05/28/2014, 10:34 PM
So why dont we have Re(x) << 0 ?
Well probably because Re(x) << 0 implies taking many logs , semilogs etc hence our value has " growth difficulties " in the sense that it gets too small.
Similarly for Re(x) >> 0 then we have the supergrowth problem , it gets too big.
I therefore guess all solutions have 0.5 < Re(x) < 4.
However for x with very (or sufficiently ) large Imaginary parts this might no longer hold as a good argument ...
Notice that log interations have stronger growth issues than exp iterations , because of the chaotic nature of exp iterates ... hence the previous sentense.
Since For x~=2.6918099719192 + 0.62660048483655i, sexp_x(x)~=-1 apparantly , I wonder about 2 things
-- apart from the fact that I need to learn more about complex base iterations ! --
1) Is this x~=2.6918099719192 + 0.62660048483655i the only solution ??
2) What happens with x~=2.6918099719192 - 0.62660048483655i ??
Also there should be a proof that x^^x = y Always has a complex solution.
Since every nonconstant analytic function takes every value apart from possibly one value the REAL ( ahum ) question is there such an y such that x^^x has no complex solution ?
Is there something intresting about oo ^ ^ oo for complex infinity ??
But the main question is : is x^^x analytically continuable to the entire complex plane or does it have a natural boundary ?
And as suggested by question nr 2 :
x^^x = conj( conj(x)^^conj(x) ) or similar ??
But thats not all , maybe I need to learn more about complex base iterations , but I know there are multiple solutions right ?
So what answers do the other methods give ?
Many questions ...
I understand the kneser method for real bases > eta , but not for complex.
for real bases we map the reals to the real line , but what do we do for the complex bases ??
Or is that an irrelevant question ?
Sorry complex bases still confuse me.
regards
tommy1729
Well probably because Re(x) << 0 implies taking many logs , semilogs etc hence our value has " growth difficulties " in the sense that it gets too small.
Similarly for Re(x) >> 0 then we have the supergrowth problem , it gets too big.
I therefore guess all solutions have 0.5 < Re(x) < 4.
However for x with very (or sufficiently ) large Imaginary parts this might no longer hold as a good argument ...
Notice that log interations have stronger growth issues than exp iterations , because of the chaotic nature of exp iterates ... hence the previous sentense.
Since For x~=2.6918099719192 + 0.62660048483655i, sexp_x(x)~=-1 apparantly , I wonder about 2 things
-- apart from the fact that I need to learn more about complex base iterations ! --
1) Is this x~=2.6918099719192 + 0.62660048483655i the only solution ??
2) What happens with x~=2.6918099719192 - 0.62660048483655i ??
Also there should be a proof that x^^x = y Always has a complex solution.
Since every nonconstant analytic function takes every value apart from possibly one value the REAL ( ahum ) question is there such an y such that x^^x has no complex solution ?
Is there something intresting about oo ^ ^ oo for complex infinity ??
But the main question is : is x^^x analytically continuable to the entire complex plane or does it have a natural boundary ?
And as suggested by question nr 2 :
x^^x = conj( conj(x)^^conj(x) ) or similar ??
But thats not all , maybe I need to learn more about complex base iterations , but I know there are multiple solutions right ?
So what answers do the other methods give ?
Many questions ...
I understand the kneser method for real bases > eta , but not for complex.
for real bases we map the reals to the real line , but what do we do for the complex bases ??
Or is that an irrelevant question ?
Sorry complex bases still confuse me.
regards
tommy1729