/sci/ - Science & Math

>>8884466
y=mx+b

>>8884466
bait

>functions are equations

[math]
y - y_{0} = m(x - x_{0})
[/math]

>>8884466
x

>>8884466
>Last week of grad complex analysis
>Talk about analytic continuation
>Professor proves -1/12 meme

What is beautiful in an equation? This is something I've never understood.

>>8884466
>beautiful
>isn't actually defined for the only part of the function that is important

c2 = a2 + b2

>>8884466
There is nothing beautiful about that equation. It's a definition. It says nothing beyond giving a name to something. It doesn't really relate two objects.

>>8884521
Concise, has a lot of unexpected complexity, seems to appear in random places hinting at deep mathematics. That kind of stuff.

A good example are the fully symmetrized Maxwell equations. Yeah, it's cliche, but for good reason. These equations are the foundation for a lot of interesting math, physics, and engineering applications. It's astonishing how much you can explain starting out with the same set of equations. These don't even represent the most sophisticated description of EM theory.

[eqn]
\begin{align}
&\nabla \times \vec{E} = -\vec{J}_m\\
&\nabla \times \vec{H} = \vec{J}_e\\
&\nabla \, \cdot \,\,\vec{D} = \rho_e\\
&\nabla \,\cdot \,\,\vec{B} = \rho_b
\end{align}
[/eqn]

1 + 1 = 2

>>8884555
Those Maxwell equations are useful, yes, but I don't think they are "beautiful". I just can't see it.

>>8884466
Selberg trace formulas and formulas from index theorems are up there for me

e^{i\π }+1=0

y = y

>>8884466
>Post beautiful equations
>proceeds to post a definition
Good start Giggly Figgly Faggington Fag

>>8884555
>>8884588
Anyone who uses those [math]\overrightarrow D[/math] and [math]\overrightarrow H[/math] fields have shit taste.

>>8884653
Anyone who only uses [math]\vec{E}[/math] and [math]\vec{B}[/math] have never built anything of practical value in their life.

>>8884615

>>8884667
The most beautiful one yet

>>8884466
A = P^-1*B*P

>>8884493
>grad complex analysis
i did this in my uk university in my second year, also in the last class

US education is really pathetic...

>>8884733
Well I am in my second year, but yeah I think only 3-4 /22 students are undergrads.

>>8884698
A = (NP)^-1*B*(NP)

f(zn) = sin(zn) + ez + c

69 * 222 * 51 *7

2+2=5

I just love it because it immediately makes you think about the nature of truth, percception and objectivity.

>>8884902
kys

>>8884902
fuck off

P=NP

x=x

>>8884466
nothing is beautiful anymore
https://www.mathjax.org/cdn-shutting-down/

>>8884667

test

[eqn]\sum_{k=1}^\infty k=-\frac{1}{12}[/eqn]

>>8884665
, said the engineering brainet

[math] \displaystyle
e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots
\\ \\
e^x=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots
\\ \\
sin(x)=\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\cdots
\\ \\
cos(x)=\frac{x^0}{0!}-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\cdots
\\ \\
cos(x)+sin(x)=1+x-\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}-\frac{x^6}{6!}-\frac{x^7}{7!}+\frac{x^8}{8!}+\frac{x^9}{9!}-\cdots
\\ \\
e^{ix}=\frac{(ix)^0}{0!}+\frac{(ix)^1}{1!}+\frac{(ix)^2}{2!}+\frac{(ix)^3}{3!}+\frac{(ix)^4}{4!}+\cdots
\\ \\
e^{ix}=1+ix-\frac{x^2}{2!}-\frac{ix^3}{3!}+\frac{x^4}{4!}+\frac{ix^5}{5!}-\frac{x^6}{6!}-\frac{ix^7}{7!}+\frac{x^8}{8!}+\frac{ix^9}{9!}-\cdots
\\ \\
e^{ix}=\left ( 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\cdots \right )
+i \left ( x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\cdots \right )
\\ \\
e^{ix}=cos(x)+i \, sin(x)
[/math]

>>8884815
You'll fill your ego faster if you don't use a anonymous board. Please stay at reddit

>>8885557
>not writing that first terms of cos+sin as x^0/0! + x^1/1!

>>8885629
there is a good reason for that, think again

>>8884466
chromsome= XX XY XYX

>>8885679
I can't see the reason. cos and sin have those too, just for cos+sin they are simplified and break the patern.

>>8884466
This is not an equation, you fucking moron.

[math]\displaystyle{\left(\sum_{k = 1}^{n} k \right)^2 = \sum_{k = 1}^{n} k^3} [/math]

>>8884667
Explain this to a brainlet that doesn't know math.
I know e is a certain constant and i is -1^(1/2). I also know that an exponent of a positive number (e) van't be negative.

To make up for OP's brain fart here's a beautiful equation involving the Riemann zeta.

>>8885692
tip:
[math] \displaystyle
\frac{(ix)^0}{0!}= \frac{x^0}{0!}
[/math]

>>8884469
I came

>>8884667
how advanced is japanese high school math?

Y=mx+b
I'm retarded

>>8885798
number theory is some crazy shit
post more

>>8884469
In Australia it's

q + xw = λ

>>8885794
>Explain this to a brainlet
no, bask in the mystery of our intellect

[eqn]\left(\pi\right)_{16} = \sum_{k=0}^{\infty} 16^{-k}\left[ \frac{4}{8k+1} -\frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right][/eqn]

2b=e

>>8885555
Bro, do you even Clausius–Mossotti?

e+pi=i+1+0

>>8886191

op said equations not transforms you fucking idiot

>>8885550
[math] \lim_{z\to 1} \left( \sum_{n=1}^\infty n^m z^n - (-1)^{m+1} \dfrac{m!}{\log(z)^{m+1}} \right) = -\dfrac{1}{m+1} B_{m+1} [/math]

where [math] B_{m+1} [/math] are the Bernoulli numbers
1/6, 0, -1/30, 0, 1/42, 0, -1/30, ...

So
[math] \sum_{n=1}^\infty n^m [/math]
gives
-1/2, 0, 1/120, 0, -1/252, ...

>>8885962
cute post senpai

>>8885550
[math] \lim_{z\to 1} \left( \sum_{n=1}^\infty n^m z^n - (-1)^{m+1} \dfrac{m!}{\log(z)^{m+1}} \right) = -\dfrac{1}{m+1} B_{m+1} [/math]

where [math] B_{m+1} [/math] are the Bernoulli numbers
1/6, 0, -1/30, 0, 1/42, 0, -1/30, ...

So
[math] \sum_{n=1}^\infty n^m [/math]
gives
-1/2, 0, 1/120, 0, -1/252, ...

>>8884466

OC

>>8886208
Chill. Just plug the second equation into the first. Think of it as a vector expansion.

Btw, that notatation is gay as fuck.

[eqn]
F(\omega) = \frac{1}{(2\pi)^\frac{n}{2}}\int_{\mathbb R^n} f(x) e^{+i \omega x} dx\\

f(\omega) = \frac{1}{(2\pi)^\frac{n}{2}}\int_{\mathbb R^n} F(\omega) e^{-i \omega x} d\omega
[/eqn]

ftw

>>8886282
>R^n
>scalar x and w

nigger

>>8885794
>I also know that an exponent of a positive number (e) van't be negative.
No, that's just wrong. Who told you that bullshit?

[math]a^{-b} = \frac1{a^b}[/math]

e^pi*i + 1 = 0

>>8884466
[math]{S_{GS}} = - T\int {{d^2}\sigma \left[ {\sqrt { - h} {h^{\alpha \beta }}{g_{\mu \nu }}\prod _\alpha ^\mu \prod _\beta ^\nu } \right] - T\int {{d^2}\sigma \left[ {{\varepsilon ^{\alpha \beta }}{\partial _\alpha }{X^\mu }\left( {{{\bar \Theta }^1}{\Gamma _\mu }{\partial _\beta }{\Theta ^1} - {{\bar \Theta }^2}{\Gamma _\mu }{\partial _\beta }{\Theta ^2}} \right) + {\varepsilon ^{\alpha \beta }}{{\bar \Theta }^1}{\Gamma ^\mu }{\partial _\alpha }{\Theta ^1}{{\bar \Theta }^2}{\Gamma _\mu }{\partial _\beta }{\Theta ^2}} \right]} } [/math]

>>8886459
the fuck I cant keep up with all the different actions for ST, which one is that? I assume GS stands for Green-Schwarz?

1

>>8886459
is that the TOS?

>>8886459
Kappa symmetry is pretty dope I can dig it.

>>8885629

You can write those terms as x^0/0! + x^1/1!, but if you do that then you also need to specify the additional requirement that 0^0 = 1.

The problem with making 0^0 = 1 a requirement is that its analysis is ambiguous. As x->0, the limit of x^0 is 1, but the limit of 0^x is 0. The fact that those two limits are different means that analysis cannot be used to conclude what 0^0 is. As a result, a special-case declaration by fiat is required to define 0^0 (if you even care about making such a definition).

Some mathematicians are fine with the assumption that 0^0 = 1, but some consider that definition to be an unnecessary complexity, and they prefer to leave 0^0 undefined.

There is no "right" answer to this question, because it depends on a person's subjective view about what is "elegant" and what is "unnecessary" in mathematics.

>>8886268
I agree but desu can there be models for the last one?

>>8886282
Brainlet here. Where on wikipedia can I go to read what this is?

>>8885557
Wow, I didn't know that.

196884 = 196883 + 1

>>8886571

The last two are inconsistent.

>>8886582
Fourier Transform.

The best way to think about it is as an expansion of a vector in a basis. It's weird because the "vectors" are functions and there are uncountably infinite dimensions.

That is, given a finite dimensional basis [math]\hat{e}_i[/math], where [math]\langle\hat{e}_i | \hat{e}_j\rangle=\hat{e}_i\cdot \hat{e}_j=\delta_{ij}[/math] (it's orthonormal w.r.t the "dot product" inner product), any vector [math]\vec{v}[/math] can be expressed like:

[eqn]
\begin{align}
\vec{v}&=\sum_i\alpha_i\hat{e}_i\\
\langle\vec{v}|\hat{e}_j\rangle&=\left\langle\sum_i\alpha_i\hat{e}_i|\hat{e}_j\right\rangle
=\left(\sum_i\alpha_i\hat{e}_i\right)\cdot\hat{e}_j=\sum_i\alpha_i(\hat{e}_i\cdot\hat{e}_j)=\sum_i \alpha_i\delta_{ij}=\alpha_j\\
\Rightarrow\vec{v}&=\sum_i\langle\vec{v}|\hat{e}_i\rangle\hat{e}_i
\end{align}
[/eqn]

Analogously, given the uncountably infinite orthonormal basis [math]e_{\omega}(x)=\frac{1}{\sqrt{2\pi}}e^{i\omega x}[/math], where [math]\langle e_{\omega}(x)|e_{\omega^\prime}(x)\rangle=\int e_\omega(x)e_{\omega^\prime}(x)^*dx=\delta(\omega-\omega^\prime)[/math] (it's orthonormal w.r.t. to a different inner product, the excercise is left to the reader), (m)any vector [math]f(x)[/math] can be expressed as

[eqn]
\begin{align}
f(x)&=\int\alpha(\omega)e_\omega(x)d\omega\\
\langle f(x)|e_{\omega^\prime}(x)\rangle&=\left\langle\int\alpha(\omega)e_\omega(x)d\omega | e_{\omega^\prime}(x)\right \rangle=\int\left(\int\alpha(\omega) e_\omega(x) d\omega \right ) e_{\omega^\prime}(x)^*dx=\int\alpha(\omega)\left ( \int e_\omega(x)e_{\omega^\prime}(x)^*dx \right )d\omega=\int\alpha(\omega)\delta(\omega-\omega^\prime)d\omega=\alpha(\omega^\prime) \\
\Rightarrow f(x)&=\int \langle f(x)|e_\omega(x)\rangle e_\omega(x)d\omega\\
&=\int\left(\int f(x)\frac{e^{-i\omega x}}{\sqrt{2\pi}} dx\right)\frac{e^{+i \omega x}}{\sqrt{2\pi}}d\omega
\end{align}
[/eqn]

The basis here are sinusoids. Thus, the inner product [math]\langle f(x)|e_\omega(x)\rangle[/math] is a spectrum.

>>8886208
If it's got an = then it's an equation

>>8886371
He meant that e^x is never negative in R^2

>>8885001
>1_r
That's the worst unit vector notation I've ever seen.

>>8884476
i bust a nut so hard i cant feel my left leg

>>8885783
so [eqn]\sum_{k=1}^\infty k^3=-\frac{1}{1728}[/eqn]

>>8885794
Alright, so in calculus, the derivative (or rate of change) of e^x is just e^x
In calculus, there is also something in calculus called taylor series, where you can write the approximate value of any function f (x) by using the formula in the picture, where f'(x) is the first derivative, f''(x) is the second derivative, and f^n(x) is the nth derivative. Xo is the number that the approximation is 'centered around', and can jjust be any number. The accepted approximation of e^x is centered around 0, and will be shown in the next post cont.

>>8886770
Clearly I fucked up the formatting int the middle. The sentence reads:

Analogously, given the uncountably infinite orthonormal basis
[math]e_\omega(x) = \frac{1}{\sqrt{2\pi}}e^{i\omega x}[/math], where [math]\langle e_\omega(x)|e_{\omega^\prime}(x) \rangle= \int e_\omega(x) e_{\omega^\prime}(x)^*dx = \delta(\omega - \omega^\prime)[/math] (it's orthinormal w.r.t a different inner product, the exercise is left to the reader, (m)any vector [math]f(x)[/math] can be expressed as

I should note that the little star is complex conjugate.

I should stress the weird "integral" inner product [math]\int f(x) g(x)^* dx[/math] I used is just as valid as a "dot" inner product [math]\vec{v}\cdot\vec{u}[/math], because both satisfy the inner product axioms:

[eqn]
\langle f(x) | g(x) \rangle = \langle g(x) | f(x) \rangle^* \\
\langle \alpha f(x) + \beta h(x) | g(x) \rangle = \alpha \langle f(x) | g(x) \rangle + \beta \langle h(x) | g(x) \rangle \\
\langle f(x) | f(x) \rangle \geq 0 \quad \vee\quad \langle f(x) | f(x) \rangle = 0 \Leftrightarrow f(x) = 0
[/eqn]

The Fourier Transform is a little hard to interpret for the uninitiated, but if you discretized it to try to approximate it (or are studying Fourier series), you will see you are building a function out of sinusoids of different frequency, amplitude, and phase.

Was this pretty clear?

>>8884466
ax + by = 1

>>8886797
So this is the expansion of e^x derived by using the equation in the previous post. Now instead of x, put in (i*x) for x, for the definition of i that you know, i^2 = -1, i^3 = -i, i^4 = 1

So some of the terms become negative and some still have i's in them and some terms have both. Now if you look at the sin x and cos x expansions, combined they form the expansion for e^x, and all of the sin terms would still have an i attached to them, and the i can be factored out, so that the equation forms e^ix = cos (x) + i*sin (x)

Instert pi into the eq, and it works

... + x3 + x2 + x + 1 + 1/x + 1/x2 + 1/x3 + ... = 0

>>8886797
>>8886816
Thanks I learnt something.

>>8886807
>Was this pretty clear?
It was mostly pretty wrong.

The orthonormal basis IS countable ( how can you even believe that it is not? Omega is in Z.), and NOT a basis of the L^2 vector space (only it's closure is) and can only approximate most L^2 functions.

1=1

bottom the page

x^2 = x + 1

>>8884466

Your mum + my dick = you

>>8885962
clever

For the Mario fans

>>8884902
Truly, you have selected the only truly beautiful equation among the infinite set of all possible equations. This has already made me ponder the question of wHY YOU HAVEN'T FUCKING KILLED YOURSELF ALREADY. FUCKING RETARDED, BRAINLET FAG.
Thank you <3

>>8888102
>floating point values
>real numbers

>>8888073
Wtf I love Mario 64 now

>>8886794
For some reason, you computed the cube and not the square.
Also, no, the answer happens to differ from (-1/12)^2 by +1/6!.

>>8885962
kekd

>>8884902
Define 2,+,=5

>>8887790
>The orthonormal basis IS countable ( how can you even believe that it is not? Omega is in Z.)
Uh, yeah, no. Frequency is in R for what I defined. I did a transform, not a series.

did /sci/ already reach 8888888?

>>8884466
Babby's first Zeta function

>>8888193
No

>>8888102
wtf is going on here.

Gamma(s) * Gamma(1 - s) = Pi / (sin pi (s))

0.99999999...=1

remember when albert asstein invented the theory of lelativity :^)
also [s4s] master race fugging happening

>>8888888
FUCK S$S DOES >ITA AGAINJ

>>8888888
Fuck you

>>8888888
fuck off you should have spammed a spectrum meme not this crap and your get doesn't count because you used multiple IPs to circumvent the post timer

>>8888888
Fuck you

>>8887978
is this gibberish?

>>8884466
[math]{\text{Ric}}\left( g \right) - \frac{S}{2}g + \Lambda g = T[/math]

>>8888888
kek is here

>>8888502
https://www.youtube.com/watch?v=kpk2tdsPh0A
It''s worth the watch

>>8888909
What's this?

>>8888908
yes, its from a website that generates random math papers

>>8888888
Gets are a spectrum, just because you got 7 8s doesnt mean it is a heptuple. I identity as a heptuple though, so checkem.

>>8888888
HIS NAME WAS ALFREB

>>8888888

>>8888917
EFEs without messy index notation

>>8888888
fucking asshole

>>8888912
The guy in the video is you isn't it kek?

>>8885962
I don't get it.

>>8888888
>>8888888

mediocre

Equation of big tits

>>8888102
this is starting to go from "huh, that's pretty cool" to "that's just sad" territory

>>8888996
it's upside down

>>8888888
THE BASTARDS FINALLY DID IT

>>8888888
what a waste

>>8888888
DELET

>>8888888
AHH>>8888888

>>8888888
Disgusting.

>>8888888
worst get i've ever seen

good one anon

>>8889114
thx

>>8889044
>finally
http://s4s.wikia.com/wiki/Stolen_GETs

>>8888888
ayyt lmao

>>8884466
[math]\sin (x) = \arcsin ^{-1} (x)[/math]
[math]\cos (x) = \arccos ^{-1} (x)[/math]
[math]\tan (x) = \arctan ^{-1} (x)[/math]

>>8889290
uh

[eqn]
\int_{\partial \Omega}\omega = \int_{\Omega}\mathrm{d}\omega
[/eqn]

mind = blown

>>8888182
Nah, you're both slightly wrong.
L^2 is a separable Hilbert space, hence it has a countable "orthonormal basis" (ie. a countable Hilbert basis, but I wanna stress that it's not an actual basis in the sense of a linearly independent spanning set, it just spans a dense subspace), that you can get by applying the Gram-Schmidt process to a countable dense set (which I think can be obtained by taking rational linear combinations of characteristic functions segments with rational ends)
Now, for the inner-product, the family [math]e^{i\omega x}[/math] is *not* a basis nor a Hilbert basis, nor anything. These functions aren't even in L^2, their dot product according to your formula doesn't make sense. You're probably mixing this up with Fourier series.

>>8889482
>(ie. a countable Hilbert basis, but I wanna stress that it's not an actual basis in the sense of a linearly independent spanning set, it just spans a dense subspace)
Just call it a Schauder basis, jesus

>>8889393
>Calculus: The Equation

>>8889482
Yes, the functions [math]e^{i \omega x}[/math] are definitely not in L^2, however the inner product I wrote in a weird sense is correct, and they are in a sense basis functions. To construct a function from L^2 out of them, you must have some spread in your spectrum, it can't be a delta function

This is something I've been trying to wrap my head around. In quantum, two famous and basic examples are the free particle (no potential) and the harmonic oscillator (quadratic potential). Applying basic QM, the free particle basis are the uncountably infinite set of functions [math]e^{i \omega x}[/math], while for the harmonic oscillator the functions are the countably infinite set of functions of exponentials raised to Hermite polynomial powers.

The mind fuck is that both can be used to describe functions in L^2. Why this is weird is because the dimensionality of the basis sets are different. It seems that vectors in infinite dimensional Hilbert spaces can be expanded w.r.t either a countably infinite set of functions in L^2, or an uncountably infinite set of functions outside of L^2 but linear combinations of them are in L^2. The choice is arbitrary.

Of course, I am hardly being mathematically rigorous, and physicist have a tendency to take vast amounts of mathematical nuance in stride, but this more or less works.

OP = fag

>>8889541
>Tautology
>Beautiful

1,1,2,3,5,8,13,21,34,55,89,144...

>>8886555
but those are used elsewhere in >>8885557 so what the fuck

>>8889519
this is an annoying feature of quantum mechanics. it has to do with the fact that the space of "physical" wave functions (namely, those which have well-defined expectation, variance, and all higher moments) is topologically incomplete, and strictly smaller than L2. So it can be shown that the space of "physical bras" (i.e., linear functions on the space of physical kets) is strictly larger than L2. The "proper" way to interpret e^{iwx} is as a physical bra, namely the map [math] |f>\to \int e^{iwx}f(x)dx[/math].This is a well-defined assuming f is a physical ket, but is not a well-defined map on L2 (since otherwise, the Riesz representation theorem would imply that e^{iwx} is in L2). The keyword here is "Rigged Hilbert Space".

>>8889519
I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it, [math]\left( \mathbf R,\, +,\, \times,\, \leqslant,\, \left| \cdot \right|,\, \tau \,=\, \left\{ A \,\subset\, \mathbf R \mid \forall x\,\in\, A,\, \exists \varepsilon \,>\, 0,\, \left] x \,-\, \varepsilon,\, x \,+\, \varepsilon\right[ \,\subset\, A \right\},\, \bigcap_{\begin{array}{c} A \,\sigma \text{-algebra of}\, \mathbf R \\ \tau \,\subset\, A \end{array}} A,\, \ell \right)[/math]-analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital [math]\mathbf R[/math]-related properties comprising a full number field as defined by pure mathematics.

Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.

There really is a calculus, and these people are using it, but it is just a part of the field they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field. Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis. All the so-called calculus problems are really problems of real analysis.

>>8889587
What's with physicists using this bra-ket nomenclature instead of using duality like any normal human being?

>>8889609
that's the way dirac did it, and it stuck

>>8889603
Can you post the pasta with the latex code? I'm too lazy to rewrite it. I'll also swap that topology definition for the one generated by open intervals

>>8889587
>physical bras

>>8889615
[math]\left( \mathbf R,\, +,\, \times,\, \leqslant,\, \left| \cdot \right|,\, \tau \,=\, \left\{ A \,\subset\, \mathbf R \mid \forall x\,\in\, A,\, \exists \varepsilon \,>\, 0,\, \left] x \,-\, \varepsilon,\, x \,+\, \varepsilon\right[ \,\subset\, A \right\},\, \bigcap_{\begin{array}{c} A \,\sigma \text{-algebra of}\, \mathbf R \\ \tau \,\subset\, A \end{array}} A,\, \ell \right)[/math]

>>8889587
I think this is what I was beginning to appreciate last time I studied QM. It's weird that the vector isn't really the function [math]e^{i\omega x}[/math], but the functional [math]\int e^{i\omega x} \cdot dx[/math] meaning that you have to roll the inner product and the "basis" function up into the mathematical object identified as a covector, aka a bra in the Riesz representation. Am I interpreting this correctly?

I've heard the phrase Rigged Hilbert Space before, but I haven't really seen it defined. Is this what is meant, that in normal Hilbert Space your basis is a set of vectors in L^2 with some inner product defined, while in Rigged Hilbert Space your basis is a uncountably infinite set of covector functionals?

Also, in what sense is QM wavefunction space topologically incomplete. Are you saying it's a subset of L^2 because of restrictions on smoothness and such? Are you saying that there are convergent series of physical wavefunctions that converge to unphysical wavefunctions?

>>8889652
>with the latex code

[math]\sqrt{-1} \frac{heqarktp}{qkp} b^{o^2+1}s[/math]

>>8889615
>>8889662

>>8884466
1 = 0.999...

>>8889925
Thanks for schooling my retarded ass, senpai

>>8884996
The binomial expansion/series method of proving this is actually much prettier.

>>8889519
>The mind fuck is that both can be used to describe functions in L^2. Why this is weird is because the dimensionality of the basis sets are different. It seems that vectors in infinite dimensional Hilbert spaces can be expanded w.r.t either a countably infinite set of functions in L^2, or an uncountably infinite set of functions outside of L^2 but linear combinations of them are in L^2. The choice is arbitrary.

I think that you are making the same mistake as people who claim that 1+2+3+...=-1/12, because the extension of of zeta function has value -1/12 at -1. Fourier transform on L2 is defined as an extension of Fourier transform on L1 intersected with L2. This is exactly the thing that Placherel theorem says.
https://en.wikipedia.org/wiki/Plancherel_theorem
So in fact when you use this extension you are talking about L2 from the perspective of the part of the dual of L1, the set {e^-itx}.

x + y = boy
x + x = girl

>>8889393
My favorite as well
https://youtu.be/QNznD9hMEh0?t=8m40s

Used to price options.

>>8890176
tell me more.

>>8890172
Nice

>>8889609
http://www.mathpages.com/home/kmath638/kmath638.htm

>>8884466

[math]F_g = \frac{Gm_1 m_2}{r^2}[/math]

[math] (A+UBV^t)^{-1} = A^{-1} - A^{-1}U(B^{-1} + V^tA^{-1}U)^{-1}V^tA^{-1}[/math]

1+1=2

1+2=3

>>8885962
No it isn't you faggot

>>8886151
that's not true

>>8888112
kekd

[math]PV=nRT[/math]

[math]0[/math]

>>8892925
fug

>>8884466

id = >>8895317

>>8892948
ʇəluıɐɹq

>>8886219
wtf who is that?

>>8889662
Right click "Show math as" -> "Tex commands"
You're welcome fagget.

>>8895694
Hawking before ALS

>>8884466
√2 ∈ Q

>>8895972
>√2 ∈ Q
That is neither true nor an equation.

>>8889658
>Is this what is meant
Just read the wikipedia on Gelfand Triples, its not at all a complicated construction.

>>8884466