(03/05/2016, 01:27 PM)tommy1729 Wrote: It seems the method has passed the Weierstrass M-test !!

Regards

Tommy1729

So ... It is analytic !

Here is a sketch of the proof.

Lets consider the half-exp for simplicity, the general case follows by analogue.

Proof sketch

1) 2sinh^[1/2] (x) = g(x) is analytic in [0,e] so it is analytic for x>= 0.

2) therefore f = ln(g(exp(x)) is analytic in [1,oo].

X* = x - eps , X** = x + eps.

Let f(x,n) = ln^[n](f( exp^[n](x) ). + continuation ( like ln(exp) = id ).

Clearly for finite n (integer >0)

F(x,n) is analytic in x for x > 1.

3) we can choose small eps > 0 , independant of n such that

For the points w* in the radius eps around x,

And points w** around exp(x) with radius exp(x**) - exp(x),

( Abs(f(w*,n )) - Abs(ln( f(w**,n ) )^2 < (Abs(f(x*,n)) - Abs(ln( f(exp(x**) ,n)) ))^2 { follows from squeeze theorem and analiticity } < 10/n^3 { follows from fast convergeance of the n th step for real x > 1 }.

Notice that for small enough eps we get a univalent radius.

4) the weierstrass M test gives the Sum S =< 10 zeta(3).

So we have passed the weierstrass M test.

5) since we have past the weierstass M test , we have uniform convergeance within the radius eps > 0.

Notice eps is independent of n so lim eps(n) > 0 , no infinitesimal.

So we have a positive radius around Some real x > 1 where the M test is satisfied and hence it is uniform convergent there.

6) if within a radius a function is the uniform convergent limit of a sequence of analytic functions ( analytic in the radius) then that limit function is analytic in that radius too.

{ a classic , but the name and inventor is uncertain to me , resembles uniform limit theorem and work of cauchy and Weierstrass. Also uncertain about the date - assume around 1874 ... Feel Free to inform me. I dont Mind naming after me , but it probably has a name already ... }

Therefore the 2sinh method is analytic !!

Q.e.d.

Tommy1729

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Im considering a geometric proof too.

I think this proof and the geometric ( yet to make) one are the only possible.

I had matrix ideas and basic real calculus ideas , but despite arguments they did not get closer to a proof.

This is probably in part due to the importance of error terms over closed forms.

Naturally , this is the beginning and not the end.

The natural idea of what else is analytic in a provable way is very inspiring.

Notice this extends to all real bases within [exp(1/2), + oo].

I believe my exp(2/5) base method is also analytic because of this proof so Maybe we have

Analytic methods for bases in [exp(2/5), + oo].

I know uniformal convergeance for matrix functions is way harder.

( Maybe one of the top 5 problems in lin alg ).

But Maybe we Will find a way around it.

Regards

Tommy1729

The master

"Truth is whatever does not go away when you stop believing in it."