Re: [math-fun] Help with some math
gosper.org/xx-2.png ries (@ -l6) failed to recognize c2(-2-) ~ -4.95208960443219142648286540385 - 1.88898604639119460912777031678 I --rwg On 2017-03-24 17:07, David Wilson wrote:
I presume you are talking about evaluating
c2(x) = c(c(x)) - x^2 + 2
with
c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2
If you start with |x| >= 2, you end up with
c(x) = y^√2 + z^√2
where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well.
If |x| < 2, however, we end up with
c(x) = y^√2 + z^√2
Where y and z are complex numbers. It then seems that y^√2 and z^√2 take on an infinitude of non-real complex values, none better than another. So how do you compute y^√2 or z^√2 with y or z complex? Is there some principle value?
At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of c2(x) on |x| < 2. I would like to see a plot of c2(x) on the real line.
-----Original Message----- [Chopped]
Thanks!
-----Original Message----- From: math-fun [mailto:math-fun- bounces+davidwwilson=comcast.net@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Friday, March 24, 2017 10:13 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Help with some math
gosper.org/xx-2.png ries (@ -l6) failed to recognize c2(-2-) ~ -4.95208960443219142648286540385 - 1.88898604639119460912777031678 I -- rwg
On 2017-03-24 17:07, David Wilson wrote:
I presume you are talking about evaluating
c2(x) = c(c(x)) - x^2 + 2
with
c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2
If you start with |x| >= 2, you end up with
c(x) = y^√2 + z^√2
where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well.
If |x| < 2, however, we end up with
c(x) = y^√2 + z^√2
Where y and z are complex numbers. It then seems that y^√2 and z^√2 take on an infinitude of non-real complex values, none better than another. So how do you compute y^√2 or z^√2 with y or z complex? Is there some principle value?
At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of c2(x) on |x| < 2. I would like to see a plot of c2(x) on the real line.
-----Original Message----- [Chopped]
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On Fri, Mar 24, 2017 at 7:13 PM, Bill Gosper <billgosper@gmail.com> wrote:
gosper.org/xx-2.png ries (@ -l6)
and -l7
failed to recognize c2(-2-) ~ -4.95208960443219142648286540385 - 1.88898604639119460912777031678 I
--rwg Duh, Limit[c2[x], x -> -2, Direction -> 1] gives -2 + (I^(2 √2) - Sqrt[-1 + I^(4 √2)])^√2 + (I^(2 √2) + Sqrt[-1 + I^(4 √2)])^√2 but the real and imaginary parts seem inexpressible with real, named functions. --rwg
On 2017-03-24 17:07, David Wilson wrote:
I presume you are talking about evaluating
c2(x) = c(c(x)) - x^2 + 2
with
c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2
If you start with |x| >= 2, you end up with
c(x) = y^√2 + z^√2
where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well.
If |x| < 2, however, we end up with
c(x) = y^√2 + z^√2
Where y and z are complex numbers. It then seems that y^√2 and z^√2 take on an infinitude of non-real complex values, none better than another. So how do you compute y^√2 or z^√2 with y or z complex? Is there some principle value?
At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of c2(x) on |x| < 2. I would like to see a plot of c2(x) on the real line.
-----Original Message----- [Chopped]
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David Wilson