04/21/2014, 10:07 PM
Consider the half-iterate of exp(x) : .)
In particular we consider the real-analytic half-iterate of exp(x) such that for all real x :
and also
.
So far so good. But then I get confused ...
}{dx} = 0 )
and for some 100 > y > -oo :
}{dx} = 1 )
SO for C in the neighbourhood of y we get that
is approximated by the linear function
f1(x) = A + (1) x.
(A =
and "x" follows from
)
Now clearly
.
By analogue let
.
be approximated by the linear function f2(x) = A_2 + B x.
Now my idea was that since exp is its own derivative and composition of linear functions is simple we get :
A_2 = exp(y) and (1) B = exp(y).
HOWEVER (!!!) this implies that we have the derivative of exp(y) at both
and
!?
This violates the initial condition (above) that for all real x :
and also
.
So this confuses me.
I had the idea this composition structure is only valid for derivatives above exp(1) but Im unable to show and understand this completely ...
I made pictures to help understand it but to my amazement that did not solve my confusion. ( pictures usually help for me )
Maybe you guys here can explain this.
regards
tommy1729
In particular we consider the real-analytic half-iterate of exp(x) such that for all real x :
So far so good. But then I get confused ...
and for some 100 > y > -oo :
SO for C in the neighbourhood of y we get that
f1(x) = A + (1) x.
(A =
Now clearly
By analogue let
be approximated by the linear function f2(x) = A_2 + B x.
Now my idea was that since exp is its own derivative and composition of linear functions is simple we get :
A_2 = exp(y) and (1) B = exp(y).
HOWEVER (!!!) this implies that we have the derivative of exp(y) at both
This violates the initial condition (above) that for all real x :
So this confuses me.
I had the idea this composition structure is only valid for derivatives above exp(1) but Im unable to show and understand this completely ...
I made pictures to help understand it but to my amazement that did not solve my confusion. ( pictures usually help for me )
Maybe you guys here can explain this.
regards
tommy1729