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

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