(02/07/2021, 05:03 PM)tommy1729 Wrote: ....
Analytic continuations are perhaps not possible for your case (or my analogue) in attempt to go from Re(a) < 0 to Re(a) > 0 or vice versa.
In fact there is a huge gap in my understanding about continuations for infinite compositions. Or Riemann surfaces of infinite compositions.
But I think a natural boundary occurs for Re(a) = 0 in both our cases.
....
tommy1729
Yes, I'd have to agree with you as it being a natural boundary. When we flip to \( \Re(a) < 0 \) all we get is the equation,
\(
\psi(s-1,a,b,c) = e^{as + b + c \psi(s,a,b,c)}
\)
From,
\(
\psi(s,a,b,c) = \Omega_{j=1}^\infty e^{a(s+j) + b + cz}\bullet z = \phi(-s,-a,b,c)\\
\)