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Jul
28
comment how to show that $\mathbb{Q}[\sqrt[3]{2}]$ is a field? (by elementary means)
See here for a proof that if $a$ is algebraic, $F(a)=F[a]$.
Jul
23
comment How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?
You seem to have forgotten the $\pi^2$ term in the final line.
Jun
28
comment Binomial Congruence
See if you can adapt this proof.
Jun
23
comment Does the index of a curve determine the asymptotic behaviour of certain vector fields?
@DavidH I think I'm talking about the index of the curve, sorry for the confusion of terminology (regardless, I'm talking about the thing Needham is talking about in Visual Complex Analysis, page $\approx 498$).
Jun
23
comment Does the index of a curve determine the asymptotic behaviour of certain vector fields?
@DavidH $$\frac{1}{2\pi} \int_{\gamma} \frac{v(p)_x}{|v(p)|^2}dy- \frac{v(p)_y}{|v(p)|^2}dx.$$
Jun
22
comment Does the index of a curve determine the asymptotic behaviour of certain vector fields?
I think Needham calls it the winding number, I'd be happy to know the proper name for this.
Jun
22
comment Does the index of a curve determine the asymptotic behaviour of certain vector fields?
@DavidH Let $v(p)$ be the vector associated with the point $p$. I define winding number to be the number of anticlockwise revolutions $\frac{v(p)}{|v(p)|}$ makes as $p$ moves along $\gamma$ in a closed loop. What you're referring to seems to be the case when $v(p)=p$.
Jun
18
comment Does the index of a curve determine the asymptotic behaviour of certain vector fields?
@HandeBruijn Thanks. I've not got the time now, but by Saturday I'll have edited it.
Jun
17
comment Difference between “Show” and “Prove”
@AndréNicolas I saw a textbook from the $1950$s with an extensive use of 'shew'.
Jun
3
comment $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
@t.b. Could you consider making an answer detailing the use of $\vartheta$ functions to solve this?
Jun
1
comment If a function's zeroes are doubly periodic, must that function be elliptic?
@QiaochuYuan Of course.
Jun
1
comment What is the sum of all complex integers?
Perhaps make it explicit in the question that you're looking for the analytic continuation of $f(s)$, not just a special case of the analytic continuation.
Jun
1
comment Is this function familiar to anyone?
It's foreseeable that you could analytically continue $f_a(z)$ for all complex $a \ne 2$.
Jun
1
comment Is this function familiar to anyone?
This one.
Jun
1
comment What is the sum of all complex integers?
I suppose this is equivalent to proving a Zeta reflection formula-like identity (see here) for $f(s)$.
May
20
comment Manifolds and magnetic potential
@GiuseppeNegro Arnold looks about right. I'll leave the question open as I don't think I could read Arnold very quickly.
May
20
comment Manifolds and magnetic potential
@GiuseppeNegro Thank you, that goes some of the way to clearing up concerns about the behaviour of the potential: I haven't seen the vector potential as applied to Lagrangian mechanics yet. However, the question concerning my hunch about manifolds still stands.
May
19
comment Evaluating $\int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x$
I'm not sure which paper it is exactly. He touches on it here, but doesn't solve it to the extent given above.
May
19
comment Evaluating $\int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x$
Should there be an equality somewhere in the second line?
May
17
comment Proving that nth derivate of $x e^{-x}$ is $(-1)^n (e^{-x})(x-n)$ by induction.
As a reply to 'always?', note that sometimes for step $1$ you will prove that the claim is true for all n up to $n$, then prove it for $n+1$ (as opposed to just for $n$). Also, you may occasionally assume it for $n,n-1$ and $n-2$ (for example), then prove it for $n+1$.