The beautiful math formula
Good evening english corner and chattering's friends 
.
I open this thread to talk (in english obviously) about a beautiful - in our opinion - math formula and describe it in some words: we are in a math forum
.
I begin with the Euler formula
$e^(iy) = cos(y)+i sin(y)$.
Apply it with the basic property of an exponential
$e^z = e^(x+iy) = e^x \cdot e^(iy) = e^x(cos(y)+i sin(y))$
forall $z=x+iy \in \CC$.
This is a great math law that, in a first look, seems to break out our high school knowledge. We grew with the axioms "exponential is one-to-one" and similar properties of the real exponential.
In complex analysis we learn - especially based on this formula - that for purely imaginary values complex exponential is limited and periodic:
$e^(iy) = cos(y)+i sin(y)$.
A first consequence of this is that the modulus of a complex exponential is equal to the modulus of the corresponding real exponential obtained elevating the base to the real part of the complex number:
$|e^z|= |e^(x+iy)| = |e^x| |e^(iy)| = |e^x| = e^x$ (because $x=Re(z)\in \RR$).
We can note that, reverting this, complex sine and cosine are not limited for purely imaginary values.
.I open this thread to talk (in english obviously) about a beautiful - in our opinion - math formula and describe it in some words: we are in a math forum
.I begin with the Euler formula
$e^(iy) = cos(y)+i sin(y)$.
Apply it with the basic property of an exponential
$e^z = e^(x+iy) = e^x \cdot e^(iy) = e^x(cos(y)+i sin(y))$
forall $z=x+iy \in \CC$.
This is a great math law that, in a first look, seems to break out our high school knowledge. We grew with the axioms "exponential is one-to-one" and similar properties of the real exponential.
In complex analysis we learn - especially based on this formula - that for purely imaginary values complex exponential is limited and periodic:
$e^(iy) = cos(y)+i sin(y)$.
A first consequence of this is that the modulus of a complex exponential is equal to the modulus of the corresponding real exponential obtained elevating the base to the real part of the complex number:
$|e^z|= |e^(x+iy)| = |e^x| |e^(iy)| = |e^x| = e^x$ (because $x=Re(z)\in \RR$).
We can note that, reverting this, complex sine and cosine are not limited for purely imaginary values.
Risposte
"apatriarca":
I like Stokes' theorem
\[ \int_{\partial \Omega} \omega = \int_{\Omega} d\,\omega \]
where \(\omega\) is a differential form over the boundary of an orientable manifold \(\Omega.\) I like its simplicity, symmetry and importance in the theory.
First time I see this theorem I fall in love with it. This generalization of the foundamental theorem of calculus is as simple as powerful. It also seems that you "move the \(d/\partial\) from the domain to the integrand". Love it!
@Friction your explanation is perfect! These arguments are so beautiful!
You're right, it was also used by Planck!
You're right, it was also used by Planck!
I like Stokes' theorem
\[ \int_{\partial \Omega} \omega = \int_{\Omega} d\,\omega \]
where \(\omega\) is a differential form over the boundary of an orientable manifold \(\Omega.\) I like its simplicity, symmetry and importance in the theory.
\[ \int_{\partial \Omega} \omega = \int_{\Omega} d\,\omega \]
where \(\omega\) is a differential form over the boundary of an orientable manifold \(\Omega.\) I like its simplicity, symmetry and importance in the theory.
"grimx":
Ah! An another really nice equation is the Einstein equation for the Photoelettric Effect!
$E=h\nu$
I think this formula was first introduced by Planck (in 1900) when he tried and succeded to explain the black-body radiation: we can write the black-body density of radiation as \(u(\nu,T)=\frac{1}{V}\times \mathcal{D}(\nu)\,d\nu\times\langle E\rangle\) where \(\mathcal{D}(\nu)\) is the 3D density of states for a classical particle. Now the 3D DOS is \(\mathcal{D}(k)=\frac{V}{2\pi^2}k^2\) which can be expressed as a function of \(\nu\) as follows:
\begin{align*}
&k=\frac{2\pi}{\lambda}=\frac{2\pi}{c}\nu\\
&\mathcal{D}(\nu)\,d\nu=\frac{V}{2\pi^2}\left(\frac{2\pi}{c}\right)^3\nu^2\,d\nu
\end{align*}
We can think of the black-body at equilibrium temperature \(T\) as a set of harmonic oscillators at frequency \(\nu\) which emit and absorb the e.m. radiation. Now we postulate that \(E_n=nh\nu\) is the quantized energy of such a system.
Let
\[Z_{cl}:=\sum_n{e^{-\beta E_n}}=\sum_n{\left(e^{-\beta h\nu}\right)^n}=\frac{1}{1-e^{-\beta h\nu}}\]
be the canonical partition function of our (classical) system, where \(\beta=(k_BT)^{-1}\), we observe that
\[
\langle E\rangle=-\partial_{\beta}\log{Z_{cl}}=\frac{h\nu e^{-\beta h\nu}}{1-e^{-\beta\nu h}}=\frac{h\nu}{e^{\beta h\nu}-1}
\]
and then we can put all together to obtain[nota]We multiply \(\mathcal{D}(\nu)\) by \(2\) in order to take into account the two directions of polarization of the electromagnetic radiation.[/nota]
\[
u(\nu,T)=\frac{1}{V}\times2\frac{V}{2\pi^2}\left(\frac{2\pi}{c}\right)^3\nu^2\times\frac{h\nu}{e^{\beta h\nu}-1}=\frac{8\pi}{c^3}\times\frac{h\nu}{e^{\beta h\nu}-1}
\]
which is the well-known Planck's formula.
At the beginning Planck was suspicious of the role of the \(h\) constant in Physics, insomuch as he considered it only a useful mathematical trick to get through his problem, but then, when Einstein used the hypothesis of "quanta of action" to explain the photoemission in metals, he realized that this constant should've had a fundamental place, as he stated in his Nobel lecture in 1918.
Now we need someone who integrates Planck's formula [nota]Which is not so trivial.[/nota] in order to obtain Stefan-Boltzmann's law so that the story can go on
\[
\int_0^{+\infty}\frac{x^3}{e^x-1}\,dx=???
\]
I really love the Mean value Theorem's formula:
Given a continuous map \( f: [a,b] \rightarrow \mathbb{R} \), which is differentiable in \( (a,b) \)
\[
\displaystyle
\exists \xi \in (a,b) : \frac{f(b)-f(a)}{b-a} = f^{\prime}(\xi)
\]
Not only for its mean-related meaning (
) but mostly for the elegance of the formula itself and for the equivalence of a "long-range" finite incremental ratio and the passage to the infinite/infinitesimal limit of a similar incremental ratio. Deep and polite. Amazing.
It can also be written as \(\displaystyle f(b)-f(a) = f^{\prime}(\xi)(b-a) \), which I find as expressive as the original formulation, but from another (and more geometrical) point of view.
Given a continuous map \( f: [a,b] \rightarrow \mathbb{R} \), which is differentiable in \( (a,b) \)
\[
\displaystyle
\exists \xi \in (a,b) : \frac{f(b)-f(a)}{b-a} = f^{\prime}(\xi)
\]
Not only for its mean-related meaning (
) but mostly for the elegance of the formula itself and for the equivalence of a "long-range" finite incremental ratio and the passage to the infinite/infinitesimal limit of a similar incremental ratio. Deep and polite. Amazing.It can also be written as \(\displaystyle f(b)-f(a) = f^{\prime}(\xi)(b-a) \), which I find as expressive as the original formulation, but from another (and more geometrical) point of view.
Ah! An another really nice equation is the Einstein equation for the Photoelettric Effect!
$E=h\nu$
The energy of a photon, the quanta of the light, is equal to the planck constant multiplied by the frequency of the incident wave!
$E=h\nu$
The energy of a photon, the quanta of the light, is equal to the planck constant multiplied by the frequency of the incident wave!
Hi!
I think that is very beautiful and also very important this equation:
$ih(partial \Psi)/(partial t) =-h^2/(2m)(partial^2 \Psi)/(partial x^2)+V(x)\Psi$ (h is equal to h-bar)
This is the schrodinger equation for the quantum wave function $Psi(x,t)$ , the importance of this equation is that resolving it we get the wave function for all $t> 0$ !
Beautiful
!
I think that is very beautiful and also very important this equation:
$ih(partial \Psi)/(partial t) =-h^2/(2m)(partial^2 \Psi)/(partial x^2)+V(x)\Psi$ (h is equal to h-bar)
This is the schrodinger equation for the quantum wave function $Psi(x,t)$ , the importance of this equation is that resolving it we get the wave function for all $t> 0$ !
Beautiful
!
"Pianoth":
$-1 = i^2 = i*i = sqrt(-1)*sqrt(-1)=sqrt((-1)*(-1))=sqrt(1)=1$
It seems to me that the equality \( \sqrt{-1} \cdot \sqrt{-1} = \sqrt{(-1) \cdot (-1)} \) doesn't hold, because \( -1 < 0 \).
"sehscharfe":
[size=85](By the way, the subject of your sentence is plural so the verb should be "have" instead of "has")[/size]
You're right, I'll correct.
"Zero87":
Very nice.
Sounds like: for two points will pass only one line but these two points has to be very very large.
Oh, well for three points will always pass one straight line...as long as it's fat enough!
[size=85](By the way, the subject of your sentence is plural so the verb should be "have" instead of "has")[/size]
"sehscharfe":
\(\displaystyle1+1=3\)
(for large values of 1)...
It's very beatiful!
"sehscharfe":
1 + 1 = 3
(for large values of 1)
...so surreal, how can you not love it?
Very nice.
Sounds like: for two points will pass only one line but these two points have to be very very large.
"Meringolo":
Well Zero,
I think that $E=mc$ is a formula that makes us reflect on the philosophical meaning of the things that surround us.
And I think most beautiful formulas are simple ones, for example $i=sqrt(-1)$
But I don't mean that the Einstein's field equation (the tensor equation that describes the geometry of space-time as a function of various parameters) is rough, on the contrary...
$R_(muv)-1/2g_(muv)R+\Lambdag_(muv)=-(8piG)/(c^4)T_(muv)$
..also, the value of lambda in Einstein's field equations is a rather messy problem.
1 + 1 = 3
(for large values of 1)
...so surreal, how can you not love it?
(for large values of 1)
...so surreal, how can you not love it?
$E = mc^2$ .
"gio73":
[quote="Zero87"][quote="Caenorhabditis"].....
I see on you profile that you are 17 years old! [/quote]
I don't believe Caenorhabditis is 17.
[/quote]
You are right, it's an approximation. Actually, I am about 17.195.
"gio73":
[quote="Zero87"]I see on you profile that you are 17 years old!
I don't believe Caenorhabditis is 17.[/quote]
I know it's hard to believe that Caenorhabditis is only 17 years old but I see in this forum many people with advanced knowledge and few years...
"Zero87":
[quote="Caenorhabditis"].....
I see on you profile that you are 17 years old! [/quote]
I don't believe Caenorhabditis is 17.
Maybe he wants to be so young; me too I want to be younger but "time runs and flies e doesn't stop an hour" like my granny said.
"Zero87":
[size=85]A formula that astonish me is the Heron's formula: if we have a triangle in which we know only the lenght of the sides, we are able to compute the area.[/size]
In elliptic and hyperbolic geometries, also the three angles are enough to determine the area.
"Caenorhabditis":
[quote="Zero87"]
I see on you profile that you are 17 years old! It's incredible that you talk of complex analysis at this age, greetings!
Don't expect me to be able to solve a problem in practice...[/quote]
But it's amazing in all case.
[size=85]A formula that astonish me is the Heron's formula: if we have a triangle in which we know only the lenght of the sides, we are able to compute the area.[/size]
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