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% AUTHOR: Amlal El Mahrouss
% PURPOSE: WG04: Integrals and Equations for Analysis.
\documentclass[11pt, a4paper]{article}
\usepackage{graphicx}
\usepackage{listings}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
\title{Integrals and Equations for Analysis.}
\author{Amlal El Mahrouss\\\texttt{amlal@nekernel.org}}
\date{February 2026}
\begin{document}
\maketitle
\begin{center}
\rule[0.01cm]{17cm}{0.01cm}
\end{center}
\section{Definitions}
We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < b, \quad t = a$
\subsection{The A-Integral:}
\begin{equation}
I_{1} \coloneqq \sum_{n}^{t} \int_{a}^{b} S(x) d \cdot x
\end{equation}
\subsection{The A-Function:}
\begin{equation}
S(x) : (x, y) \to \mathbb{R}
\end{equation}
\subsection{The A-Equation:}
\begin{equation}
I_{2} \coloneqq \int_{a}^{b} E_{1}d \cdot x, \quad E_{1} = Z(x) := \frac{(S(x) \cdot y)}{\Delta t \cdot z}
\end{equation}
Where $\Delta t \in \mathbb{R}$ such that $0 < \delta t$.
\subsection{The K-Equation:}
\begin{equation}
K_{n} \coloneqq \int_{a}^{b} S_{n}(u_{n}) \quad d \cdot u_{n}, \quad \forall u_{n} \in \mathbb{R}
\end{equation}
Where $K_{n} > 0$ for $z \in \mathbb{R}$. \\
Where $S_{n}(u)$ is the A-Function at index $n$ of $u$.
\end{document}
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