I thought I'd write some plain examples using \( \phi(s,z) = z \); in which we are reduced into the exponential case.
\(
\int_\gamma f(s)z \,ds\bullet z = ze^{\displaystyle \int_\gamma f(s)\,ds}\\
\)
Now when we write,
\(
\int_\gamma (f(s) + g(s))z\,ds\bullet z = \int_\gamma f(s)z \,ds \bullet \int_\gamma g(s)z\,ds\bullet z\\
\)
We mean that if,
\(
F(z) = ze^{\displaystyle \int_\gamma f(s)\,ds}\\
\)
And,
\(
G(z) = ze^{\displaystyle \int_\gamma g(s)\,ds}\\
\)
Then,
\(
F(G(z)) = ze^{\displaystyle \int_\gamma f(s) + g(s)\,ds}\\
\)
And the residue theorem is,
\(
\int_\gamma f(s)z\,ds\bullet z = \Omega_j \text{Rsd}(s = \zeta_j,f(s)z;z)\bullet z = \Omega_j ze^{2 \pi i\text{Res}(s=\zeta_j,f(s))}\,\bullet z = z e^{2 \pi i \sum_j \text{Res}(s=\zeta_j, f(s))}
\)
The thesis of this paper was that in some modded out space; it works exactly the same. In fact, for any function \( \phi = \phi(z) \), we can prove this result using all of the old analysis,
\(
\int_\gamma (f(s) + g(s))\phi(z)\,ds\bullet z = \int_\gamma f(s)\phi(z) \,ds \bullet \int_\gamma g(s)\phi(z)\,ds\bullet z\\
\)
Which is exactly what mphlee has been talking about lately; and the content of the YT video he posted. It's just written very strangely here.
The benefit of writing it this strangely; is that it generalizes in an algebraic way much better than the typical vector space mumbo jumbo.
In which we can now talk about,
\(
\int_\gamma f(s)\phi(s,z)\,ds\bullet z\\
\)
And if we conjugate these things; and "hide" the conjugations using \( \oint \); we get,
\(
\oint_\gamma (f(s)+g(s))\phi(s,z)\,ds\bullet z = \oint_\gamma f(s)\phi(s,z) \,ds \bullet \oint_\gamma g(s)\phi(s,z)\,ds\bullet z\\
\)
This means; explicitly; if \( f,g \) are meromorphic and \( \gamma \) is a Jordan curve:
\(
F = \int_\gamma f(s)\phi(s,z) \,ds \bullet z\\
G = \int_\gamma g(s)\phi(s,z)\,ds\bullet z\\
H = \int_\gamma (f(s)+g(s))\phi(s,z)\,ds\bullet z\\
\)
And there exists functions \( a,b,c \) such that,
\(
a(F(a^{-1}(b(G(b^{-1}(z)))))) = c(H(c^{-1}(z)))\\
\)
And these functions are always solvable as compositional contour integrations.
\(
\int_\gamma f(s)z \,ds\bullet z = ze^{\displaystyle \int_\gamma f(s)\,ds}\\
\)
Now when we write,
\(
\int_\gamma (f(s) + g(s))z\,ds\bullet z = \int_\gamma f(s)z \,ds \bullet \int_\gamma g(s)z\,ds\bullet z\\
\)
We mean that if,
\(
F(z) = ze^{\displaystyle \int_\gamma f(s)\,ds}\\
\)
And,
\(
G(z) = ze^{\displaystyle \int_\gamma g(s)\,ds}\\
\)
Then,
\(
F(G(z)) = ze^{\displaystyle \int_\gamma f(s) + g(s)\,ds}\\
\)
And the residue theorem is,
\(
\int_\gamma f(s)z\,ds\bullet z = \Omega_j \text{Rsd}(s = \zeta_j,f(s)z;z)\bullet z = \Omega_j ze^{2 \pi i\text{Res}(s=\zeta_j,f(s))}\,\bullet z = z e^{2 \pi i \sum_j \text{Res}(s=\zeta_j, f(s))}
\)
The thesis of this paper was that in some modded out space; it works exactly the same. In fact, for any function \( \phi = \phi(z) \), we can prove this result using all of the old analysis,
\(
\int_\gamma (f(s) + g(s))\phi(z)\,ds\bullet z = \int_\gamma f(s)\phi(z) \,ds \bullet \int_\gamma g(s)\phi(z)\,ds\bullet z\\
\)
Which is exactly what mphlee has been talking about lately; and the content of the YT video he posted. It's just written very strangely here.
The benefit of writing it this strangely; is that it generalizes in an algebraic way much better than the typical vector space mumbo jumbo.
In which we can now talk about,
\(
\int_\gamma f(s)\phi(s,z)\,ds\bullet z\\
\)
And if we conjugate these things; and "hide" the conjugations using \( \oint \); we get,
\(
\oint_\gamma (f(s)+g(s))\phi(s,z)\,ds\bullet z = \oint_\gamma f(s)\phi(s,z) \,ds \bullet \oint_\gamma g(s)\phi(s,z)\,ds\bullet z\\
\)
This means; explicitly; if \( f,g \) are meromorphic and \( \gamma \) is a Jordan curve:
\(
F = \int_\gamma f(s)\phi(s,z) \,ds \bullet z\\
G = \int_\gamma g(s)\phi(s,z)\,ds\bullet z\\
H = \int_\gamma (f(s)+g(s))\phi(s,z)\,ds\bullet z\\
\)
And there exists functions \( a,b,c \) such that,
\(
a(F(a^{-1}(b(G(b^{-1}(z)))))) = c(H(c^{-1}(z)))\\
\)
And these functions are always solvable as compositional contour integrations.
