fake id(x) for better 2sinh method.

[tommy1729]

For improving the 2sinh method so that it works for all bases > eta and other improvements I consider :

2sinh(fake id(x))

fake id(x) behaves like e/2 x for small x, like x for large real and like 0 for large imaginary part.

SO what is the best fake id(x) ? Remember we only want a single fixpoint (2sinh(fake id(x))=x) for any base.

maybe id(x) = exp(fakelog(x)) ...

Thinking...

regards

tommy1729

[tommy1729]

There is a problem with this particular type of fake function.
Although it may be fixed as stated in post 1 it seems a big problem.

Perhaps the biggest problem of " fake function theory ".

id(x) as desired in post nr 1 : " fake id(x) behaves like e/2 x for small x, like x for large real and like 0 for large imaginary part "

is in contradiction with the well known Lindelöf theorem.

It seems the natural fix is : " fake id(z) behaves like e/2 z for small z and like z for large z "

regards

tommy1729

[tommy1729]

Hmm this new id(z) also seems in trouble because

id(z) -z goes to 0 for large z ... but stays bounded for small z : so id(z) - z is not entire.

So maybe use a function that grows like exp(x) near the real line and grows like z otherwise ?

Notice if id(z) is not entire , then neither is id(exp(z)) or even id(fakeexp(z)).

Hmm.

Seems really an issue.

regards

tommy1729

[tommy1729]

Wait a sec , it seems carlson's theorem applies here !

id(z) must grow fast enough otherwise id(z) - z is "close" to 0 for integer z , hence to avoid id(z) being flat it seems id(z) must grow fast enough in the imaginary direction.

The problem is " close " is not the exact condition of the theorem
(exactly 0 is).

To make ln(id(z)/z) an entire function this also suggests id(z) is of exp type.

Hence I write id(z) = z exp(t(z)) where t(z) is a taylor series.

Now id(z)/z IS NEVER ALLOWED TO BE EQUAL TO 1.

and exp(t(x)) for real x is a fake id(1) function.

Thus t(x) is a fake id(0) function.

However saying t(z) = id(z) - z brings us back to the Original problem ...

Hmm.

This is getting stranger by the sec.

Maybe t(z) should be a fake id(0) for reals and a fake exp^[1/2](z) in the upper half plane ( in terms of absolute value ).
But that brings us back again to carlson so I guess its better to have

f(z) = fake id(0) around the real axis.
f(z) = fake exp(z) near the imag axis.

I finally see less objections.
But what is f(z) ?


regards

tommy1729

[tommy1729]

Now id(z)/z IS NEVER ALLOWED TO BE EQUAL TO 1.

and exp(t(x)) for real x is a fake id(1) function.

This is still a big objection.

so exp(t(z)) is not allowed to be 0 nor 1.
This violates the little picard theorem.

Hmm

Seems all theorems of complex analysis are against me !

regards

tommy1729

[tommy1729]

A good id(z) has been found !

id(z)/z = 1 sometimes but that is no big problem.

id(z) = z * fake(1) on the real line
id(z) = z * fake(exp(exp(i z))^O(z)) on the imag line.

id(z) is of the type z * f(z i) where f(z i) is similar too

\( f(z i)=\int_0^\infty \frac{e^{zt i}}{t^t}\mathrm{d}t \)

Yes the same function that occured in the fake exp^[1/2] thread.

It seems " fake function theory " is developping fast.

regards

tommy1729