darn. i wanted to post that today.
f(x) = x^2
f(x) = x^3
f(x) = x^sqrt(2)
are imho the best examples to understand and investigate the seemingly contradiction.
i often argue about branches and (-1)^sqrt(2).
( related to the irrationality of sqrt(2) and thompsons lamp paradox )
sometimes even consider (-1)^(irrational real) to be the unit circle , which is controversial ( branches are considered 'countable' even when 'dense' but binary , ternary or continued fraction expansions may lead to another ' interpretation ' ).
afterall after considering the superfunctions of x^sqrt(2) , x^2 , x^3 and x^sqrt(3) or their inverses , we see a striking resemblence that cannot be ignored !!
( an animation of changing riemann surfaces with a real parameter would be nice )
square roots and cube roots are school examples of functions that are complex abs continu , but not complex differentiable. ( more *nice* examples are welcome , where *nice* probably means analytic almost everywhere , however i suppose al *nice* examples are inverses of polynomials or similar looking .. not ? )
im very intrested in complex abs continu functions and i believe they are important and natural.
but its a long time ago i studied these , is there a terminology for complex abs continu functions defined on all of z ( as analogue of entire function ) ?
i especially like complex abs continu functions that are still continu at oo too.
furthermore
x^a has a superfunction agreeing on all fixpoints !!
( or at least for integer a and especially 0 and 1 )
( i notice that what makes this rare property may be the combination of complex abs continu + = to secondary , ternary , ... fixpoint expansion ?!? )
i would like to see more superfunctions agreeing on their fixpoints !
btw , maybe g^[-1]((g(x))^a) for suitable g(x) can be used to construct all superfunctions that agree upon fixpoints and are complex abs continu ???
( like we did for solving the iterations of exp(x)+x ; replacing the fixpoint at - oo to finite )
however if im not mistaken in my rapid brainstorm , this puts restrictions on the fixpoints ( repelling in direction region Q etc ) so maybe there may be other examples of superfunctions agreeing on fixpoints that match the other possible properies of the fixpoints ...
btw , how does one link the disagreement of half-iterates on multiple fixpoints to the superfunction with the fixpoints at complex oo ?
i assume thats simple : either the superfunction does not go to oo ( not defined beyond abs(z) = X ) , or it is not constant near the fixpoint but periodic , since if at fixpoint 2 the halfiterate gives value B then the half-iterate of B = fixpoint 2 ...
intresting stuff. basic yet intresting.
forgive my brainstorm type of writing ...
regards
tommy1729
f(x) = x^2
f(x) = x^3
f(x) = x^sqrt(2)
are imho the best examples to understand and investigate the seemingly contradiction.
i often argue about branches and (-1)^sqrt(2).
( related to the irrationality of sqrt(2) and thompsons lamp paradox )
sometimes even consider (-1)^(irrational real) to be the unit circle , which is controversial ( branches are considered 'countable' even when 'dense' but binary , ternary or continued fraction expansions may lead to another ' interpretation ' ).
afterall after considering the superfunctions of x^sqrt(2) , x^2 , x^3 and x^sqrt(3) or their inverses , we see a striking resemblence that cannot be ignored !!
( an animation of changing riemann surfaces with a real parameter would be nice )
square roots and cube roots are school examples of functions that are complex abs continu , but not complex differentiable. ( more *nice* examples are welcome , where *nice* probably means analytic almost everywhere , however i suppose al *nice* examples are inverses of polynomials or similar looking .. not ? )
im very intrested in complex abs continu functions and i believe they are important and natural.
but its a long time ago i studied these , is there a terminology for complex abs continu functions defined on all of z ( as analogue of entire function ) ?
i especially like complex abs continu functions that are still continu at oo too.
furthermore
x^a has a superfunction agreeing on all fixpoints !!
( or at least for integer a and especially 0 and 1 )
( i notice that what makes this rare property may be the combination of complex abs continu + = to secondary , ternary , ... fixpoint expansion ?!? )
i would like to see more superfunctions agreeing on their fixpoints !
btw , maybe g^[-1]((g(x))^a) for suitable g(x) can be used to construct all superfunctions that agree upon fixpoints and are complex abs continu ???
( like we did for solving the iterations of exp(x)+x ; replacing the fixpoint at - oo to finite )
however if im not mistaken in my rapid brainstorm , this puts restrictions on the fixpoints ( repelling in direction region Q etc ) so maybe there may be other examples of superfunctions agreeing on fixpoints that match the other possible properies of the fixpoints ...
btw , how does one link the disagreement of half-iterates on multiple fixpoints to the superfunction with the fixpoints at complex oo ?
i assume thats simple : either the superfunction does not go to oo ( not defined beyond abs(z) = X ) , or it is not constant near the fixpoint but periodic , since if at fixpoint 2 the halfiterate gives value B then the half-iterate of B = fixpoint 2 ...
intresting stuff. basic yet intresting.
forgive my brainstorm type of writing ...
regards
tommy1729
