I only looked at it with Maple, which finds c2(x) = c(c(x)) - x^2 + 2 real for x>=-2, and makes a nice plot, which I can put somewhere (e.g. dropbox) if anyone wants. On Fri, Mar 24, 2017 at 6:07 PM, David Wilson <davidwwilson@comcast.net> 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----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Friday, March 24, 2017 6:06 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Help with some math
c2(x) := c(c(x)) - x² + 2 is fascinating. Continuous and real for x > -2, where c2(-2+) = - 4 sin(π √2)² ~ -3.7164323713376353833, followed immediately by a minimum at x = -√(2 - 2 sin(π √2)) ~ -1.981869083895239 where c2(x) = 4 sin(π √2) ~ -3.8556101313995. c2(x) first reaches its permanent value of 0 at x = 2 cos(π/√2) ~ -1.2113997341576, where the slope is discontinuous. All these magic numbers are technically conjectural, from a spectacular performance by Munafo's ries. As David suggests, Mathematica's simplifiers were embarrassed at several points but excelled at others. Likewise the numerics. There are some bugs to report! --rwg
(Note: David is using b^3, e.g., to mean b(b(b())).) On 2017-03-23 15:40, David Wilson wrote:
In the fortuitous case of
b(x) = x^2 - 2
b is conjugate to
a(x) = x^2
by the simple mapping
f(x) = x + x^-1 f^-1(x) = (x + sqrt(x^2 - 4))/2
with
b^n = f o a^n o f^-1
Specifically, if
c(x) = b^(1/2)(x) = ((x + sqrt(x^2 - 4))/2)^sqrt(2) + ((x - sqrt(x^2 - 4))/2)^sqrt(2)
then
c(c(x)) = x^2 - 2.
It's a nice problem to give the c(x) formula and ask for c(c(x)). It might be fun to blindside a math forum with this problem. I can't get Wolfram Alpha to answer it.
At any rate, the form of this f(x) led me to postulate the power series form for the f mapping in the x -> x^2 + 1 case.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Thursday, March 23, 2017 12:42 AM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: Re: [math-fun] Help with some math
Dan's worked with flows quite a bit.
Some of the rest of us have played around a little, looking for functional square-roots or real-indexed iterated functions. There are dragons lurking.
Long ago, I wrote several flow programs in Lisp, using the nice feature of exact rational arithmetic. I tried to find a power series for the half flow of x+x^2, expanded around 0. I used f(x) = x+x^2, g(x) = x + x^2/2 + tbd, and the equation f(g(x)) = g(f(x)), matching the power series on each side. This led to a relatively simple formula for computing terms of g(). Things looked good at the beginning, but I got greedy and went out to a hundred terms. The numbers looked nice for a while, but soon the numerators & denominators ballooned. To make sense of things, I floated the rational coefficients, and took the nth root of |g_n| to estimate the reciprocal of the radius of convergence. A nasty surprise: the numerical evidence was that the radius of convergence seemed to be heading for 0.
Some of our other experiments had odd results.
Good luck with your approach!
Rich
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun