Real Analysis — Lecture Notes
Jane Smith · March 2025
1 Introduction
Let f : ℝ → ℝ be continuous. We define the Riemann integral as:
2 Riemann Integrability
A bounded function f is integrable on [a,b] iff:
where L and U denote the lower and upper Darboux sums for partition P.
Found a typo on line 8:
The integral command is \int, not \intergral. Fix applied automatically.
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[pdflatex] Output: document.pdf (24,891 bytes)
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Typo: \intergral → should be \integral. Here's the fix:

We examine the Lorentz
factor in special relativity...
\end{abstract}
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\documentclass[12pt]{article}
\usepackage{amsmath, amssymb}
\usepackage[margin=1in]{geometry}
\title{Fourier Analysis}
\author{Jane Smith}
\date{\today}
\begin{document}
\maketitle
\section{Transform Definition}
The Fourier transform of $f(t)$ is:
\[
\hat{f}(\xi) = \int_{-\infty}^{\infty}
f(t)\, e^{-2\pi i \xi t}\, dt
\]
\section{Parseval's Theorem}
For square-integrable functions:
\[ \int_{-\infty}^{\infty} |f(t)|^2\, dt
= \int_{-\infty}^{\infty} |\hat{f}(\xi)|^2\, d\xi \]
\end{document}\section{Use cases}
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