[math-fun] ries strikes again. Mathematica answers!
Date: 2017-03-27 18:21 From: "David Wilson" <davidwwilson@comcast.net> rwg, apologies for filling your head with all these √2's.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, March 27, 2017 8:25 PM To: math-fun@mailman.xmission.com Subject: [math-fun] ries strikes again. Mathematica answers!
In[69]:= ((1/2 ((-I)^(1 + √2) + I^(1 + √2)) - Sqrt[-1 + 1/4 ((-I)^(1 + √2) + I^(1 + √2))^2])^(-1 + √2) + (1/2 ((-I)^(1 + √2) + I^(1 + √2)) + Sqrt[-1 + 1/4 ((-I)^(1 + √2) + I^(1 + √2))^2])^(-1 + √2)) // FullSimplify
Out[69]= 2 Sin[π 2√2]
(How??) --rwg
CNS Crash Report: Brain capacity exceeded. OS version: √2 Reboot. OS version √3 In[518]:= %367 /. m -> 2 - Sqrt[3] /. n -> Sqrt[3] + 2 // Simplify // tim During evaluation of In[518]:= 20.750373,2 Out[518]= 1/2 ((1/2 ((x - Sqrt[-1 + x^2])^(2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^( 2 - Sqrt[3])) - Sqrt[-1 + 1/4 ((x - Sqrt[-1 + x^2])^( 2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^(2 - Sqrt[3]))^2])^( 2 + Sqrt[3]) + (1/ 2 ((x - Sqrt[-1 + x^2])^(2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^( 2 - Sqrt[3])) + Sqrt[-1 + 1/4 ((x - Sqrt[-1 + x^2])^( 2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^(2 - Sqrt[3]))^2])^( 2 + Sqrt[3])) == x ∀ complex x ! (according to Plot3D, which can miss isolated point discontinuities, which I think can only occur along ridges of infinite curvature. --rwg)
On Tue, Mar 28, 2017 at 8:43 AM, Bill Gosper <billgosper@gmail.com> wrote:
Date: 2017-03-27 18:21 From: "David Wilson" <davidwwilson@comcast.net>
rwg, apologies for filling your head with all these √2's.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Monday, March 27, 2017 8:25 PM To: math-fun@mailman.xmission.com Subject: [math-fun] ries strikes again. Mathematica answers!
In[69]:= ((1/2 ((-I)^(1 + √2) + I^(1 + √2)) - Sqrt[-1 + 1/4 ((-I)^(1 + √2) + I^(1 + √2))^2])^(-1 + √2) + (1/2 ((-I)^(1 + √2) + I^(1 + √2)) + Sqrt[-1 + 1/4 ((-I)^(1 + √2) + I^(1 + √2))^2])^(-1 + √2)) // FullSimplify
Out[69]= 2 Sin[π 2√2]
(How??) --rwg
CNS Crash Report: Brain capacity exceeded. OS version: √2 Reboot. OS version √3
In[518]:= %367 /. m -> 2 - Sqrt[3] /. n -> Sqrt[3] + 2 // Simplify // tim
During evaluation of In[518]:= 20.750373,2
Out[518]= 1/2 ((1/2 ((x - Sqrt[-1 + x^2])^(2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^( 2 - Sqrt[3])) - Sqrt[-1 + 1/4 ((x - Sqrt[-1 + x^2])^( 2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^(2 - Sqrt[3]))^2])^( 2 + Sqrt[3]) + (1/ 2 ((x - Sqrt[-1 + x^2])^(2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^( 2 - Sqrt[3])) + Sqrt[-1 + 1/4 ((x - Sqrt[-1 + x^2])^( 2 - Sqrt[3]) + (x + Sqrt[-1 + x^2])^(2 - Sqrt[3]))^2])^( 2 + Sqrt[3])) == x
∀ complex x ! (according to Plot3D, which can miss isolated point discontinuities, which I think can only occur along ridges of infinite curvature. --rwg)
OUCH! Here is the error plot-- gosper.org/b0gus.png , a glorious expanse of floating point noise. But plugging in numbers, this should look instead like the 2nd two plots in gosper.org/cheb1.png, a knife edge of discrepant points ! I don't know why the knife edge is invisible. (Swapping n and m leaves a 2-dimensional area of discrepancy.) --rwg
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Bill Gosper