Post Beautiful Equations

[eqn]1 - 1= 0[/eqn]

>>8824911
Is that even true? The Maclaurin Series of a function does not necessarily converge to the value of the function for all values of x.

>>8824924
2deep4me

>>8824940
(z-z0)^n+1

>>8824928
[eqn]\simeq* or \approx[/eqn] would probably be better

>>8824928
its tailored around zero.
the finite sum approximates the function best around 0. but the infinite sum should indeed equal the function

>>8824944
:$ sharp eye

>>8824911
[eqn]\bigg(\sum\limits_{i=0}^ni\bigg)^2 = \sum\limits_{i=0}^ni^3 \qquad (1)[/eqn]
[eqn]\cos\left(z\right) = \frac{e^{iz}+e^{-iz}}{2} \qquad(2)[/eqn]
[eqn]\sin\left(z\right) = \frac{e^{iz}-e^{-iz}}{2i} \qquad (3)[/eqn]
[eqn]f^{(n)}\left(z_0\right)=\frac{n!}{2\pi i}\int_\Gamma \frac{f\left(z\right)}{(z-z_0)^{n+1}}dz \qquad (4)[/eqn]

>>8824928
>maclaurin

fucking 10/10 fell for the bait

>>8824994
taylor series centered around zero are often called maclaurin series

>>8824950
>the infinite sum should indeed equal the function
No
https://en.wikipedia.org/wiki/Non-analytic_smooth_function

>>8825012
thanks for pointing that out

>>8824911
[eqn] \tan \alpha \tan \beta + \tan \beta \tan \gamma + \tan \gamma \tan \alpha = 1 [/eqn]
if [math] \alpha + \beta + \gamma = \frac{\pi}{2} [/math]

[math]i \partial_t \psi = \hat{H} \psi[/math]

>>8824911

y=mx+b; y2-y1/x2-x1

>>8824924
weak...

[eqn]|G|=[G:H]|H| [/eqn]

File: taylor.png (165KB, 939x761px) Image search: [Google]

165KB, 939x761px

>>8824911
How about when f is of two variables
[eqn]f(x, y) = \sum _{i=0}^{\infty } \frac{1}{i!}\sum _{j=0}^i \binom{i}{j} \left(y-y_0\right){}^j \left(x-x_0\right){}^{i-j} \frac{\partial f^i}{\partial x^{i-j}\partial y^{j}}\left(x_0,y_0\right)[/eqn]
Here's a picture of [eqn]\exp \left(-\frac{1}{4} (x-2)^2-\frac{1}{4} (y-3)^2\right) \cos (2 x+y-7)[/eqn] and its 4th degree Taylor expansion about (2, 3) in orange and blue, respectively.

How about isomorphisms instead?

[math]{\operatorname{Hom} _{{\mathcal{O}_X}}}\left( {\mathcal{F},\mathcal{G}} \right) \cong {\operatorname{Hom} _{{\mathcal{O}_X}^{an}}}\left( {{\mathcal{F}^{an}},{\mathcal{G}^{an}}} \right)[/math]

>>8825267
What does ^{an} do

>>8825273
It is GAGA. Says (over a complex projective variety) there is a correspondence between sheaves in the Zariski topology (standard topology for a variety) and sheaves in the analytic topology (the more traditional topology of a complex manifold).

>>8824911
Being analytic is a very beautiful property to have

>>8825283
That's just a more "complicated" way to state the properties of logarithms.

>>8825232
Use multiindices and it will not look like such a mess.

>>8825363
I don't know how.

>>8825334
Everything is a more complicated way of saying something else

[math]\frac {\mathrm d} {\mathrm d x} x^2 = 2 x[/math]

File: tau_over_2.png (2KB, 147x57px) Image search: [Google]

2KB, 147x57px

*blocks your path*