04/22/2008, 12:59 AM
Hello, bo198214; you have cathced the important point! I see, in my paper, I have to write
Theorem 0. There exist function \( F(z) \), analytic in the whole complex \( z \) plane except \( z<-2 \), satisfying
(100) \( ~~~~ \exp(F(z))=F(z+1) \),
(101) \( ~~~~ F(0)=1 \)
and
(102) \( ~~~~ F(z)=L + {\mathcal O}\Big( \exp( L z ) \Big)~ \) at any fixed \( \Re(z) \) and \( \Im(z) \rightarrow +\infty \)
Theorem 1. There exist only one such function.
I hope, the reviewer catches this point, and I already have the corresponding correction above.
By the way, your deduction gives the hint, how to prove the Theorem 1. (However, we have to scale the argument of sin function.) How about the collaboration?
Theorem 0. There exist function \( F(z) \), analytic in the whole complex \( z \) plane except \( z<-2 \), satisfying
(100) \( ~~~~ \exp(F(z))=F(z+1) \),
(101) \( ~~~~ F(0)=1 \)
and
(102) \( ~~~~ F(z)=L + {\mathcal O}\Big( \exp( L z ) \Big)~ \) at any fixed \( \Re(z) \) and \( \Im(z) \rightarrow +\infty \)
Theorem 1. There exist only one such function.
I hope, the reviewer catches this point, and I already have the corresponding correction above.
By the way, your deduction gives the hint, how to prove the Theorem 1. (However, we have to scale the argument of sin function.) How about the collaboration?