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100 Years to Solve an Integral (2020)

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Re: 100 Years to Solve an Integral (2020)

#31

Post by GeoAtreides »

Sesse__ wrote: post #25
cl3misch wrote: post #11

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).)

speak for your own european country, in my neck of the woods (EE) we were taught and we worked with both secant and cosecant.

Re: 100 Years to Solve an Integral (2020)

#33

Post by sebzim4500 »

Sesse__ wrote: post #25
cl3misch wrote: post #11

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).)

In the UK we certainly use sec(x)

Re: 100 Years to Solve an Integral (2020)

#35

Post by oersted »

Sesse__ wrote: post #25
cl3misch wrote: post #11

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).)

They were taught to us in Spain, I suppose they don't make an appearance often, but they are perfectly familiar.

Re: 100 Years to Solve an Integral (2020)

#36

Post by Rexxar »

seanhunter wrote: post #16
scotty79 wrote: post #13

Earlier quoted context omitted.

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.

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…

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.

Re: 100 Years to Solve an Integral (2020)

#37

Post by perihelions »

mppm wrote: post #22
ziofill wrote: post #3

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.

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.

Re: 100 Years to Solve an Integral (2020)

#38

Post by mrybczyn »

anthk wrote: post #23
2b3a51 wrote: post #21

Earlier quoted context omitted.

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 cours…

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.tandfonline.com/doi/full/10.1080/00295450.2021.1...

Re: 100 Years to Solve an Integral (2020)

#40

Post by pyfon »

LegionMammal978 wrote: post #4

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:…

https://xkcd.com/977/
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