Where are you?
And I'm so sorry.
I cannot sleep, I cannot dream tonight.
I need somebody and always
This sick strange darkness
Comes creeping on so haunting every time.
I need to stop.
Mental Amusements
Where are you?
And I'm so sorry.
I cannot sleep, I cannot dream tonight.
I need somebody and always
This sick strange darkness
Comes creeping on so haunting every time.
I need to stop.
Say you are talking with someone and you want to tell her something important but not directly. You encrypt it.
How? Well we have to reject simple "shifted" ciphers, that is the ones in which you just shift the letters: $A = C$, $B = D$, and generally $\alpha = \alpha + 2$, which is a "wrong" way to write it but programmer use this way all the time (also it's the correct form to use when you write a program), but it defines that the $\alpha$-th variable is equal to itself plus two.
Here the variables are the letters, so the shift is clear. In general we might have $\alpha = \alpha + n$ provided that $n$ is such that $\alpha - (\text{mod n}) \alpha \neq 0$ otherwise you just get the original alphabet and... well not smart!
Considering that I do love cryptography and number theory too, I of course thought about a personal encryption: a "dynamical shift over the integers" (obviously, since we cannot deal with any other set of numbers, or... can we :D ). I fabricated a very interesting key (because yes, I sleep little and and I really want to think about a way to encrypt my feelings) which associated to a single modified letters, its true letter in a way that is dynamical within the word itself and within the words. For the sake of my future brain:
$$\text{ERAERAT YU KEMAN}$$
I am perhaps going nuts.
But as far as I know, to dream is not forbidden yet.
Greetings humans,
This is an experiment for me as much as it is an experiment for you, who are reading this.
The ones of you who already knows me (or "knows" me), are already familiar with the fact that I create and put challenges of physics and mathematics on my instagram account (@theartoffugue [my username changes sometimes]).
Travellers who randomly stepped into this blog, well... Just read the rows above to quickly get on the road.
The aim of this blog, which I started as a sort of conversation with myself which ended up in silence, and hoping it will have a future, is now clear: to write here the challenges I put on IG with the solutions too, in order to create a record of funny useful interesting problems and, above all, to have them written down in a clear and more suitable way (through the help of HTML code instead of my hands with pen and paper).
All the past challenges were once highlighted on IG, but I have recently removed them all so this is also a way to make a jump in the past in order to restore all them. I will indeed start with the oldest ones, and then back to the future!
So, what else to say... Let's go!
Back to basics, I want to talk about polynomials and roots.
Pages I need to store.
https://en.wikipedia.org/wiki/List_of_HTTP_status_codes
https://en.wikipedia.org/wiki/Separation_axiom
https://en.wikipedia.org/wiki/Pietro_Annigoni%27s_portraits_of_Elizabeth_II
https://it.wikipedia.org/wiki/Esistenzialismo
https://en.wikipedia.org/wiki/List_of_tests
https://it.wikipedia.org/wiki/Categoria:Figure_retoriche
https://it.wikipedia.org/wiki/Prima_lettera_ai_Corinzi
https://en.wikipedia.org/wiki/Norbert_Wiener
https://www.youtube.com/watch?v=9KnUGMHKec4
https://en.wikipedia.org/wiki/Jacques_de_Vaucanson
https://en.wikipedia.org/wiki/Machine_code
https://en.wikipedia.org/wiki/Synesthesia
https://edge.pokerlistings.it/assets/photos/cani-poker-Cassius-Coolidge.jpg?t=1407762730
https://www.arteworld.it/wp-content/uploads/2014/10/autoritratto-van-gogh-1887.png
https://it.wikipedia.org/wiki/Fenomenologia_dello_spirito
https://en.wikipedia.org/wiki/Symphony_No._7_(Beethoven)
https://en.wikipedia.org/wiki/EPA_list_of_extremely_hazardous_substances
I have decided to start a subsection (subset) of this blog by talking, and solving, the most beautiful (yes, aesthetically pleasant) integrals I have come across to during my "having fun time" periods, or generally during my studying periods. There are some of the which are really amazing, and they are amazing also because me myself and I have solved them. Sure, I don't claim to be the first one who solved them, since they are nothing special. But they are "hard enough" to make me compliment to myself for my courage and my will to attack them :)
THE ZEROTH API (Aesthetically Pleasing Integral)
$$J = \int_0^{+\infty} \dfrac{\sin(\pi x^2)}{\sinh^2(\pi x)}\ \text{d}x$$
Whose numerical result is $\dfrac{2 - \sqrt{2}}{4}$
PROOF
First of all we notice the integrand is even, hence we can write
$$\dfrac{1}{2}\int_{-\infty}^{+\infty}\dfrac{\sin(\pi x^2)}{\sinh^2(\pi x)}\ \text{d}x$$
Let's now define
$$f(z) = \dfrac{\cos(\pi z^2)}{\sinh(2\pi z) \sinh^2(\pi x)}$$
And not that because
$$f(x\pm i) = \dfrac{-\cos(\pi x^2)\cosh(2\pi x) \pm i\sin(\pi x^2)\sinh(2\pi x)}{\sinh(2\pi x)\sinh^2(\pi x)}$$
we have
$$\int_{\gamma} f(z)\ \text{d}z = \int_{-\infty}^{+\infty} \left(f(x-i) + f(x+i)\right)\ \text{d}x$$
That is $2\pi i \times \text{Sum of the residues}$, which becomes $-\dfrac{\pi}{2} \times$ Sum of residues.
The poles are at $0$ and $\pm \dfrac{i}{2}$ and $\pm i$ hence we have:
RESIDUES
If you are able enough to calculate the residues, which are rather easy, you'll find that the residue near zero is $-\dfrac{1}{2\pi}$
Instead we have $\dfrac{\sqrt{2}}{4\pi}$ near $\pm \dfrac{i}{2}$, whereas the residues around $\pm i$ are $-\dfrac{1}{2\pi}$.
Hence
$$\int_0^{+\infty} \dfrac{\sin(\pi x^2)}{\sinh^2(\pi x)}\ \text{d}x = \dfrac{\pi}{2}\left(-\dfrac{1}{2\pi} - \dfrac{1}{2\pi} + \dfrac{\sqrt{2}}{4\pi} + \dfrac{\sqrt{2}}{4\pi}\right) = \dfrac{2 - \sqrt{2}}{4}$$
As wanted.