Searching for an asymptotic to exp[0.5]

[tommy1729]

Ok time to get more formal.

As mentioned before there are links between fake and multisections.
That will be more clearly with the method presented here.

Also this method - with a little twist - gives the CORRECT values if all derivatives are already positive. ( such as exp )

So as a general case method I think this is one of the better ones.

Consider f(x) with the conditions :

f(x) is real-analytic for x >= -1.

for x > 0 we have

f(x) > 0 , f ' (x) > 0 and f " (x) > 0 and also

f(x) grows larger/faster then any polynomial :

lim x-> +oo x^t / f(x) = 0 for all real t >= 0.

So this f(x) satisfies the conditions needed to get a fake [f(x)] = g(x).

g(x) = a_0 + a_1 x + a_2 x^2 + ... ~ f(x).
with a_j >=0

We need a method to find the a_k.

Let n > i > 2 such that for all x > 0 : D^i f(x) > 0.

Define G_n(x) = SUM_i a_i x^i.

Our equations for finding a_j are then :

for all x > 0.
a_0 x^0 =< f(x)

or equivalent
for x > 0
a_0 = inf( f(x) )

here we simply get a_0 = f(0).

further
( x > 0 Always so I will stop mentioning this )

a_1 x = inf( f(x) )

=>

a_1 = inf( f(x) / x )

( notice the similarity to the derivative f ' (0) = lim ( f(x) - f(0) ) / x. )

a_2 x^2 = inf ( f(x) )

=> a_2 = inf ( f(x) / x^2 )

( this looks simpler then all those logs in post 9 ... although it has its use there for tetration type functions ofcourse )

a_3 x^3 = inf( f(x) )

=> a_3 = inf( f(x) / x^3 )

a_4 x^4 + G_4(x) =< f(x)

this is a bit more complicated but notice we already have a_0 , a_1 , a_2 and a_3 from the above equations so G_4 is known(*).

( * assuming the condition for i is understood , see def for G_n above )

and it continues like

a_n is computed from

a_n x^n + G_n(x) =< f(x)

or equivalent

a_n x^n + G_n(x) = inf( f(x) )

Notice how this works PERFECT for exp giving fake[exp] = exp if we replace the equations for a_1 and a_2 with a_1 = 1 and a_2 = 1/2 instead and then go on to solve the others.
Similar good results for sinh(x) for instance.

This method is never worse then the method from post 9.

The G_n is an important concept , because equations like

a_0 + a_1 x + a_2 x^2 + a_3 x^3 =< f(x)

could FAIL for some f(x).

The G_n guarantees non-negativity of the a_j.

This captures most of my ideas here in this thread so I will simply call this :

tommy's fake method.

I will be using this in the future and base my conjectures on this.

Error terms depend on f(x) alot and I do not yet understand them.

But fake function theory is advanced by this.

My friend mick will post the following problem to MSE about tommy's fake method.

( an old problem considered by me for the record )

Let f(x) be as defined above.

Let F(x) := integral_0^x f(t) dt.

Then

conjecture : Fake[ F(x) ] - integral_0^x Fake[ f(t) ] dt = O(1 + x^3)
where O is big-O notation.

A weaker (related !) version ( post 9 method ) is

n>3

a_n(f) = inf( f(x) / x^n )
b_(n+1)(F) = inf( F(x) / x^{n+1} )

[ inf( f(x) / a_n(f) x^n ) ] / [ inf( F(x) / ( (a_n(f) x^{n+1}) /n) ) ] ~ 1 +/- O(1/n).

suggesting that integral_0^x a_n(f) t^n dt ~ b_(n+1)(F) x^{n+1}.

Regards

tommy1729

[tommy1729]

quote : Let n > i > 2 such that for all x > 0 : D^i f(x) > 0.

If we are not completely sure which i satisfy it we can consider the ones we are certain of and use these. Similar ideas might work under some conditions. ( such as a counting function approximation like PNT )

You can call that new function G*_n.

***

Notice if we know the i's for f(x) we also know them for F(x) !

Regards

tommy1729

[tommy1729]

Finally closer to proving TPID 17.

Notice that the entire function we want a fake of could itself be a fake or even the best fake ... Therefore we get insight in comparing methods.

The idea fake f = fake fake f needs more consideration though.
( dont ask me about fake^[1/2] f plz !! The horror ! )

Key ideas are that entire functions converge fast in Taylor form and the derivatives (or fake ones) go to 0 fast. ( not using that makes it impossible to prove ?? )

Notice fake x^n f ~ x^n fake f.

Regards

Tommy1729

[tommy1729]

Hadamard is also useful for TPID 17.
Useful once again.

Regards

Tommy1729

[tommy1729]

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

[tommy1729]

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

[tommy1729]

Exercise 1 a)

What entire functions are asymptotic to exp^[0,5](x) for positive real x and has no zero in the complex plane ?

Answer : g(z) = exp(fake_ln(+fake_semi_exp(z)))


Exercise 1 b)

Show growth(g) is always between 0,5 and 2,5 for fixed arg(z).

Hint : investigate fake_ln.

Plot g(g)

Exercise 2)


What entire functions h are asymptotic to exp^[0,5](x) for positive real x and satisfy that h(s) = 0 IFF s = L or L* ?

I think we are close to finishing page 1 in the " fake function theory book " .

Regards

Tommy1729

[tommy1729]

Exercise 2 is not so hard as it looks.
In other words, i know the solution and its easy.

Many many more questions exist that are harder.

Regards

Tommy1729

[tommy1729]

I wanted to write a long post about that idea to write the min as An integral.

But then I found mick's post that captured the idea !!

This unfamous post might have more importance then at first look.

Ideas of continuum Sum and double integrals also cross my mind.
And even the min,- algebra from zeration.

Here is the link



http://math.stackexchange.com/questions/...n-integral

Regards

Tommy1729

[tommy1729]

For clarity im not looking for the rootfinding iteration methods like newton's method or Steffensens method.

Regards

Tommy1729