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Feb
7
comment Show that $J_n(x)$ satisfies Bessel equation $ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $
also math.stackexchange.com/questions/359107/…
Jan
29
comment Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \ $?
The paper that I am reading is unusual as they are cutting along $\mathbb{R}^+$ instead of $\mathbb{R}^-$ I should have mentioned that thanks.
Jan
29
comment Does $\overline{ \sqrt{1 + i}} = \sqrt{1-i} \ $?
what if you cut at positive reals?
Jan
26
comment express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $
Yeah this could be the one
Jan
25
comment express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $
@tired yes. the $e^{\frac{1}{2g}}$ is an instanton correction
Jan
25
comment express as contour integral $ f(x) = \int_0^\infty dt \; e^{-t/g} \; \frac{1}{\sqrt{1 - 2 t x}} $
@tired you get the error function if you don't add any correcting terms wolframalpha.com/input/… that's why I started making stuff up. it comes from a physics paper
Jan
24
comment How to prove that $f$ is integrable if $\forall \epsilon, \ \exists$ partition $M\in [a,b]$ such that $U_f(M) - L_f(M)\lt\epsilon$?
In practice these partitions might get pretty exotic. What partition makes the upper and lower Riemann sums close when $f(x)=\sum_{n=0}^{100}\frac{1}{n}\sin nx$ ? This converges to a sawtooth wave
Jan
22
comment How is the B-Spline definition constructed?
Please look at these notes pomax.github.io/bezierinfo I can provide more discussion later
Jan
21
comment Proof verification for $A \subset B$ iff $A - B = \varnothing$
I always found $A-B$ to be a finmy notation for sets since we're showing $A$ is inside $B$. Technically A "subset" of B
Jan
19
comment Cartan Lie Algebra of the Unitary Group $U(N)$?
I also found this resource astro.sunysb.edu/steinkirch/books/group.pdf
Jan
15
comment What are the dual polyhedra of the face-centered cubic lattice?
do those things really tile 3D space?? I'd love to see a picture of that!
Jan
14
comment What does it mean to solve an equation?
Sometimes people give up on solving $f(x) = 0$ and just solve $f(x) \approx 0$ and are happy
Jan
13
comment How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$?
Not exactly the same math.stackexchange.com/questions/217240/…
Jan
4
comment What is the minimum polynomial of $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$?
the computation of the minimal polynomial is pretty mechanical... but you can still get it wrong :-) I am asking if we pick out some deeper geometric meaning or symmetric to help us understand better. In the same way roots of unity correspond to regular polygons.
Dec
30
comment calculating $(1-i\sqrt{3})^{1+i}$
This would be useful if you study the zeta function over the imaginary numbers $\zeta_{\mathbb{Z}[i]}(s)$
Dec
29
comment How to write the Adeles over $\mathbb{Q}(i)$?
@anomaly if I understand correctly than $\mathbb{Q}_5 \otimes \mathbb{Q}(i) = \mathbb{Q}_{2+i} \oplus \mathbb{Q}_{2-i}$ ? And a similar story for $p = 13 = 2^2 + 3^2$ ? The case $p = 2+i$ seems funny... it's probably not true that $\mathbb{Q}_2 \otimes \mathbb{Q}(i) = \mathbb{Q}_{1+i} \oplus \mathbb{Q}_{1-i}$
Dec
26
comment How do I simplify: $ \exp \left(-a \frac{\log x}{\log Q} \right) $?
It only took 5 tries! This is from a research paper :-) It's full of these weird notations.
Dec
22
comment Show $a^2 + b^2 + 1 \equiv 0 \mod p$ always has a solution if $p = 4k+3$
Homomorphism. Then pigeonhole.
Dec
22
comment How do I get $ \int_0^1 \frac{dz}{\sqrt{z(z - 1\,)(z+1\,)}} = \frac{\sqrt{\pi}}{2} \frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{9}{4})}$?
I found this similar problem math.stackexchange.com/questions/867676/…
Dec
22
comment In Euclid' s Elements, why is $\frac{\sqrt{a^ 2 - b^2}}{a}$ important in the definition of “apotome”?
There's two definitions. The apotome which requires $( \frac{a}{b})^2 \in \mathbb{Q}$ - that rules out a lot of numbers - and confusingly, the first apotome which involves the ratio $ \sqrt{1 - (\frac{a}{b})^2} \in \mathbb{Q}$.