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f( a , x ) C oo
#1
many ideas of mine about tetration are related to talking the limit

lim a -> 0 g(g(x)) = f ( a , x )

where g has a single real fixed point for all a > 0 but not for a = 0 and the half iterate is thus computed " regularly ".

now the 2 big questions are :

in such cases as above ( such as 1 real fixed point for a > 0 )

g(x) depends on "a".

lets say : g(x,a).

IS g(x,a) Coo with respect to a ?

IS the radius of convergeance with respect to x Coo with respect to a ?

regards

tommy1729
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#2
(06/12/2009, 12:27 PM)tommy1729 Wrote: many ideas of mine about tetration are related to talking the limit

lim a -> 0 g(g(x)) = f ( a , x )

where g has a single real fixed point for all a > 0 but not for a = 0 and the half iterate is thus computed " regularly ".

now the 2 big questions are :

in such cases as above ( such as 1 real fixed point for a > 0 )

g(x) depends on "a".

lets say : g(x,a).

IS g(x,a) Coo with respect to a ?

IS the radius of convergeance with respect to x Coo with respect to a ?

regards

tommy1729

the answers are yes.
(06/17/2009, 04:57 PM)tommy1729 Wrote:
(06/12/2009, 12:27 PM)tommy1729 Wrote: many ideas of mine about tetration are related to talking the limit

lim a -> 0 g(g(x)) = f ( a , x )

where g has a single real fixed point for all a > 0 but not for a = 0 and the half iterate is thus computed " regularly ".

now the 2 big questions are :

in such cases as above ( such as 1 real fixed point for a > 0 )

g(x) depends on "a".

lets say : g(x,a).

IS g(x,a) Coo with respect to a ?

IS the radius of convergeance with respect to x Coo with respect to a ?

regards

tommy1729

the answers are yes.

matrix thinking was key...
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