[HN Gopher] 100 Years to Solve an Integral (2020)
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100 Years to Solve an Integral (2020)
Author : blobcode
Score : 198 points
Date : 2025-04-20 03:16 UTC (19 hours ago)
(HTM) web link (liorsinai.github.io)
(TXT) w3m dump (liorsinai.github.io)
| ziofill wrote:
| 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.
| dataflow wrote:
| The 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 wrote:
| 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 regard?
| fxj wrote:
| Yes the dirac comb for example. Actually there are infinitely
| many.
|
| https://en.wikipedia.org/wiki/Dirac_comb
|
| and for other:
|
| http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/.
| ..
| abetusk wrote:
| A 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-
| th...
|
| [1] https://en.wikipedia.org/wiki/Hermite_polynomials#Hermi
| te_fu...
| perihelions wrote:
| As 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 wrote:
| 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.
| kkylin wrote:
| 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.
| JoshTriplett wrote:
| 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.
| LegionMammal978 wrote:
| If 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
| rini17 wrote:
| You probably don't live in New Zealand? (Yes I know it's there.
| Barely.)
| pyfon wrote:
| https://xkcd.com/977/
| scythe wrote:
| The 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 wrote:
| This 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.
| billab995 wrote:
| About how long it'd take me to solve the integral in my calculus
| finals.
| whatever1 wrote:
| It feels like LLMs could be good contenders for solving
| symbolically integrals. After spending some time, it really feels
| like translating between two languages.
| 331c8c71 wrote:
| Wolfram engine was taking integrals just fine way before LLMs
| were even a thing.
| anthk wrote:
| And Lisps too, fitting a sector from a disk:
|
| 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 wrote:
| Derive 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 wrote:
| 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.
| mrybczyn wrote:
| 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.tandfonline.com/doi/full/10.1080/00295450.202
| 1.1...
| anthk wrote:
| Electronics themselves _work_ by understanding
| integration.
|
| 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.
| cl3misch wrote:
| 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?
| scotty79 wrote:
| I'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 wrote:
| The secant is the reciprocal of a cosine - the hypotenuse
| over the adjacent
| anyfoo wrote:
| 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.
| grandempire wrote:
| It is a function. What do you mean?
| fiddlerwoaroof wrote:
| The weird thing about 1/cos is it's discontinuous
| wherever cos is 0 but, yes, it's a function.
| seanhunter wrote:
| I know what you mean, but as a sibling pointed out for
| everyone else's benefit, parent is using the word inverse
| where they mean reciprocal.
|
| The inverse of cosine is arccosine (sometimes written acos or
| cos^{-1}). Secant is the reciprocal of cos ie sec x =
| 1/cos(x)).
|
| Likewise cotan is the reciprocal of tan (1/tan). The inverse
| of tan is atan/arctan/tan^{-1}.
|
| This is confusing for a lot of people because if you write
| x^{-1} that means 1/x. If you write f^{-1} and f is a
| function, then _generally_ it means the inverse of f. In the
| case of trig functions this is doubly confusing because
| people write sin^2 theta meaning (sin theta)^2 but sin^-1
| theta means arcsin theta.
|
| That's why in my maths studies they started by teaching you
| to do the inverse with a -1 so when you see it you don't get
| confused but changed to preferring arcsin etc as this is
| unambiguous and if you learn to write this way you won't
| confuse others.
| Rexxar wrote:
| It does not help that both reciprocal and inverse come from
| French, and that their common meanings are reversed in
| English. I'm not sure whether the meaning of both words has
| remained constant over time in these two languages, as they
| both roughly mean "the opposite" and if you want to avoid
| ambiguity, you simply add context. For example, if you say
| "inverse function" or "multiplicative inverse" it's not
| ambiguous.
|
| Inverse function:
| https://en.wikipedia.org/wiki/Inverse_function /
| https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque
|
| Reciprocal:
| https://en.wikipedia.org/wiki/Multiplicative_inverse /
| https://fr.wikipedia.org/wiki/Inverse
|
| Wikipedia seems to have chosen "multiplicative inverse"
| over "reciprocal" for title, even though they are clearly
| indicated as synonymous.
| seanhunter wrote:
| 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
| 867-5309 wrote:
| iirc, haversine is useful for transforming 2-d "as the crow
| flies" coords to their 3-d equivalents. at longer distances a
| body's curvature is really noticeable and often overlooked
| jjgreen wrote:
| See R.W. Sinnott, "Virtues of the Haversine", Sky and
| Telescope, vol. 68, no. 2, 1984, p. 159
| Suppafly wrote:
| >Neither in (German) high school nor in the many math courses
| of a physics B.Sc. have I ever used the secant function
|
| I think we used it in geometry in US high school, but only to
| complete an assignment or two to show we could use trig
| functions correctly. I had to relearn how all of them worked to
| help my kid with homework, it's mostly look at the angles and
| sides you have available and pick which trig function is
| necessary to figure out which one you're solving for. I'm sure
| there are real life uses for trig functions, and I hate to be
| one of those "when are we ever going to use this" types, but
| I've never used any of them outside of math classes.
| kevin_thibedeau wrote:
| Law of sines is useful when constructing things with a known
| angle outside of CAD.
| Sesse__ wrote:
| 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).)
| RichardLake wrote:
| I imagine it was more useful when using tables to
| lookup/approximate the values before calculators with trig
| support were a thing.
| GeoAtreides wrote:
| speak for your own european country, in my neck of the woods
| (EE) we were taught and we worked with both secant and
| cosecant.
| sebzim4500 wrote:
| In the UK we certainly use sec(x)
| oersted wrote:
| They were taught to us in Spain, I suppose they don't make an
| appearance often, but they are perfectly familiar.
| fsckboy wrote:
| there are these old-fashioned looking drawings...
|
| (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"
| thelaxiankey2 wrote:
| the only other one is cot, actually.
|
| Personally I thought they were nice to have because coming up
| with the integral of 1/cos on the fly is pretty brutal in a
| long integral
| redbell wrote:
| Oh! This was already discussed five years ago with 77 pts and 40
| comments (https://news.ycombinator.com/item?id=24304311)
| ljsprague wrote:
| >[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 wrote:
| I 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 wrote:
| A refreshing Hacker News article after a week of repetitive
| political garbage. Thank you!
| ForOldHack wrote:
| Pure math never ceaceses to amaze, even if the title was a
| misnomer.
| ForOldHack wrote:
| spell checkers are taking a full week off.
| rurban wrote:
| I've learned it in Austrian highschool, but then from university
| on nobody needed it anymore.
| Imustaskforhelp wrote:
| 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
| 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 wrote:
| So 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_ wrote:
| Remarkably, 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.
| thelaxiankey2 wrote:
| symbolically no (in fact I believe it can be proven that it's
| impossible)
|
| numerically sure (ie definite integrals can be evaluated for
| given values)
| ForOldHack wrote:
| Which is course leads to the misnomer in the title: The
| integral was long solved by numeric means, more easily so
| with the inventions, but the proof of the solution, took a
| while... and as some other brilliant hacker-newsestition,
| pointed out, it because even easier with an ingenious
| u-substitution, related to the solution of the integral of
| 1/x discovered in the late 1930s...
|
| ( The solution is both possible, and proved, and there is a
| goddamned youtube video about the trick, and its not a
| minor trick either, like the proofs of int (sec (x)) or int
| (1/x ). )
|
| In my text book, and current text books, it is said it
| cannot be resolved by elementary means, and it cannot, but
| it can be solved, and proven by one whopper of an idea.
|
| The research is left as an exercise.
| ChuckMcM wrote:
| It 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 wrote:
| This is also the inverse Gudermannian function [1]. That
| Wikipedia page has some nice geometrical insights.
|
| [1] https://en.m.wikipedia.org/wiki/Gudermannian_function
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