chore: update papers, add assumptions in 'Lemmas for Integrals.' paper.

Signed-off-by: Amlal El Mahrouss <amlal@nekernel.org>

1 files changed, 3 insertions, 2 deletions

diff --git a/source/dn001.07/paper.tex b/source/dn001.07/paper.tex
index d2fea32..659aea8 100644
--- a/source/dn001.07/paper.tex
+++ b/source/dn001.07/paper.tex
@@ -6,6 +6,7 @@
 \usepackage{listings}
 \usepackage{xcolor}
 \usepackage{hyperref}
+\usepackage{mathtools}
 \usepackage{amsfonts}
 \usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
 
@@ -25,8 +26,8 @@
 We assume that $u \in \mathbb{R}, \quad \forall u \in \mathbb{R} : u < \pi, \quad 0 < u < \pi$
 
 \begin{equation}
-	I_{1} = \int_{u_{0}}^{u} S(t) dt, \quad t = u, \quad u > 0
+	I_{1} \coloneqq \int_{u_{0}}^{u} S(t) dt, \quad t = u, \quad u > 0
 \end{equation}
-Where $u > 0, \quad u \in \mathbb{R}$, let $S : (0, \pi) \to \mathbb{R}$.
+Where $u, u_{0} > 0, \quad u, u_{0} \in \mathbb{R}$, such that $S : (u, \pi) \to \mathbb{R}$.
 
 \end{document}