[math-fun] What's the proof?
Did you mean that "." at the end of the LHS to be "..."? Jim Propp On Thu, Apr 16, 2015 at 11:23 PM, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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Jim, you have only begun to nitpick. You forgot to carp about the period after sin(x), which is redundant, since the equal sign already indicates an independent clause. --Dan
On Apr 16, 2015, at 8:38 PM, James Propp <jamespropp@gmail.com> wrote:
Did you mean that "." at the end of the LHS to be "..."?
Jim Propp
On Thu, Apr 16, 2015 at 11:23 PM, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
I just tossed that one off before I went to bed. Sorry I neglected to adhere to the international standards for ASCII mathematics. -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Thursday, April 16, 2015 11:43 PM To: math-fun Subject: Re: [math-fun] What's the proof? Jim, you have only begun to nitpick. You forgot to carp about the period after sin(x), which is redundant, since the equal sign already indicates an independent clause. --Dan
On Apr 16, 2015, at 8:38 PM, James Propp <jamespropp@gmail.com> wrote:
Did you mean that "." at the end of the LHS to be "..."?
Jim Propp
On Thu, Apr 16, 2015 at 11:23 PM, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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David, I hope you realize I was not nitpicking at all. Just pointing out more *potential* nitpicking that *could have been* inflicted on you, had anyone wished to be that cruel. Which fortunately they weren't. --Dan
On Apr 17, 2015, at 2:11 PM, David Wilson <davidwwilson@comcast.net> wrote:
I just tossed that one off before I went to bed. Sorry I neglected to adhere to the international standards for ASCII mathematics.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Thursday, April 16, 2015 11:43 PM To: math-fun Subject: Re: [math-fun] What's the proof?
Jim, you have only begun to nitpick. You forgot to carp about the period after sin(x), which is redundant, since the equal sign already indicates an independent clause.
--Dan
On Apr 16, 2015, at 8:38 PM, James Propp <jamespropp@gmail.com> wrote:
Did you mean that "." at the end of the LHS to be "..."?
Jim Propp
On Thu, Apr 16, 2015 at 11:23 PM, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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I meant it as humor, but I does come off as rather snippy. Apologies. This was something I worked out in my head staring off into the darkness. -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Friday, April 17, 2015 5:15 PM To: math-fun Subject: Re: [math-fun] What's the proof? David, I hope you realize I was not nitpicking at all. Just pointing out more *potential* nitpicking that *could have been* inflicted on you, had anyone wished to be that cruel. Which fortunately they weren't. --Dan
On Apr 17, 2015, at 2:11 PM, David Wilson <davidwwilson@comcast.net> wrote:
I just tossed that one off before I went to bed. Sorry I neglected to adhere to the international standards for ASCII mathematics.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Thursday, April 16, 2015 11:43 PM To: math-fun Subject: Re: [math-fun] What's the proof?
Jim, you have only begun to nitpick. You forgot to carp about the period after sin(x), which is redundant, since the equal sign already indicates an independent clause.
--Dan
On Apr 16, 2015, at 8:38 PM, James Propp <jamespropp@gmail.com> wrote:
Did you mean that "." at the end of the LHS to be "..."?
Jim Propp
On Thu, Apr 16, 2015 at 11:23 PM, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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On 17/04/2015 04:23, David Wilson wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
The LHS is an entire function (analytic on the whole of the complex plane). To prove this it suffices that the sum of |1-cos(x/2^n)| be locally uniformly convergent, which it is because for x in any compact region the series looks like a convergent g.p. once n is large enough. It further follows that the only zeros of the LHS are those of its factors -- i.e., simple zeros at 0 (from the initial x) and at odd multiples of 2^n pi/2 (from cos x/2^n). So the LHS is an entire function with simple zeros at all integer multiples of pi and nowhere else. And so is the RHS. Therefore the ratio of the two sides is an entire function with no zeros and no poles. Or, to say that in simpler terms, a constant. If x is extremely small then the LHS is x + o(x). So is the RHS. So the constant ratio is in fact 1, and we're done. -- g
Is there a Greek-style geometric proof that interprets the nth partial product (after some minor adjustments perhaps) as the perimeter of a regular 2^n-gon inscribed in the unit circle? Jim Propp On Thursday, April 16, 2015, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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Yes! Think about the ratio between the length of an arc and the straight cord that connects its ends. Very nice! - Cris On Apr 17, 2015, at 6:45 AM, James Propp <jamespropp@gmail.com> wrote:
Is there a Greek-style geometric proof that interprets the nth partial product (after some minor adjustments perhaps) as the perimeter of a regular 2^n-gon inscribed in the unit circle?
Jim Propp
On Thursday, April 16, 2015, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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I meant chord, of course... On Apr 17, 2015, at 9:08 AM, Cris Moore <moore@santafe.edu> wrote:
Yes! Think about the ratio between the length of an arc and the straight cord that connects its ends.
Very nice! - Cris
On Apr 17, 2015, at 6:45 AM, James Propp <jamespropp@gmail.com> wrote:
Is there a Greek-style geometric proof that interprets the nth partial product (after some minor adjustments perhaps) as the perimeter of a regular 2^n-gon inscribed in the unit circle?
Jim Propp
On Thursday, April 16, 2015, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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You lost your thread? WFL On 4/17/15, Cris Moore <moore@santafe.edu> wrote:
I meant chord, of course...
On Apr 17, 2015, at 9:08 AM, Cris Moore <moore@santafe.edu> wrote:
Yes! Think about the ratio between the length of an arc and the straight cord that connects its ends.
Very nice! - Cris
On Apr 17, 2015, at 6:45 AM, James Propp <jamespropp@gmail.com> wrote:
Is there a Greek-style geometric proof that interprets the nth partial product (after some minor adjustments perhaps) as the perimeter of a regular 2^n-gon inscribed in the unit circle?
Jim Propp
On Thursday, April 16, 2015, David Wilson <davidwwilson@comcast.net> wrote:
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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sin(x) = 2 cos(x/2) sin(x/2) = 4 cos(x/2) cos(x/4) sin(x/4) = 2^n cos(x/2) ... cos(x/2^n) sin(x/2^n). In the limit, 2^n sin(x/2^n) = x. -- Gene From: David Wilson <davidwwilson@comcast.net> To: 'math-fun' <math-fun@mailman.xmission.com> Sent: Thursday, April 16, 2015 8:23 PM Subject: [math-fun] What's the proof? x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
That identity is at the start (page 1, line 1) of one of the true great math books, Mark Kac's Statistical Independence in Prob., Analysis and Number tTheory. He attributes it to Vieta. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, Apr 17, 2015 at 11:05 AM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
sin(x) = 2 cos(x/2) sin(x/2) = 4 cos(x/2) cos(x/4) sin(x/4) = 2^n cos(x/2) ... cos(x/2^n) sin(x/2^n). In the limit, 2^n sin(x/2^n) = x. -- Gene
From: David Wilson <davidwwilson@comcast.net> To: 'math-fun' <math-fun@mailman.xmission.com> Sent: Thursday, April 16, 2015 8:23 PM Subject: [math-fun] What's the proof?
x*cos(x/2)*cos(x/4)*cos(x/8)*. = sin(x).
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I would like to second Neil's recommendation of this book. It is indeed a great book, totally readable and totally fascinating. I seem to recall it arose from a series of lectures Kac gave at Haverford College. --Dan
On Apr 17, 2015, at 8:11 AM, Neil Sloane <njasloane@gmail.com> wrote:
That identity is at the start (page 1, line 1) of one of the true great math books, Mark Kac's Statistical Independence in Prob., Analysis and Number Theory. He attributes it to Vieta.
For anyone motivated to dip in non-committally --- http://download.pdf5.org/s/statistical-independence-in-probability,-analysis... WFL On 4/17/15, Dan Asimov <asimov@msri.org> wrote:
I would like to second Neil's recommendation of this book. It is indeed a great book, totally readable and totally fascinating.
I seem to recall it arose from a series of lectures Kac gave at Haverford College.
--Dan
On Apr 17, 2015, at 8:11 AM, Neil Sloane <njasloane@gmail.com> wrote:
That identity is at the start (page 1, line 1) of one of the true great math books, Mark Kac's Statistical Independence in Prob., Analysis and Number Theory. He attributes it to Vieta.
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participants (8)
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Cris Moore -
Dan Asimov -
David Wilson -
Eugene Salamin -
Fred Lunnon -
Gareth McCaughan -
James Propp -
Neil Sloane