This distinction is even important for polynomials. Working over $\Bbb Z_p$, for example, the polynomial $f(x)=x^p-x$ is definitely not the zero polynomial, and yet all its values are just $0$.
Think about what I just said, rather than tangling up with power series.
I think you can find examples of Laurent series converging to $1$ inside the unit circle and to $0$ outside, but 1) I'm not sure I'm remembering right, 2) it's constant inside and outside of the unit circle, but different constants
I suppose that, when I'm working in the ring of formal power series, I should write $e\exp(\exp(x)-1)$ rather than $\exp(\exp(x))$, where by $\exp$ I mean the power series corresponding to $e^x$.
well, for one, $f_n(z)$ is just a rational function with poles on the unit circle. so if one takes the domain to be, say, the interior of the unit disk, then as $n\to\infty$ one has a rational approximation to $f(z)=0$.
Meanwhile, on FB, one of my old friends and math geeks has discovered the definition "A set M in a metric space R is said to be compact if every sequence of elements in M contains a subsequence which converges to some x in R" in Kolmogorov/Fomin. He's apoplectic (understandably).
I guess what's interesting (I don't know why the $z^{-1}$ has to be there in the numerator) is that $f_n$ has no pointwise limit as $n\to\infty$ when $|z|<1$.
suppose you've got a function with a convergent Taylor series inside the unit circle $u(z)=\sum_{j=0}^\infty c_j z^n$
suppose you wanted to compute one of those $c_j$ numerically rather than analytically. one way to approach that is to use contour integration i.e. $\displaystyle c_j = \frac{1}{2\pi i}\oint_C z^{-j-1}u(z)\,dz$
to make things simple in the bit ahead, i'll assume that the radius of convergence is strictly bigger than $1$ so that I can take $C$ to wind just around the unit circle.
BTW, @Semiclassic, this reminds me of something that came up when I taught graduate complex the last time. If $f_n\to f$ pointwise, can you say anything about the convergence of the Taylor coefficients (or vice versa)?
It turns out to be very interesting, @Semiclassic. One of my students made a claim in homework, and I struggled to find a counterexample, but it was a false claim.
the reason that i bring that up, btw: if $u(z)$ has radius of convergence $R>1$, then the error of those estimates $c_j^{[n]}$ from the true $c_j$ is on the order of $R^{-n}$
and the reason that's convenient is because each of those estimates can be computed by applying the residue theorem to the $n$ poles on the unit circle of $f_n(z)$.
and that turns out to just be a trapezoidal approximation :)
Let $u(z)$ be some function defined on the unit circle $|z|=1$. Let $I=\frac{1}{2\pi i}\int_{|z|=1} u(z)\frac{dz}{z}$ and $I_n=\frac{1}{n}\sum_{j=1}^n u(e^{2\pi i j/n})$.
Then Theorem 2.1 is: Suppose $u$ is analytic and satisfies $|u(z)|\leq M$ in the disk $|z|<r$ for some $r>1$. Then for any $n\geq 1$, $|I_n - I |\leq \frac{M}{r^n-1}$ and the constant $1$ is as small as possible.
i have in mind the scenario where you've got two branch points inside the unit circle and two outside
in that case you can draw one branch cut entirely inside the unit circle and the other entirely outside, so one can pick an annulus that doesn't touch either cut
fun fact: in the p-adic world it can happen that $f\circ g$ evaluated at $x$ does not equal $f$ evaluated at $g(x)$ [where composition is done in formal power series ring]
Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in the The Nine Chapters on the Mathematical Art,[27] much earlier than the fifteenth century when they became well established in Europe.
@SoumyoB do I actually have to say that computers aren't nearly as new as you think and what you probably think of as computers is just the electrotechnically advanced model of Turing Machines and basic boolean algebra?
Rightist ideas are hardly seen as civil, I've been kind of ostracized from my social circle since most of my friends identify themselves as 'progressive' leftist liberals
If I have a probability question that my c++ program that simulates the system 100000 times to give an approximation of the probability(I'm using the law of large numbers) differs substantially from my mathematical results, do I post the mathematical part here and the program over at stackoverflow?
This weekend I am going to see Andrew Granville - The pretentious approach to analytic number theory, that provide us the Institut des Hautes Études Scientifiques from its official channel in YouTube. I will understand only a few facts , but will be about 200 minutes eating popcorns. I say it if some user want to see these videos (1/3), (2/3) and (3/3). Good weekend.
Given that $\frac{d^n}{dx^n}$ is the $n$-th derivative, perhaps $\frac{d^{-1}}{dx^{-1}}$ could signify integration? Not that I'm proposing to actually use this, but I find the idea somewhat funny
@SteamyRoot The "inverse" of the operator $d/dx$ is indeed integration (once you fix some integration constant), so that is a probably valid - although nonstandard - notation.
@SteamyRoot I don't think I've ever seen that usage, but I have seen people write $D=\dfrac{d}{dx}$ and then take $D^{-1}$ as notation for antidifferentiation
@Anubhav 1) A 3-manifold that has a Morse function with (1,g,g,1) critical points is the same thing as a 3-manifold with a Heegaard splitting of genus $g$. 2) Yes. Try to do it for surfaces first to see the idea.
@Semiclassical I guessed that might be an integral you wanna calculate. Not sure 100% (I have to check that) but I think that the integral can be reduced to one that can also be found in Paul Nahin's book.
@Semiclassical My teacher gave me Griffith's book on quantum mechanics. This stuff seems to me to be mostly mathematics, but rigorous in the sense of a physicist :P
Weird to see the insight/intuition getting pushed below the technicalities in physics books too.
@Semiclassical Yes, I'm right. I think my memory is still in a good shape, but what I find fantastic is that in the first second I saw it I established the connection with that integral, without thinking over it, all happening instantaneously.
As opposed to the professor who taught Quantum Physics at my university. He introduced a "Hilbert Space" as "A vector space such that the sum of two vectors lies in the same vector space again."
It would be easier for me to visualize it if I had a concept of what imaginary distance worked like. But all the foundations for distances are in real units.
