Posted on 11/27/2007 7:00:02 PM PST by snarks_when_bored
In my space, a closed form solution is nice if you can get something for it like performance speed or whatever, but it's by no means a requirement. Usually it's better to be correct if a little sloppy and "elegant" is just the icing on the cake.
Still sounds like a fun class though.
Reminds me of Math 201 at OU in the sixties. Differential Equations for Scientists and Engineers. The prof started the class with “It’s a pleasure to see so many of you back from last semester. I fully expect to see 80% back again next semester.”
Essentially, there were three sections and two teachers. Professor Lafond taught two. You repeated the class until you got someone other than Lafond, then you passed the class.
I used to have aspirations to be an astrophysicist until I took the physics equivalant of this at Cornell. I did OK in the course but there were some students who were frighteningly brilliant. I just wasn’t in their league. It forced me to rethink my career goal.
Math knowledge has a halflife of like 4 months. I got out of the airforce and wanted to continue with Junior level math classes (300+?), and couldn’t do it. (That and the pro-fessor was using radically different syntax)
Today I couldn’t simplify a polynomial equation to save my life.
My favorite math was geometry in high school. I was behind in high school, between 10th and 11th grade, so I bought myself a summer geometry class. Its amazing how much you can learn without being burdened with 8 other different subjects all at once.
School is structured all wrong. You can’t force a kid to be interested in a subject, you can’t learn a subject unless you are interested in it.
But, thanks for the Plato quote--not surprising.
Perhaps you did not note or understand my "/sar" tag meaning that I was being sarcastic.
I noticed your /sar tag. I was just buttressing your point...
You are correct. How long did it take for you to reason this out?
It didn't require reasoning; just a bit of recollection, followed by some verification.
Sum[Cos[x*n]/n]=?, where n=1,...,Infinity.
That is,
Sum[Cos[x*n]/n]= -Ln[2*Sin[x/2]] where x is bounded between 0 and 2*Pi.
The problem is that there's no "standard" way that I'm aware of to prove this. I had to resort to some older techniques.
Yes, Versions 5.0 and later will. Earlier versions will inform the user that the seriers fails to converge. I just checked.
Also, Mathematica 6.0 doesn't yield the "nice" answer I provided unless you set x to a specific value (e.g. x=1).
In general, Mathematica 6.0 will yield:
(1/2)(-Ln[1-exp{-i*x}] - Ln[1-exp{i*x}])
Artefacts of an incomplete basis set, then...
Is there any practical interest in either differing convergence rates with the type of discontinuity, the characteristics of the partial series (e.g. if FFT shows different ringing than a conventional Fourier), or the behaviour as you include more and more terms?
...or did I just re-invent a well-known square wheel from the 1800's?
Cheers!
Cheers!
EVERYONE, and I mean EVERYONE (including historians and linguists)had to take 3 semesters of Calculus and 1 semester of Prob/ Stat.
Anyone not ready to take college calculus was in "Ranger Math."
Yes, there is a practical interest, it's called ERROR in the approximation of an infinite series. ;-)
Is there any practical interest in either differing convergence rates with the type of discontinuity, the characteristics of the partial series (e.g. if FFT shows different ringing than a conventional Fourier), or the behaviour as you include more and more terms?
The Wikipedia entry on the Gibbs phenomenon is pretty informative, g_w. As for practicalities, the entry points out that there's no overshoot/undershoot if wavelet transforms are used. Something else to study (tentatively scheduled for the spring of 2093)...
Cheers!
I’d love to take it. I would probably take me a week to understand every hour of the class, however.
I know I’m late, but here’s my stab. Since you’re summing as n goes from 1 to infinity, doesn’t that just converge to an integral over n? In that case, you can integrate Cos (nx)/n using the quotient rule, yielding something like x*(sin x + cos x).
Am I even close?
That's an odd analogy (the Survivor TV Show one). The people who dropped the course self-selected to drop. They were not "voted off the island" at all.
You should get the result that I provided in Post #68, which, of course, can be equivalently expressed as
Sum[Cos[n*x]/n]= -(1/2)*Ln[2 - 2*Cos[x]]
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