periodic ...

[tommy1729]

i had some ideas relating tetration and superfunctions in general to the following properties :

A) f(z) is Coo and Im f(z) is periodic.

B) f(z) is Coo and Re f(z) is periodic.

C) A) and B) but f(z) is not periodic.

what is known about functions satisfying A) B) or C) ?

anything special about them ?

are there series expansions only valid for such functions ?

[tommy1729]

nothing known about them ?

i know my question is a bit vague , but im looking for questions rather than answers i guess

[sheldonison]

i had some ideas relating tetration and superfunctions in general to the following properties :

A) f(z) is Coo and Im f(z) is periodic.

B) f(z) is Coo and Re f(z) is periodic.

C) A) and B) but f(z) is not periodic.

what is known about functions satisfying A) B) or C) ?

anything special about them ?

are there series expansions only valid for such functions ?

I think the Taylor series expansions are usually generated by a matrix. When the superfunction is generated from the fixed point of L for that base, then you have a limit equation, which can indirectly be used to generate a Taylor series. Henryk posted something the other day, that might be relevant (true or false logarithm). But in general, I don't think much is known about generating the Taylor series terms in a closed form.

I don't know any cases where Re f(z) is periodic. Im f(z) is periodic for bases<eta.
- Sheldon

[tommy1729]

I don't know any cases where Re f(z) is periodic. Im f(z) is periodic for bases<eta.
- Sheldon

im i going crazy or is it really that simple ?

you say you dont know a case with Re periodic.

but you do know a case where Im is periodic.

if you multiply your Im solution by i , you get a Re periodic.

shoot me if that is wrong.

thanks for your reply though

the rest made sense to me :p

i was thinking about the fixpoints being all conjugate of each other and expansion of fixp1 = expansion of fixp2 as some sort of condition ...

also the above comment by you and me seem to be focused on superfunctions.

but im also intrested in the functions not defined as superfunctions ( or even NO superfunctions [ not every f is a superfunction ] ) and how THEIR superfunctions behave if they have re/im periodic.

regards

tommy1729

[bo198214]

im i going crazy or is it really that simple ?

you say you dont know a case with Re periodic.

but you do know a case where Im is periodic.

if you multiply your Im solution by i , you get a Re periodic.

and which is then no more a superfunction.

[tommy1729]

for clarity , im not considering superfunctions with a Re or Im period , but a general entire function with Re or Im periodic.

also i do mean that Im(f(z)) is periodic with Im periodic , not a function that has q i as a period.