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 ...

and for some 100 > y > -oo :

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 : and also .

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 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