Yes, the Computer Algebra viewpoint is that multivalued functions are a thought crime, and we must all agree where the branch cuts are in all the named functions. This just gave me an isolated point singularity in a merely algebraic function that I almost reported as a bug: Evidently, -2 z + (-3 z + 4 z^3 + (1 - 4 z^2) Sqrt[-1 + z^2])^( 1/3) + (-3 z + 4 z^3 + Sqrt[-1 + z^2] (-1 + 4 z^2))^(1/3) is 0 within the semihyperbola Re@z > Sqrt[1/4 + (Im@z)^2/3], discontinuous on the boundary, and smooth elsewhere. But boundary pt z=1/2 is discontinuous with both regions. Almost impossible to find with Plot... --rwg -------- Original Message -------- Date: 2017-03-26 08:43 From: "David Wilson" <davidwwilson@comcast.net> To: "'math-fun'" <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com> Apologies if I am seem pedantic I programmed this in C++, and got essentially the same values for c(c(x)) plotted by rwg. For any x < 2, the evaluation of c(c(x)) ultimately ends up computing a^√2 for complex a. Unfortunate, for complex a, a^√2 = exp(√2 (log(a) + 2πi k)) has an infinitude of values (for integer k). For C++, and presumably Mma, log(a) returns the principle value with -π < Im(log(a)) <= π. This happily coincides with real log(a) for positive real a. Regarding the c(c(x)) calculation: If x >= 2, we can restrict calculations to the real realm, take the principle (real) value of a^√2, and end up with c(c(x)) = x^2 -2. If x < 2, however, a^√2 must be evaluated for non-positive-real x, and it is not clear that the principle value is appropriate in any particular calculation. I submit that, when c(c(x)) differs from x^2 - 2, we might restore c(c(x)) = x^2 -2 by using the appropriate (non-principle) value of a^√2 in the appropriate calculations, though it is unclear which value of a^√2 (if any) works for any particular evaluation of c(x).

-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Saturday, March 25, 2017 11:35 PM To: Wolfram Technical Support Cc: math-fun@mailman.xmission.com Subject: [math-fun] Brutal FullSimplify challenge II

In[540]:= $Version

Out[540]= "11.0.1 for Mac OS X x86 (64-bit) (September 21, 2016)"

cc[x_] := Nest[((# + √(#^2 - 4))/2)^√2 + ((# - √(#^2 - 4))/2)^√2 &, x, 2]

In[653]:=FullSimplify@cc@-Sqrt[2 (1 - Sin[√2 π])] == 2 Sin[√2 π]

Out[653]= 2^(-1 - Sqrt[ 2]) (((-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] - \[Sqrt](-2^( 1 + Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^( 2 Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^(2 Sqrt[2])))^Sqrt[ 2] + ((-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^Sqrt[ 2] + \[Sqrt](-2^( 1 + Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] - I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^( 2 Sqrt[2]) + (-Sqrt[1 - Sin[Sqrt[2] \[Pi]]] + I Sqrt[1 + Sin[Sqrt[2] \[Pi]]])^(2 Sqrt[2])))^Sqrt[2]) == 2 Sin[Sqrt[2] \[Pi]]

In[654]:= Chop@N@# & /@ %

Out[654]= True --Bill Gosper