mersenneforum.org Does Anyone Know how to Simplify the Following Expression?
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 2004-03-26, 11:29 #1 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts Does Anyone Know how to Simplify the Following Expression? x - sqrt(x^2 - 1) Thanks
 2004-03-26, 13:25 #2 antiroach     Jun 2003 22·61 Posts as x gets very large you get 0 :)
 2004-03-26, 14:00 #3 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts I mean, an exact simplification.
 2004-03-30, 00:42 #4 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 22·3·641 Posts This is almost certainly not what you want, and is not exactly a simplification, but "Factoris" at http://wims.unice.fr/~wims/wims.cgi?...ebra/factor.en gives: The factorization of x-(x2-1)0.5 = (0 - i) + x + (0 + 0.5i) x2 + (0.0 + 0.125i) x4 + (0.0 + 0.0625i) x6 + O(x8) ;-) Last fiddled with by cheesehead on 2004-03-30 at 00:45
 2004-03-30, 02:41 #5 rogue     "Mark" Apr 2003 Between here and the 172416 Posts It appears to be something out of trigonometry, so I looked it up and found: arccosh(x) = ln(x - sqrt(x² - 1)) so e^arccosh(x) = x - sqrt(x² - 1)
 2004-03-30, 09:35 #6 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts Thanks for the help, but rogue's formula doesn't work for x = 2.0996011
 2004-03-30, 14:24 #7 rogue     "Mark" Apr 2003 Between here and the 172416 Posts I don't have a calculator that can do arccosh. What do both terms evaluate to?
 2004-03-30, 14:42 #8 jinydu     Dec 2003 Hopefully Near M48 2×3×293 Posts LHS ~= 3.945765984 RHS ~= 0.2534362159
 2004-03-30, 17:40 #9 rogue     "Mark" Apr 2003 Between here and the 172416 Posts Oddly, the website I found this in has '-', but others have '+'. Using '+' instead of '-', you get 1) arccosh(x) = ln(x + sqrt(x^2 - 1)) 2) e^arccosh(x) = x + sqrt(x^2 - 1) 3) x - sqrt(x^2 - 1) = (x - sqrt(x^2 - 1)) * (x + sqrt(x^2 - 1))/(x + sqrt(x^2 - 1)) 4) x - sqrt(x^2 - 1) = (x^2 - (x^2 - 1))/(x + sqrt(x^2 - 1)) 5) x - sqrt(x^2 - 1) = 1/(x + sqrt(x^2 - 1)) 6) x - sqrt(x^2 - 1) = 1/(e^arccosh(x)) 7) x - sqrt(x^2 - 1) = e^-arccosh(x) I suspect that e^-arccosh(x) might be simplified as well, but I'll leave that for the next person. Last fiddled with by rogue on 2004-03-30 at 17:42
2004-04-02, 01:03   #10

"Richard B. Woods"
Aug 2002
Wisconsin USA

11110000011002 Posts

Quote:
 Originally Posted by jinydu Thanks for the help, but rogue's formula doesn't work for x = 2.0996011 . . . LHS ~= 3.945765984 RHS ~= 0.2534362159
Did you forget to try the negative square root on the RHS? :)
Using that one, my calculator shows LHS = RHS = 3.94576598405592039632709017264425

Alternatively, you can try the negative arccosh with the positive square root, in which case LHS = RHS = 0.253436215944079603672909827355751

Last fiddled with by cheesehead on 2004-04-02 at 01:11

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