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(Yes, Henry, ln phi = asinh 1/2 .-)
I'm not even managing to clamber in on the ground floor here. _Why_ is it true?
Lessee: sinh(x) := (e^x - e^-x)/2, so one branch (the usual one) of the inverse function is asinh(y) = ln(x + sqrt(x^2 + 1). So asinh(1/2) = ln( 1/2 + sqrt(5/4) ) = ln(phi). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On Sat, Jul 26, 2008 at 10:42 PM, Dan Asimov <dasimov@earthlink.net> wrote: [...]
Lessee:
sinh(x) := (e^x - e^-x)/2,
so one branch (the usual one) of the inverse function is
asinh(y) = ln(x + sqrt(x^2 + 1).
So asinh(1/2) = ln( 1/2 + sqrt(5/4) ) = ln(phi).
--Dan
More surprisingly (13/17)ln(phi) = 14700630551 / 39948938261, a ratio of two primes:)
I humbly beg to disagree, most of us knew ;-)) that James' thing was only a crude approximation: ArcCsch[2] - (14700630551 *17 / 39948938261 /13) = ~ 1.087951735001*10^-24 Logs of surds seldom go rational (?) W. ----- Original Message ----- From: "Fred lunnon" <fred.lunnon@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Sunday, July 27, 2008 8:54 PM Subject: Re: [math-fun] dilog puzzle
On 7/27/08, James Buddenhagen <jbuddenh@gmail.com> wrote:
More surprisingly (13/17)ln(phi) = 14700630551 / 39948938261, a ratio of two primes:)
Of which one might indubitably observe "not many people know that" ...
WFL
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