[math-fun] IMPORTUNATE PERMUTATION
While flagitating some kids with my usual sutra of always replacing Bernoulli numbers with Bernoulli polynomials in some new variable, I derived (c47) intosum(sum(2*(-1)^i*x^(2*i-1)*bernpoly(y,2*i)/(2*i)!,i,0,inf) = sin(x*y)+cot(x/2)*cos(x*y)) inf ==== i 2 i - 1 \ 2 (- 1) x bernpoly(y, 2 i) (d47) > ---------------------------------- = / (2 i)! ==== i = 0 x sin(x y) + cot(-) cos(x y) 2 and (c48) intosum(sum(2*(-1)^i*x^(2*i)*bernpoly(y,2*i+1)/(2*i+1)!,i,0,inf) = cot(x/2)*sin(x*y)-cos(x*y)) inf ==== i 2 i \ 2 (- 1) x bernpoly(y, 2 i + 1) (d48) > ---------------------------------- = / (2 i + 1)! ==== i = 0 x cot(-) sin(x y) - cos(x y) 2 generalizing the expansions of cot(x/2) and -+1(!) respectively. Putting y=0 and y=1 in the first one must be equal, giving the cot halfangle formula. --rwg COURT PAINTERS PERSCRUTATION
On 4/21/10, Bill Gosper <billgosper@gmail.com> wrote:
While flagitating some kids with my usual sutra of always replacing Bernoulli numbers with Bernoulli polynomials in some new variable, ...
Don't people get locked up for that sort of thing these days? (Or is that only in Switzerland ...)
Putting y=0 and y=1 in the first one must be equal, giving the cot halfangle formula. --rwg
Maybe should have read "y=0 and y=pi" ? A university lecturer of yore, while teaching a course in complex analyis, was alleged to have become so tired of writing 1/(2 pi) in front of integrals, that he took to announcing at the start of the course that 2 pi = 1. Even more drastic than decreeing pi = 3, methinks ... WFL
On Wednesday 21 April 2010 17:24:11 Fred lunnon wrote:
A university lecturer of yore, while teaching a course in complex analyis, was alleged to have become so tired of writing 1/(2 pi) in front of integrals, that he took to announcing at the start of the course that 2 pi = 1.
I always thought that the "correct" representation of angles on a computer was as a fixed-point fraction with the binary point all the way to the left. So 2 pi would indeed be 1 (mod 1) = 0. --ms
="Fred lunnon" <fred.lunnon@gmail.com> A university lecturer of yore, while teaching a course in complex analyis, was alleged to have become so tired of writing 1/(2 pi) in front of integrals, that he took to announcing at the start of the course that 2 pi = 1.
Or ape the notation for Planck's constant and "bar" the integral sign a tad?
Particle physicists have for years set units to hbar=c=1. This makes relativistic quantum mechanics calculations much easier. But, in this case, the redefinition actually makes some logical sense, unlike the pi case. Rowan. On Wed, Apr 21, 2010 at 5:04 PM, Marc LeBrun <mlb@well.com> wrote:
="Fred lunnon" <fred.lunnon@gmail.com> A university lecturer of yore, while teaching a course in complex analyis, was alleged to have become so tired of writing 1/(2 pi) in front of integrals, that he took to announcing at the start of the course that 2 pi = 1.
Or ape the notation for Planck's constant and "bar" the integral sign a tad?
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This reminded me of an item from Littlewood's Miscellany: The spoken word has dangers. A famous lecture was unintelligible to most of its audience because ' Harnoo ', clearly an important character in the drama, failed to be identified in time as hv. On Wed, Apr 21, 2010 at 5:04 PM, Marc LeBrun <mlb@well.com> wrote:
="Fred lunnon" <fred.lunnon@gmail.com> A university lecturer of yore, while teaching a course in complex analyis, was alleged to have become so tired of writing 1/(2 pi) in front of integrals, that he took to announcing at the start of the course that 2 pi = 1.
Or ape the notation for Planck's constant and "bar" the integral sign a tad?
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participants (6)
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Bill Gosper -
Fred lunnon -
Marc LeBrun -
Mike Speciner -
Rowan Hamilton -
Thane Plambeck