07/18/2015, 09:34 PM

(07/17/2015, 01:45 PM)tommy1729 Wrote: It might be important to rewrite min since min is crucial in fake function theory.

Sheldon is correct ofcourse ; min f(x) = f(w) , f ' (w) = 0.

But the inverse of f ' can be complicated.

So the idea here is to estimate min with other tools.

For instance

G (f) = f/(f+1)

T = growth f

Min f ~ integral_1^oo 1/f dx

Or more advanced

Min f(x)/x^n ~ lim m -> oo q(m)

Where

Q(m) = exp^[m+T]( integral_1^(n m) ln^[m+T](G ( f(x)/x^n )) dx / (n m) ).

There are probably better ones.

But that is the idea.

For proofs that may be important.

Maybe there is An argument principle equivalent ... But im concerned about the nonreal w messing up.

Regards

Tommy1729

Another way is Laplace_method[ G(f) ] to get close to f(w).

Ofcourse the Laplace method and its variants are just the " main idea " and other and better methods will very likely exist.

For clarity , the Laplace_method :

https://en.m.wikipedia.org/wiki/Laplace%27s_method

All these integral Ideas must relate to Sheldon's IV method but how exactly is not clear.

I think this Laplace thing might get us closer to a proof of TPID and a deeper understanding of fake function theory.

Regards

Tommy1729