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