meromorphic idea

[tommy1729]

i was thinking about tetration and came up with important questions.

for the 3rd time !!

are there meromorphic functions f(x) so that they commute with exp(x) ?

thus :

exp(f(x)) = f(exp(x))


related 2nd question :

t ( t(x) ) = exp(x) ( any real analytic t(x) )

g(x) a meromorphic function


t(x) = inverse_g( 1-fixpoint-regular-half-iterate[ exp(g(x)) ] )


related 3rd question :

q(x) a meromorphic function

t ( t(x) ) = exp(x) ( any real analytic t(x) )

1-fixpoint-regular-half-iterate[ exp(q(x)) ] = t( q(x) )



thanks in advance

regards

tommy1729