@Semiclassical Oh I agree. I think you've raised some good points.
@TedShifrin This is the research that UWO offers for MSc and PhD Mathematics students... imgur.com/a/lhF5Y are those pretty broad topics, or is this a sign that UWO doesn't have a strong graduate math program?
I saw that 30+ years ago, Balarka. I barely remember anything.
That alone isn't enough to decide anything, Nate, and it's honestly not that relevant when you're going to be an undergraduate. Plenty of people go to good liberal arts colleges with no graduate program and turn into very good mathematicians.
I take it Balarka has given up on un-un-un-sleeping yet again.
In the text "Theory of Functions of a Complex Variable" I'm having trouble verifying the power series expansion for the theorem within $(1.)$ my initial attack can be seen in $(2.)$
$(0)$
$$(\frac{\partial}{\partial z})^{k}f(z)= \frac{k!}{2 \pi i} \oint_{|\zeta - p|=r}\frac{f(\zeta)}{|\zeta -z|...
I have many long equations with so many parameters in it , in order to solve for a particular unknown it is little difficult to do by hand pages after pages, but yes it is ok though but still I need to verify it so
I know that Wolfram is supporting symbolic expressions so any other way how I can handle these ?
$e^{(\cos(\frac{1}{n}) - 1)^ {n^3}} = e^{e^{n^3.\log{(\cos{\frac{1}{n} - 1)}}}}$ next when $n \rightarrow \infty$ this goes to $e^{e^{(n^3)\log(-\frac{1}{2n^2}+\frac{1}{24n^4})-...}}\approx e^{e^{(n^3(-\frac{1}{2n^2})}} = e^{e^{(\frac{-n}{2})}}$,
now this tends to 1 (doesnot tend to $0$) as $n \rightarrow \infty$, hence the series diverges.
Also, on the same matter - if anyone is interested in functional analysis, I posted a question where it's just what I was writing about: https://math.stackexchange.com/questions/2298459/spectral-representation-integral-operator
That's actually nice. I often find these results all to be an extension of the ability to represent vectors, generally, as combination of orthonormal vectors.
@Studentmath Results from finite dimensional linear algebra tell you that diagonalizable operators are simultaneously diagonalizable iff they commute. So this basically works on Hilbert spaces too, and there's this principle that if you have an operator which commutes with taking second derivatives (or the Laplacian in higher dimensions) than a spectral decomp is provided by Fourier series (the eigenfunctions for the Laplacian)
Of course this interpretation really isn't rigorous if you're looking at $L^{2}$
Guys, we know that for a function with period $2\pi$, it holds that $$ c_n=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)\exp(-int)\,dt. $$ I would like to show that for a function with period $T$, it holds that $$ c_n=\frac{1}{T}\int_{-T/2}^{T/2} f(t)\exp\left(\frac{-2\pi int}{T}\right)\,dt. $$ I was considering the following substitution: $u(t)=\frac{2\pi}{T}t$. It seems to me that the only way for this to work is if we have the following: $$ c_n=\frac{1}{2\pi}\int_{-\pi}^\pi f\left(\frac{T}{2\pi}u\right)\exp(-inu)\,du=\frac{1}{T}\int_{-T/2}^{T/2} f(t)\exp\left(\frac{-2\pi int}{T}\right)\,dt.
@Semi ah right, well I'm only supposed to read chapter 1 and 2 from Griffiths for my final this week, which means I know nothing about "formal" quantum mechanics yet. The formalism happens in chapter 3.
Hey guys! needed some help in simplifying the homogenous gravitational field metric to the Rindler metric. Suppose we have the metric: (1+bk)^(2)dt^(2)-dk^(2)=ds^(2), under the condition that bk<<1, i.e., in the non-relativistic limit, the metric represents the homogeneous gravitational field. I am trying to change this via coordinate transforms to the rindler metric: ds^(2)=r^(2)dt^(2)-dr^(2).
How did you define the Fourier series? Because what you want to prove is a consequence of $\left\{ \frac1{\sqrt{L}}\text{exp}\left(i\frac{2\pi n}{L}x\right)\right\}_{n\in\Bbb N}$ being a complete orthonormal family in $L^2([-\frac L2,+\frac L2])$ @sha
@Alessandro yea I've searched for a different source, where they actually start of with the more general case, so I'll just take that as the main "definition"/theorem
@user314159 Don't get me wrong---I appreciate you telling me if you think it's something I might find interesting. But how did you come up with this? :P
Question: does $l_2$ is locally compact? i'm not sure about the answer so i tried proving it. if we take $x = (x_n)_{n \in \Bbb N} \in l_2$ , and a nbhd of $x $ , it contains a ball $B(x,r)$ , so i thought that if $\overline{B(x , r/2)}$ is compact then $l_2 $ is. So , does $$\overline{B(x , r/2)}$$ compact?
or if the space is Hausdorff, for each $x \in X$ and for each $U$ nhbd of $x$ there is a nhbd $V$ of x s.t $\overline V \subset U$ , and $\overline V$ is compact
I stumbled upon this exercise recently : We know the generalised Bertrand Series, that is $$\sum{1\over n\ln (n) \ln(\ln(n)) \dots \ln^{\circ k}(n)^\alpha}$$ converges iff $\alpha \gt 1$ for all $k\in\Bbb N$. But what about $$\sum{1\over n\ln (n) \ln(\ln(n)) \dots \ln^{\circ f(n)}(n)^\alpha}$$ where $f(n)$ is the greatest integer such that $\ln^{\circ f(n)}(n) \ge 1$ ?
@Astyx Does the convergence fact about the first series come from comparing with the integral? (antiderivative of that thing is something like log(loglog...blah...log(x)) I think)
I want to determine the flow of $\mathbf{A}=(x^2,2y,z)$ through the sphere $|r|=R$ by introducing the normal vector $\mathbf{n} = (x,y,z)/R$. This gives $$\iint_S \mathbf{A} \cdot \mathbf{n}dS = \dfrac{1}{R} \iint_S x^3+2y^2+z^2 dS = \dfrac{3}{R} \iint_S y^2 dS$$ due to symmetry. But how can I obtain $dS$ without changing coordinate system?
@EricSilva because if $x$ is the limit, then $||e_n -x|| \to 0$ , so $\sqrt((1 - x_n) \ ^ 2 + \sum_{i \ne n} x_i \ ^ 2 ) \to 0$ but this term is greater then $|x_n -1| $ so we have $x_n = 1$ for all $n$
alright, so the unit ball is not compact , how do i conclude that each ball is not compact too? it seems right but im not sure how to justify it. maybe all closed balls are homeo' ?
related @Daminark, last year in the bootcamp someone tried to do away with all the limit exchanging theorems in the complex analysis book by applying Fubini's theorem :P totally not in the right spirit
In the text "Theory of Functions of a Complex Variable" I'm having trouble verifying the power series expansion for the theorem within $(1.)$ my initial attack can be seen in $(2.)$
$(0)$
$$(\frac{\partial}{\partial z})^{k}f(z)= \frac{k!}{2 \pi i} \oint_{|\zeta - p|=r}\frac{f(\zeta)}{|\zeta -z|...
