05/11/2021, 09:48 PM
OK, I'll take time to digest this. But to be clear, I don't still understand then how can you use the omega notation on that set you define.
If the first is a set of complex numbers, the c-integrals evaluated at z. And the second set you define, the one depending only on the function and the singularity chosen is a set of functions...
how can you put that object inside the omega notation? As z varies we get a bunch of sets of functions \(
z\mapsto \text{Rsd}(f,\zeta_j;z)\subseteq \mathbb C\\
\) how do you compose them in the z?
That is: let \( {\rm loops}(\zeta_j):=\{\gamma\,|\,\gamma \,\text{is Jordan, p. oriented, closed and encloses only}\zeta_j \} \)
Do we have \( \text{Rsd}(f,\zeta_j):=\{\int_\gamma f(s,z)\,ds\bullet -:{\mathbb C}\to{\mathbb C}\,|\,\gamma \in {\rm loops}(\zeta_j) \} \) and
\( \text{Rsd}(f,\zeta_j;z):=\{\int_\gamma f(s,z)\,ds\bullet z\in {\mathbb C}\,|\,\gamma \in {\rm loops}(\zeta_j) \} \)?
Maybe it's just an abuse of notation? When I see that Omega notation I picture it as
![[Image: Annotazione-2021-05-11-224552.jpg]](https://i.ibb.co/K7PX5Cv/Annotazione-2021-05-11-224552.jpg)
If the first is a set of complex numbers, the c-integrals evaluated at z. And the second set you define, the one depending only on the function and the singularity chosen is a set of functions...
Quote:\(
\int_\gamma f(s,z)\,ds\bullet z = \Omega_j \text{Rsd}(f,\zeta_j;z)\bullet z\\
\)
how can you put that object inside the omega notation? As z varies we get a bunch of sets of functions \(
z\mapsto \text{Rsd}(f,\zeta_j;z)\subseteq \mathbb C\\
\) how do you compose them in the z?
That is: let \( {\rm loops}(\zeta_j):=\{\gamma\,|\,\gamma \,\text{is Jordan, p. oriented, closed and encloses only}\zeta_j \} \)
Do we have \( \text{Rsd}(f,\zeta_j):=\{\int_\gamma f(s,z)\,ds\bullet -:{\mathbb C}\to{\mathbb C}\,|\,\gamma \in {\rm loops}(\zeta_j) \} \) and
\( \text{Rsd}(f,\zeta_j;z):=\{\int_\gamma f(s,z)\,ds\bullet z\in {\mathbb C}\,|\,\gamma \in {\rm loops}(\zeta_j) \} \)?
Maybe it's just an abuse of notation? When I see that Omega notation I picture it as
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)