Recent content by Wingeer

I have a suggestion: Let ##f \in L^2(0, \pi)##, then we have: $$ f' = \left\{ \begin{array}{l} f(x) \text{ if } x \in (0, \pi)\\ -f(-x) \text{ if } x \in (-\pi,0) \end{array} \right.$$ and define f'(0) to be the average of the limiting processes of each of the piecewise functions (or...

Wow. Great answer again. Really clarifying, but still leaves me curious. I recently bought the real analysis book by McDonald and Weiss and I really enjoy reading in it. It is indeed meant to be the open interval, yes.

Thank you for your answer and time! We are actually using the book by Stein and Shakarchi in this course. I am not sure if I like it or not; It is certainly something else from what I've read before. Also I am just recently diving into analysis, so I might need to get used to this way of...

Hi. Just going through my notes from the last lecture I remember having some troubles understanding the proof the lecturer gave for the following theorem: Suppose that f is Riemann integrable and that all its Fourier coefficients are equal to 0, then f(x)=0 at all points of continuity. The...

I actually solved this "theoretically". What I was doing wrong was using the argument principle where it cannot be used. The problem is that the function has a zero on the path of integration, so to say. If one translates the area by an epsilon value everything works out. :)

One could of course use Rouche's theorem, but this exercise is given in the section before Rouche's theorem. So it is possible to solve it without. Also, I've dived so deep into this exercise that I really want to know what I've done wrong. Thanks for your answer, though!

Bump/revive from the dead.

Homework Statement Gamelin VIII.1.6 (8.1.6) "For a fixed number a, find the number of solutions of z^5+2z^3-z^2+z=a satisfying Re z > 0" Homework Equations The argument principle relating the change in the argument to the number of zeros and poles of the function on the domain. The...

Could you kindly elaborate a bit on the bold text in the quote? Or check if I got it right. Say, for instance that we are working with p(x)=x^3-2 \in Q[x] the roots of this polynomial are: x=\sqrt[3]{2},\omega \sqrt[3]{2},\omega^2 \sqrt[3]{2} Where, \omega = e^{\frac{2\pi i}{3}} First...

Hello, I have a quick question about extension fields. We know that if E is an extension field of F and if we have got an irreducible polynomial p(x) in F[x] with a root u in E, then we can construct F(u) which is the smallest subfield of E containing F and u. This by defining a homomorphism...

Anyone? I actually have another question about the quaternions. I am asked to show that: \mathbf{H'} = \{ a+bi+cj+dk | a,b,c,d \in \mathbf{R} \} with: i^2=j^2=k^2=-1, ij=k=-ji, ik=-j=-ki and jk=i=-kj. is isomorphic as rings to the quaternions defined in the previous post. I started by...

But you have to define f(x) for this to make sense. There are a lot of functions that does not go to infinity as x goes to infinity. For instance: f(x)=1/x.

Homework Statement What is the centre of the ring of the quaternions defined by: \mathbf{H}=\{ \begin{pmatrix} a & b \\ -\bar{b} & \bar{a} \end{pmatrix} | a,b \in \mathbf{C} \}? Homework Equations The definition of the centre of a ring: The centre Z of a ring R is defined by Z(R)=\{A...

I actually deduced that result earlier today. Should have thought of it when regarding multiple ideals comaximal to another one. Thanks. :-) What, however, if the ring has no unity?

Another question somewhat related: I just read in my book that "An ideal A in a ring R is maximal if and only if the pair X,A, for all ideals X not a subset of A, is comaximal". What does it mean for two ideals to be comaximal? That X+A=R. Is this just taking every element in X, and every...