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Old 2004-03-26, 11:29   #1
jinydu
 
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Default Does Anyone Know how to Simplify the Following Expression?

x - sqrt(x^2 - 1)

Thanks
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Old 2004-03-26, 13:25   #2
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as x gets very large you get 0 :)
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Old 2004-03-26, 14:00   #3
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I mean, an exact simplification.
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Old 2004-03-30, 00:42   #4
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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
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Old 2004-03-30, 02:41   #5
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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)
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Old 2004-03-30, 09:35   #6
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Thanks for the help, but rogue's formula doesn't work for x = 2.0996011
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Old 2004-03-30, 14:24   #7
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I don't have a calculator that can do arccosh. What do both terms evaluate to?
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Old 2004-03-30, 14:42   #8
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LHS ~= 3.945765984

RHS ~= 0.2534362159
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Old 2004-03-30, 17:40   #9
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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
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Old 2004-04-02, 01:03   #10
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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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