41–50 of 66 posts
I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.
Makes sense, given that the definition of e goes hand in hand with its property of e^x being its own integral and derivative.
Earlier quoted context omitted.
That’s right, it’s a distribution. And that fact has me, a non-mathematician, personally caused some huge headaches, because I thought I could treat it just like a function… Yeah, turns out really weird things happen if you try to do so without knowing what you’re doing. For example, taking its square does not make sense.
It is a function. What do you mean?
I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.
It still involves e, though: sech(x) = 2 * e^x / (e^(2x) + 1) Makes sense, given that the definition of e goes hand in hand with its property of e^x being its own integral and derivative.
Earlier quoted context omitted.
Learned something new today, thank you! If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that…
There has to be a link to the harmonic oscillator here. That's the Hamiltonian that's symmetric under exchange of position and momentum, and the Hermite functions are its eigenfunctions.
Earlier quoted context omitted.
There has to be a link to the harmonic oscillator here. That's the Hamiltonian that's symmetric under exchange of position and momentum, and the Hermite functions are its eigenfunctions.
Indeed, the (quantum) harmonic oscillator Hamiltonian (with suitable scalings) commutes with the Fourier transform. Since the former has the Hermite functions as eigenbasis, the Hermite functions also form an eigenbasis for the latter.
Earlier quoted context omitted.
Macsyma, PDP10 + ITS under Maclisp. https://en.m.wikipedia.org/wiki/PDP-10 https://en.m.wikipedia.org/wiki/Incompatible_Timesharing_Sys... https://en.m.wikipedia.org/wiki/Macsyma Fun fact: old Macsyma's math code still runs at is on modern Linux'/BSD's with Maxima. Even plots work the same, albeit in a different output format. A 386 it's far more powerful than this.
At the 1940s Manhattan project, back when computer meant a job: "person who computes mathematical statements", major advancements were made in the integration of hyperbolic PDEs, by substituting electro-mechanical and then vacuum-tube machines to do the job. You know, those hard-wired vacuum tube monsters like ENIAC. You could argue that the First useful thing electronic computers did was integration... https://www.t…
It's full circle. But with Lisp and Lambda Calculus even an Elementary school kid could understand integration, as you are literally describing the process as if they were Lego blocks.
Albeit in Forth would be far easier. It's almost telling the computer that multiplying it's iterated addition, and dividing, iterated substraction.
Floating numbers are done with specially memory 'blocks', and you can 'teach' the computer to multiply numbers bigger than 65536 in the exact same way humans do with pen and paper.
Heck, you can set float numbers by yourself by telling Forth how to do the float numbers by following the standard and setting up the f, f+, f/... and outputting rules by hand. Slower than a Forth done in assembly? Maybe, for sure; but natively, in old 80's computers, Forth was 10x faster than Basic.
From that to calculus, it's just telling the computer new rules*. And you don't need an LLM for that.
Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?
Trig is full of functions that fall into disuse and are forgotten. For example "versine" versin theta = 1-cos theta. There is also "haversine" which is (1-cos theta)/2. Which is used in navigation apparently https://en.wikipedia.org/wiki/Versine
Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?
It's a US thing. Europeans just write 1/cos(x) instead of treating it as a special thing with its own name. The Americans have sec, csc, and a bunch of others I never bothered to learn. It doesn't seem to add all that much to me? (Of course, it's a bit hypocritical since I gladly use tan(x).)
(quick search, didn't find the old ones, but similar to these)
https://mathematicaldaily.weebly.com/secant-cosecant-cotange...
https://www.pinterest.com/pin/enter-image-description-here--...
... which were not used in my education but whenever i saw them i wished they had been, they lay out a geometric interpretation of all of them. by "old" i mean "look like Leonardo drew them"
Basically our sir told us to multiply / divide by sec + tan and observe that its becoming something like integration f(x)^(-1) f'(x) * dx and if we let f(x) as t and this f'(x) * dx becomes dt Actually we can also prove the latter and I had to look at my notes because I haven't revised them yet but its basically f(x) = t
so f'(x) = dt/dx so f'(x)* dx = dt then we get
so integration f(x)^n * f'(x) * dx = integral t^n * dt (where t = f(x)) integral t^-1 dt so we get ln(t) and this t or f(x) was actually sec x + tan x so its ln(sec + tan) and in fact by doing some cool trigonometry we can say this as ln(tan(pi/4 + x/2)) + c
also cosec x integration is ln(tan(x/2)) + c
I haven't read the article but damn, HN, this feels way too specific for me LOL.
Dude I am not joking but today was the day that we were introduced to indefinite integration as a formal chapter in maths at my coaching and we did secx integration. Basically our sir told us to multiply / divide by sec + tan and observe that its becoming something like integration f(x)^(-1) f'(x) * dx and if we let f(x) as t and this f'(x) * dx becomes dt Actually we can also prove the latter and I had to look at my…
and in fact our sir himself told us that he would've also let us do this if we were in normal batches (we are in a slightly higher batch, but most students are still normal and it was easy to digest to be honest except when I was writing this previous comment, I actually found that our sir had complicated the step of f'(x) = df(x)/dx by letting us assume f(x) as t and so on..,maybe it makes it easier to understand considering f(x) to be its own variable like t instead, but that actually confused me a little bit when I was writing the previous comment) , still nothing too hard.
I actually want to ask here because I was too afraid to ask this to sir, but is there a way, a surefire way to solve any integral , like can computers solve any integral?