Hey, Tommy
I apologize if this is out of nowhere. This notation was developed in a paper of mine.
Suppose \( \mathcal{S} \subseteq \mathbb{C} \) is a domain. Suppose that \( \phi(s,z) : \mathcal{S} \times \mathbb{C} \to \mathbb{C} \) is a holomorphic function. Suppose that \( \gamma \subset \mathcal{S} \) is an arc.
Then the contour integration (wrt the compositional integral) is written,
\(
\int_\gamma \phi(s,z)\,ds\bullet z\\
\)
The direct way to calculate this is to write the following. Let \( \gamma : [0,1] \to \mathcal{S} \); then first we parameterize,
\(
\int_0^1 \phi(\gamma(x),z)\gamma'(x)\,dx\bullet z\\
\)
Take a descending partition \( 1 =x_0 > x_{1} > x_{2} >...>x_{n-2} > x_{n-1} > x_n = 0 \) with sample points \( x_{j} \ge x_{j}^* \ge x_{j-1} \). Let, \( \Delta \gamma_j = \gamma(x_j) - \gamma(x_{j+1}) \) and \( \gamma_j^* = \gamma(x_j^*) \). Let \( ||\Delta|| = \max_j |\Delta \gamma_j| \). Then the integral can be written,
\(
\int_0^1 \phi(\gamma(x),z)\gamma'(x)\,dx\bullet z = \lim_{||\Delta|| \to 0}\Omega_{j=0}^{n-1} z + \phi(\gamma_j^*,z)\Delta \gamma_j \bullet z\\
\)
Where in more familiar notation, if I write \( q_{jn}(z) = z + \phi(\gamma_j^*,z)\Delta \gamma_j \); this just means,
\(
\int_0^1 \phi(\gamma(x),z)\gamma'(x)\,dx\bullet z = \lim_{n\to\infty} q_{0n}(q_{1n}(...q_{(n-1)n}(z)))\\
\)
This is what I call the Riemann-Stieljtes decomposition; as it looks precisely like the Riemann-Stieljtes integral. This thing has it's place in traditional analysis, where it is better known as Euler's method (though he never used contours as I am). If I write,
\(
Y_{ba}(z) = \int_a^b f(s,z)\,ds\bullet z\\
\)
Then,
\(
\frac{d}{db}Y_{ba}(z) = f(b,Y_{ba}(z))\\
Y_{cb}(Y_{ba}(z)) = Y_{ca}(z)\\
Y_{bb}(z) = z\\
\)
And it respects substitution. Where if \( u(\alpha) = a \) and \( u(\beta) = b \);
\(
\int_a^b f(s,z)\,ds\bullet z = \int_{\alpha}^\beta f(u(x),z)u'(x)\,dx\bullet z\\
\)
This is essentially alternative notation for first order differential equations. So now we're going to do contour integration, but with first order differential equations, rather than a primitive. What we've been talking about here is about closed contours.
if \( \phi(s,z) : \mathcal{S} \times \mathbb{C} \to \mathbb{C} \) is holomorphic, and \( \mathcal{S} \) is simply connected. Then for all jordan curves \( \gamma \),
\(
\int_\gamma \phi(s,z)\,ds\bullet z = z\\
\)
Which is the equivalent of Cauchy's integral theorem.
I'll attach here a table of integrations for some simple cases. Let \( p \) be holomorphic.
\(
\int_\gamma p(s) z \,ds\bullet z = z e^{\int_\gamma p(s)\,ds}\\
\int_\gamma p(s)z^2 \,ds\bullet z = \frac{1}{\frac{1}{z} - \int_\gamma p(s)\,ds}\\
\int_\gamma p(s)z^3 \,ds\bullet z = \frac{1}{\sqrt{\frac{1}{z^2} - 2\int_\gamma p(s)\,ds}}\\
\)
And for example, if we take the unit disk as our contour \( \gamma \) and we let \( |\zeta| < 1 \); we get,
\(
\int_\gamma \frac{p(s)z}{s-\zeta}\,ds\bullet z = z e^{2\pi i p(\zeta)}\\
\int_\gamma \frac{p(s)z^2}{s-\zeta}\,ds\bullet z = \frac{1}{\frac{1}{z} - 2\pi i p(\zeta)}\\
\int_\gamma \frac{p(s)z^3}{s-\zeta} \,ds\bullet z = \frac{1}{\sqrt{\frac{1}{z^2} - 4\pi i p(\zeta)}}\\
\)
I wrote an 80 page treatise analyzing these objects. Mphlee and I are discussing a manner of classifying conjugate classes of holomorphic functions. Which, naively, one would write,
\(
\[F,G\] = \{f \|\, f(F(z)) = G(f(z))\}\\
\)
In this paper I created a class of these functions by using contour integration.
Also, it's important to note that this is a STRICT generalization of Cauchy's contour integration. If I take \( p(s) \) which is constant in \( z \) we get,
\(
\int_\gamma p(s)\,ds\bullet z = z + \int_\gamma p(s)\,ds\\
\)
And now the algebra reduces to the abelian algebra of contour integration as you're used to it. Essentially, you can think of this as non-abelian contour integration
.
I apologize if this is out of nowhere. This notation was developed in a paper of mine.
Suppose \( \mathcal{S} \subseteq \mathbb{C} \) is a domain. Suppose that \( \phi(s,z) : \mathcal{S} \times \mathbb{C} \to \mathbb{C} \) is a holomorphic function. Suppose that \( \gamma \subset \mathcal{S} \) is an arc.
Then the contour integration (wrt the compositional integral) is written,
\(
\int_\gamma \phi(s,z)\,ds\bullet z\\
\)
The direct way to calculate this is to write the following. Let \( \gamma : [0,1] \to \mathcal{S} \); then first we parameterize,
\(
\int_0^1 \phi(\gamma(x),z)\gamma'(x)\,dx\bullet z\\
\)
Take a descending partition \( 1 =x_0 > x_{1} > x_{2} >...>x_{n-2} > x_{n-1} > x_n = 0 \) with sample points \( x_{j} \ge x_{j}^* \ge x_{j-1} \). Let, \( \Delta \gamma_j = \gamma(x_j) - \gamma(x_{j+1}) \) and \( \gamma_j^* = \gamma(x_j^*) \). Let \( ||\Delta|| = \max_j |\Delta \gamma_j| \). Then the integral can be written,
\(
\int_0^1 \phi(\gamma(x),z)\gamma'(x)\,dx\bullet z = \lim_{||\Delta|| \to 0}\Omega_{j=0}^{n-1} z + \phi(\gamma_j^*,z)\Delta \gamma_j \bullet z\\
\)
Where in more familiar notation, if I write \( q_{jn}(z) = z + \phi(\gamma_j^*,z)\Delta \gamma_j \); this just means,
\(
\int_0^1 \phi(\gamma(x),z)\gamma'(x)\,dx\bullet z = \lim_{n\to\infty} q_{0n}(q_{1n}(...q_{(n-1)n}(z)))\\
\)
This is what I call the Riemann-Stieljtes decomposition; as it looks precisely like the Riemann-Stieljtes integral. This thing has it's place in traditional analysis, where it is better known as Euler's method (though he never used contours as I am). If I write,
\(
Y_{ba}(z) = \int_a^b f(s,z)\,ds\bullet z\\
\)
Then,
\(
\frac{d}{db}Y_{ba}(z) = f(b,Y_{ba}(z))\\
Y_{cb}(Y_{ba}(z)) = Y_{ca}(z)\\
Y_{bb}(z) = z\\
\)
And it respects substitution. Where if \( u(\alpha) = a \) and \( u(\beta) = b \);
\(
\int_a^b f(s,z)\,ds\bullet z = \int_{\alpha}^\beta f(u(x),z)u'(x)\,dx\bullet z\\
\)
This is essentially alternative notation for first order differential equations. So now we're going to do contour integration, but with first order differential equations, rather than a primitive. What we've been talking about here is about closed contours.
if \( \phi(s,z) : \mathcal{S} \times \mathbb{C} \to \mathbb{C} \) is holomorphic, and \( \mathcal{S} \) is simply connected. Then for all jordan curves \( \gamma \),
\(
\int_\gamma \phi(s,z)\,ds\bullet z = z\\
\)
Which is the equivalent of Cauchy's integral theorem.
I'll attach here a table of integrations for some simple cases. Let \( p \) be holomorphic.
\(
\int_\gamma p(s) z \,ds\bullet z = z e^{\int_\gamma p(s)\,ds}\\
\int_\gamma p(s)z^2 \,ds\bullet z = \frac{1}{\frac{1}{z} - \int_\gamma p(s)\,ds}\\
\int_\gamma p(s)z^3 \,ds\bullet z = \frac{1}{\sqrt{\frac{1}{z^2} - 2\int_\gamma p(s)\,ds}}\\
\)
And for example, if we take the unit disk as our contour \( \gamma \) and we let \( |\zeta| < 1 \); we get,
\(
\int_\gamma \frac{p(s)z}{s-\zeta}\,ds\bullet z = z e^{2\pi i p(\zeta)}\\
\int_\gamma \frac{p(s)z^2}{s-\zeta}\,ds\bullet z = \frac{1}{\frac{1}{z} - 2\pi i p(\zeta)}\\
\int_\gamma \frac{p(s)z^3}{s-\zeta} \,ds\bullet z = \frac{1}{\sqrt{\frac{1}{z^2} - 4\pi i p(\zeta)}}\\
\)
I wrote an 80 page treatise analyzing these objects. Mphlee and I are discussing a manner of classifying conjugate classes of holomorphic functions. Which, naively, one would write,
\(
\[F,G\] = \{f \|\, f(F(z)) = G(f(z))\}\\
\)
In this paper I created a class of these functions by using contour integration.
Also, it's important to note that this is a STRICT generalization of Cauchy's contour integration. If I take \( p(s) \) which is constant in \( z \) we get,
\(
\int_\gamma p(s)\,ds\bullet z = z + \int_\gamma p(s)\,ds\\
\)
And now the algebra reduces to the abelian algebra of contour integration as you're used to it. Essentially, you can think of this as non-abelian contour integration
.