbook on math reasoning will definitely cover the paradigm "for all blank there exists some other blank." It's a very common theme. You should try not to be intimidated by the very sight of epsilons and deltas. =) Most of the time it's enough to write down what you're given methodically and what you're being asked to demonstrate will pop out. Perhaps you could give us an example of a proof that baffles you? I'd be happy to try and make it clearer.

Cheers

Hi eeyore, I'll definitely check that book out, thanks a lot! Unfortunately none of our classes thus far have dealt with methods of proof, but I heard that Real Analysis, which I'll be taking next semester, should have been taken before taking Complex Analysis as it would have given a clearer picture. The only problem is that we weren't even given the option of taking it in semester 1 which is pretty weird, so it almost seems like there's a fault in the course structure.

One such example of a question that I would seem was "simple" is the following:

Prove that conjugate(cos(z)) = cos(conjugate(z)). I dunno if that makes sense like that but we had to prove that the conjugate over the whole expression was equal to cos of the conjugate of z.

I went about it by saying that since cos(z) = 1/2e^iz + 1/2e^-iz, then conjugate(1/2e^iz + 1/2e^-iz) = 1/2e^i(conjugate(z)) + 1/2e^i(conjugate(z)) and therefore proves that conjugate(cos(z)) = coz(conjugate(z)).

I realised that that was only one step in proving the statement, which turned out to be pretty damn complicated and involved the dreaded epsilons and deltas.

Originally Posted by Alexrey

Hi eeyore, I'll definitely check that book out, thanks a lot! Unfortunately none of our classes thus far have dealt with methods of proof, but I heard that Real Analysis, which I'll be taking next semester, should have been taken before taking Complex Analysis as it would have given a clearer picture. The only problem is that we weren't even given the option of taking it in semester 1 which is pretty weird, so it almost seems like there's a fault in the course structure.

One such example of a question that I would seem was "simple" is the following:

Prove that conjugate(cos(z)) = cos(conjugate(z)). I dunno if that makes sense like that but we had to prove that the conjugate over the whole expression was equal to cos of the conjugate of z.

I went about it by saying that since cos(z) = 1/2e^iz + 1/2e^-iz, then conjugate(1/2e^iz + 1/2e^-iz) = 1/2e^i(conjugate(z)) + 1/2e^i(conjugate(z)) and therefore proves that conjugate(cos(z)) = coz(conjugate(z)).

I realised that that was only one step in proving the statement, which turned out to be pretty damn complicated and involved the dreaded epsilons and deltas.

sometimes it's easier to see the structure of what something says, when what it is you're saying it about, is something simple:

for example, here is a proof that the function f(z) = z is continuous on all of C:

first, let's pick some arbitrary complex number. we'll call it z0, to remind us it's not a variable, but some definite complex number,

chosen at will from the rather large set of all possible complex numbers. intuitively, we want to say f(z) is near f(z0) whenever z

is near z0. but how can we express what we mean by "near"? the usual way is to take the modulus of the difference.

suppose we pick ε > 0. what i mean is that i am thinking of a very small positive real number, maybe 0.0000001 (as an example).

maybe even smaller, but still, some positive real number. now what i'd like to do, is find some δ > 0 so that when

|z - z0| < δ, |f(z) - f(z0)| < ε. "ε" is just a way of measuring "nearness" of f(z) to f(z0), and we want to ensure that z near z0

means f(z) near f(z0). another way of saying this, is that if the error in our input is small enough, the error in the output is also small,

that f doesn't do something really unpredictable (at least not near z0, which could be any point in C, so not anywhere in C).

well, as luck would have it, f(z) is nice and simple, the output IS the input, so we can use δ = ε. because if |z - z0| < ε,

then |f(z) - f(z0)| = |z - z0| < ε. what we have shown, is that lim z-->z0 f(z) exists, and is in fact equal to f(z0).

this really is just a precise way of saying: "when z is near z0, z is near z0", which we would certainly expect to be true.

now, the above proof is not earth-shattering, but the simplicity of f makes it a bit clearer what is happening with the ε's

and the δ's. we establish a bound (ε) for how far from f(z0) our function f is allowed to go, and then we seek a distance

from z0 we can allow (δ), to make sure f stays within the established range.

******

i find it sometimes helpful to draw pictures, one copy of C for the domain, and another for the co-domain, of f. "nice" complex

functions result in "nice" geometrical correspondences between the two.

Originally Posted by Prove It

Wow that's a nice way of proving it . I'm still not 100% sure why my way of proving it is not correct, could you maybe elaborate where my pitfalls are?

Thanks Deveno, that made a bit more sense after I had read through the chapter in James Stewart's Calculus on delta-epsilon proofs.

Thanks for your help so far guys, I really appreciate it. I looked at our latest tutorial that was handed to us on Wednesday and it seems we are getting less and less proofs which has made me, and the rest of my class, feel a lot more comfortable. I think our lecturer realised that giving us tons of proofs would be unfair since we have not done Real Analysis yet (which I found out will be purely proof based). I'm holding thumbs that this will be the case for the exams as well.

I don't see how you can be sure that

I know this may sound "basic" but aren't conjugates only applied to z by definition?

Originally Posted by Alexrey

I know this may sound "basic" but aren't conjugates only applied to z by definition?

Only when they're already in terms of their real and imaginary parts. In this case, your complex function is not...

Ah I see, now I know where I went wrong, thanks.

Well i'm not really smart and I was almost scared for the epsilons and deltas, but now I kinda love them.
What really helped me with understanding epsilon-delta proofs were the books Analysis I & II by Terence Tao, and literally googling: epsilon-delta proofs. Also when facing mathematical theorems and proofs that baffle you, just go back and try to understand and prove theorems you already use alot. It is a good training to try to prove things you use as 'trivial'.