(07/24/2010, 10:58 PM)tommy1729 Wrote: i have to say that the last limit formula ( recurrence ) on the bottom of page 20 is not clear to me.
i mean , how does one solve this ?
the tetration is on both sides of the equation , the integral is related to values of the tetration itself ??
Like in every recurrence you have the searched thing at both sides of the equation, you iterate. and if the recurrence is convergent, it gets closer and closer to searched for solution.
In our case we compute at the left side f with arguments i*y (let x_0=0) and on the right side there is an integral involving f applied on i*t or on i*(-t). And that is the way you iterate:
You compute values of f along suitably dense grid on the y-Axis by using these values on the y-Axis with a numeric integration.
And - amazingly enough - the values on the y-Axis converge.
I never verified it myself yet, but the convergence seems not very robust in certain cases (from some comments on this forum).
Quote:i dont see why sexp(x + 1periodic(x) ) wont satisfy the same equation , though maybe there is no uniqueness claim to the cauchy integral approach ? ( but kouznetsov claims uniqueness ... based upon a non-unique cauchy integral equation ?? )
Of course the equation is just a reformulation of the superfunction equation, however it converges to one specific solution.
(Btw see also this thread)