02/23/2016, 01:23 PM
About sheldon's pics for fake exp^[1/2].
I believe fake exp^[1/n] satisfies the functional equation Well in the zone going from positive reals > n to arg(z) = arg( (-1)^(2/n) ).
That is imho intuitive.
And it clearly holds for n = 2 , Well according to the plots at least.
I believe this property carries over to most solutions exp^[1/n] ( no fake , actual ).
This imho suggest that most analytic exp^[1/n] are analytic in the same zone described above.
A strong conjecture is that if the fake holds the functional equation in its zone , the the corresponding actual function ( of which the fake is considered ) is NEC analytic in that zone.
This makes me wonder about iterations of similar functions like sinh ; if they have that property too.
Probably another way to say it is :
" the fake function is not only a good approximation on the positive reals , but also on the region near it ! "
Although that is open for interpretation and debate I assume.
Regards
Tommy1729
I believe fake exp^[1/n] satisfies the functional equation Well in the zone going from positive reals > n to arg(z) = arg( (-1)^(2/n) ).
That is imho intuitive.
And it clearly holds for n = 2 , Well according to the plots at least.
I believe this property carries over to most solutions exp^[1/n] ( no fake , actual ).
This imho suggest that most analytic exp^[1/n] are analytic in the same zone described above.
A strong conjecture is that if the fake holds the functional equation in its zone , the the corresponding actual function ( of which the fake is considered ) is NEC analytic in that zone.
This makes me wonder about iterations of similar functions like sinh ; if they have that property too.
Probably another way to say it is :
" the fake function is not only a good approximation on the positive reals , but also on the region near it ! "
Although that is open for interpretation and debate I assume.
Regards
Tommy1729
