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math challenge

narrowbandpaul

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skye at night

skye at night

Posted
Posted

S@N, well well well what were these science degrees and why do u feel that paul should not work in science??

I feel ur agreements about admin etc are very funny, given Pauls pervious history with some people in here and the fact u seem to be trying to be a more arrogant *&^(*^& than him

How can u be soooo sure of ur high status in this thread when u know nothing of who u are speaking too

Sorry...

Me no speaky text!

(Oh No! Narrowmind Paul's gone and got his Mum):icon_eek:

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George Jones

George Jones

Posted
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Not much of a challenge

I have to disagree here; at least for me, it was a challenge. My first reaction was to try and integrate the thing by hand. After a substitution, I thought a further integration by parts might lead to an elliptic integral. But I was wrong.

Then, it was fun think about some of the pure maths concepts that might apply.

So the real challenge is to present an interesting real-world problem in which you might actually need to integrate cos(sinx).

Again, I have to disagree. This depends on what Paul meant as the challenge. Maths problems/concepts/solutions are interesting to some people as maths problems/concepts/solutions. But not interesting to everyone. I am interested in visual astronomy and have no interest in imaging. Lots of people's interests are the other way round. Makes life interesting.

A real-world problem involving the integral might be interesting, but it is not necessarily the "real challenge."

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skye at night

skye at night

Posted
Posted

Well Paul..

If you look back at the thread..

the personal (ish) stuff began with YOU singling me out for your aggression..

by selectively (which you are entitled to do) picking out my particular humorous (well I thought so anyway) post as 'disagreeable' while the other humorous replies were OK for you (again you are entitled to do this)..

However.. you should be prepared to take a bit back and not 'go-off-on-one' and get your mates involved just because you hear something you don't like...

I can assure you that in future I will simply avoid any threads that you start and perhaps any that you have a 'heavy' involvement in. Perhaps you might do the same.

I do this having now experienced your aggression on more than one occasion and having also been warned of your 'previous' by more than one forum member..

This should hopefully avoid a further clash of personalities (:icon_eek:)

Thanks

Steve

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acey

acey

Posted
Posted

Calculating the Airy disc - that's an interesting real-world problem. And I hadn't realised it involves Bessel functions, and hence cos(sinx). So thanks for educating me with regard to that.

This paper might be the sort of thing you're looking for:

http://www.iop.org/EJ/article/0959-5309/55/2/305/prv55i2p116.pdf?request-id=6239d21b-21e7-44ec-9eda-8d02672f424c

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George Jones

George Jones

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This all start when i posed the question of the nature of gaussian beams to paul this morning, they have an approximation to the data made of two seperate equations one for near field the other far.

Are the two separate equations derived using completely different methods, possibly a convergent power series for the near and a divergent asymptotic expansion for the far?

Very roughly, a truncated portion of a convergent power series for a function f(x) for fixed x gives a better approximation as the number of terms n increases. This often works well as long as one doesn't stray too far from the point of expansion.

Again very roughly, an asymptotic series for fixed n gives a better approximation as x increases.

Here's a story from one of my mathematical physics books,

"Airy himself, using a convergent series expansion of Ai at 0, managed fairly easily to compute the position of the first band, and with considerable difficulty, found the second one. ... Stokes used instead the 'devlish' method of divergent series and, after bypassing some non-trivial difficulties, obtained all the bands with a prercision of 10-4."

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David Galvin

David Galvin

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He's asked for a formula that describes the area under a curve defined by the relation y=cos(sin(x)) where x,y are cartesian coordinates.

If one doesn't exist, then the only way to calculate the area is to number-crunch it by splitting the area into very thin, almost rectangular, strips and add the little areas all up.

erm........Newton Raphson method...ooh where did that come from. My brain is hurting now.:icon_eek:

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Jamie

Jamie

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jamie, I named the thread math challenge, and I kept it short using standard mathematical terminology (a bit waffly with analytic)

clearly a math challenge is going to be difficult and if maths isnt your thing then this thread isnt suited to you.

so really cant make it dumbed down.

Point taken...:icon_eek:

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narrowbandpaul

narrowbandpaul

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ashenlight...sssshhhhhh dont tell anyone, but the math mentioned here is beyond me!

still thats what wikipedia is for.....

but seriously guys, thanks for the replys, I enjoy reading about the maths that I dont yet know, but hope to someday when I have time.

i do really enjoy maths.

was that obvious?

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661-pete

661-pete

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In reading (or trying to!) this thread I get the sense of all the stuff I've forgotten: in my student days I recall a project in thermodynamics, analysis of heat flow through a kind of heat exchanger, that got me tangled up in Bessel functions too (I think they were of the second kind). Means nothing whatever to me now, I'm afraid, but that's down to braincell degeneration not a dumbing-down obsession. :icon_eek:

But one thing occurs to me. Your function cos(sinx) looks like a sort of 'special case' of a more general class of functions which seem to have rather odd properties. Why not tackle the more general case cos(a*sin(x)) where a is a constant? cos(pi * sin(x)) might be an interesting one?

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FLO

FLO

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Posted

I enjoy the 'neatness' of maths when it solves a particular problem but my brain doesn't revel in it. I guess some here see poetry at work when mathematics confirms planetary orbits and the like but my brain isn't wired that way. Life would be dull if we were all the same :icon_eek:

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themos

themos

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Why not tackle the more general case cos(a*sin(x)) where a is a constant? cos(pi * sin(x)) might be an interesting one?

If the special case for a=1 isn't expressible in terms of elementary functions then the general case certainly wouldn't be. The Risch algorithm accepts parameters so it would have found a formula if one existed.

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MG1

MG1

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Paul (or indeed others)

could you suggest any accessible reading material got somone who did A-level maths and physics and is interested in getting back into appreciating and understanding some of these abstract problems.

Thanks

Michael

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