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100 Years to Solve an Integral (2020)

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Re: 100 Years to Solve an Integral (2020)

#51

Post by abetusk »

fxj wrote: post #28
mppm wrote: post #22

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…

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/...

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#Hermite_fu...

Re: 100 Years to Solve an Integral (2020)

#52

Post by purplehat_ »

Imustaskforhelp wrote: post #50
Imustaskforhelp wrote: post #49

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…

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…

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.

Re: 100 Years to Solve an Integral (2020)

#53

Post by kevin_thibedeau »

Suppafly wrote: post #19
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?

>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 n…

Law of sines is useful when constructing things with a known angle outside of CAD.

Re: 100 Years to Solve an Integral (2020)

#54

Post by thelaxiankey2 »

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).)

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

Re: 100 Years to Solve an Integral (2020)

#55

Post by thelaxiankey2 »

Imustaskforhelp wrote: post #50
Imustaskforhelp wrote: post #49

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…

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…

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)

Re: 100 Years to Solve an Integral (2020)

#56

Post by ForOldHack »

thelaxiankey2 wrote: post #55
Imustaskforhelp wrote: post #50

Earlier quoted context omitted.

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…

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)

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.

Re: 100 Years to Solve an Integral (2020)

#59

Post by ChuckMcM »

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.

Re: 100 Years to Solve an Integral (2020)

#60

Post by perihelions »

abetusk wrote: post #51
fxj wrote: post #28

Earlier quoted context omitted.

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/...

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#Hermite_fu...

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.
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