6 ms·
100 Years to Solve an Integral (2020)
- rurban 1y agoI've learned it in Austrian highschool, but then from university on nobody needed it anymore.
- deleted 1y ago[deleted]
- ziofill 1y agoI 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.
- dataflow 1y agoThe impulse train is another well-known one, though I suppose someone will chime in here to rebut that it's arguably not a function.
- mppm 1y agoLearned 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 regard?
- fxj 1y agoYes the dirac comb for example. Actually there are infinitely many. https://en.wikipedia.org/wiki/Dirac_comb https://en.wikipedia.org/wiki/Dirac_comb and for other: http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/PPVIETEeigenFT.pdf http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...
- abetusk 1y agoA question I hadn't even thought to ask, thanks. So, basically, the eigenfunctions of the Fourier transform are Hermite polynomials times a Gaussian [0] [1]. [0] https://math.stackexchange.com/questions/728670/functions-that-are-their-own-fourier-transform https://math.stackexchange.com/questions/728670/functions-th... [1] https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions_as_eigenfunctions_of_the_Fourier_transform https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_fu...
- perihelions 1y agoAs well as the linear combinations (including infinite sums!) of Hermite functions with the same eigenvalue under the Fourier transform. (Those eigenvalues are infinitely degenerate). You could express sech(x) as such a sum.
- perihelions 1y agoThere 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.
- kkylin 1y agoIndeed, 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.
- deleted 1y ago[deleted]
- JoshTriplett 1y agoIt 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.
- deleted 1y ago[deleted]
- LegionMammal978 1y agoIf we're playing the map-projection-advocacy game, I'd say the Mollweide projection is underrated among equal-area maps [0]. (For local maps, use whatever you want, appropriately centered.) Sure, it distorts shapes away from the central meridian, but locally it only adds a simple horizontal skew. I'm not a big fan of how many equal-area 'compromise' projections lie about how long the lines of latitude are. [0] https://en.wikipedia.org/wiki/Mollweide_projection https://en.wikipedia.org/wiki/Mollweide_projection
- scythe 1y agoThe most elegant proof IMHO is the one that avoids the original problem entirely. Int[csc(x) dx] = 2 Int[csc(2u) du] = 2 Int[du / (2 cos(u) sin(u))] = Int[sec^2(u) du / tan(u)] = log(tan(u)) + C = log(tan(x/2)) + C Then Int[sec(x)] = Int[csc(u)] = log(tan(u/2)) + C = log(tan(pi/4 - x/2)) + C. Of course, this was no use to Mercator, because the logarithm hadn't been invented yet. But you aren't just pulling a magic factor out of nowhere. There is definitely a bit of cleverness in rearranging the fraction — you have to be used to trying to find instances of the power rule when dealing with integrals of fractions.
- qbane 1y agoThis was the one I was taught in my high school. It has some cleverness (e.g., some trig. transformations) but looks less like coming out of nowhere than the original.
- caminanteblanco 1y ago[flagged]
- billab995 1y agoAbout how long it'd take me to solve the integral in my calculus finals.
- whatever1 1y agoIt feels like LLMs could be good contenders for solving symbolically integrals. After spending some time, it really feels like translating between two languages.
- 331c8c71 1y agoWolfram engine was taking integrals just fine way before LLMs were even a thing.
- anthk 1y agoAnd Lisps too, fitting a sector from a disk: https://justine.lol/sectorlisp2/ https://justine.lol/sectorlisp2/ And probably a small forth too, with a dictionary defining every math word, something not so different to Lisp. LLM's? 4GB of RAM? Your grampa's 486 with 16MB of RAM can do calculus too.
- 2b3a51 1y agoDerive 2 for Dos. Green Screen 286 I think or 386 computers in a small side room. Later Windows version was better. Then there was the DOS version of Minitab 5 I think that came as floppy disks in the back of a spiral bound book which I used to generate data sets for students to process for homework so everyone got a slightly different sample. You can do a lot of numerical maths just with a noddy spreadsheet of course.
- anthk 1y agoMacsyma, PDP10 + ITS under Maclisp. https://en.m.wikipedia.org/wiki/PDP-10 https://en.m.wikipedia.org/wiki/PDP-10 https://en.m.wikipedia.org/wiki/Incompatible_Timesharing_System https://en.m.wikipedia.org/wiki/Incompatible_Timesharing_Sys... https://en.m.wikipedia.org/wiki/Macsyma 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.
- cl3misch 1y agoNeither 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?
- scotty79 1y agoI'm sure you used inverse of a cosine multiple times. Didactic math today is just not bothering to give it a name. Probably because people think that sin, cos and tan is enough. Even ctg which is just inverse of tan is often skipped.
- asplake 1y agoThe secant is the reciprocal of a cosine – the hypotenuse over the adjacent
- anyfoo 1y agoThat’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.
- grandempire 1y agoIt is a function. What do you mean?
- fiddlerwoaroof 1y agoThe weird thing about 1/cos is it’s discontinuous wherever cos is 0 but, yes, it’s a function.
- anyfoo 1y ago
- redbell 1y agoOh! This was already discussed five years ago with 77 pts and 40 comments (https://news.ycombinator.com/item?id=24304311 https://news.ycombinator.com/item?id=24304311)
- ljsprague 1y ago>[the Mercator projection] unnecessarily distorts shapes and in particular makes the Americas and Europe look much larger than they actually are. This has been linked, not without rational, to colonialism and racism. The fact that on many maps Europe is much smaller that it appears should just make you all the more impressed by its achievements.
- charlieyu1 1y agoI remember teaching integral of sec x to high schoolers with multiplication of sec x + tan x. I mean it is not obvious but it is not like something that would take 100 years. And the author talks like logarithm was invented long after integration
- glimshe 1y agoA refreshing Hacker News article after a week of repetitive political garbage. Thank you!
- ForOldHack 1y agoPure math never ceaceses to amaze, even if the title was a misnomer.
- ForOldHack 1y agospell checkers are taking a full week off.
- Imustaskforhelp 1y agoDude 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 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.
- Imustaskforhelp 1y agoSo I just started reading the article and it seems that it mentions a point about teachers telling their students to verify it by differentiating the value of integral of secx ie. ln(| tan x + secx|) and it equals secx 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?
- purplehat_ 1y agoRemarkably, there isn’t a way to solve most integrals symbolically. We say that the set of “elementary functions”, i.e. ordinary looking symbolic functions, is not closed under integration. Even if you try to add special functions in you cannot feasibly make it closed under integration. I’ll try to write something more detailed later but in the meantime you should look up Liouville’s theorem and non-elementary antiderivatives.
- ChuckMcM 1y agoIt amuses me that doing software and hardware engineering for decades and never once thinking about trigonometric functions other than perhaps sine and cosine, and then I get interested in software defined radio and find myself running into all of the functions! That's especially true with the discreet mathematics that SDR uses.
- jwmerrill 1y agoThis is also the inverse Gudermannian function [1]. That Wikipedia page has some nice geometrical insights. [1] https://en.m.wikipedia.org/wiki/Gudermannian_function https://en.m.wikipedia.org/wiki/Gudermannian_function