\documentclass[a4paper,12pt]{article}
\title{Mathematical Equations of Physics}
\date{\today}
\author{Mark Allen}
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\begin{document}


\maketitle

\section{Introduction}
This is an attempt to create a list of all possible ``give definition'' or ``prove theorem'' type questions that can be asked based on the notes for the course so far. It includes all exercise/problem type questions given so far in lectures, but doesn't include any homework exercises (yet). The emphasis is on providing a complete overview of what has been covered in the course so far, in chronological order, as well as an easy way to test how well the material is understood, and so I make no attempt to mimic realistic `exam' questions, nor do I attempt to divide the questions into easy and hard. For the type of questions you'll actually see on an exam, see one of the several other lists of questions available.

I intend to make one of these for at least each of the Maths courses, and possibly for the Physics classes too.
\subsection{Week 1}
Todo
\begin{enumerate}
\item
Give the formula for the Fourier Series.
\item
Why is the Fourier series better for periodic functions than the Taylor Series?
\item
Explain how $ \cos $ and $ \sin $ are the basis vectors, in some sense, of the Fourier Series.
\item
Derive the formulae for $a_0$, $a_n$ and $b_n$
\item
Find the Fourier Series for the function
\[
f(x) = \left\{ \begin{array}{ll}
		1 & \mbox{if $0<x\leq\pi$}\\
	     -1 & \mbox{if $-\pi < x \leq 0$} \end{array} \right. \]
\item
What is the Gibbs phenomenen?
\item
What is Dirichlet's Theorem proper?
\item
What is the common extension of Dirichlet's Theorem? (which is often just called Dirichlet's Theorem)?
\item
Give the expressions for complex $\sin$ and $\cos$.
\item
What is the complex dot product?
\item
Give the formula for the complex Fourier Series.
\item
Derive the forumla for $c_n$.
\item
What conditions on $c_n$ must be met if we wish the complex Fourier series to be real valued?
\end{enumerate}
\subsection{Week 2}
Todo
\begin{enumerate}
\item
Using the real Fourier series for the function
\[
f(x) = \left\{ \begin{array}{ll}
		1 & \mbox{if $0<x\leq\pi$}\\
	     -1 & \mbox{if $-\pi < x \leq 0$} \end{array} \right. \]
find the complex Fourier coefficients.
\item
Derive the complex Fourier coefficients for the formula above using the formula for $c_n$.
\item
Find the complex Fourier Series for the function
\[ f(t) = e^t, -\pi < t < \pi \]
\item
Is the function above real valued? (told from the coefficients)
\item
State and prove Parseval's Theorem.
\item
Show that the space of all periodic functions is a vector space. (prove axioms)
\item
Define an inner product space.
\item
What is the inner product we use? Show it satisfies the axioms.
\item
Give the formula for the Fourier transform.
\item
Give the function for the Fourier transform coefficients.
\end{enumerate}
\subsection*{Week 3}
Todo
\begin{enumerate}
\item
How does the Fourier transform functions of time and (say) position?
\item
What condition must be met for $\tilde{f}(k)$ to be well defined?
\item
Define a Schwartz function
\item
Integrate the following function:
 \[
\int\limits_{-\infty}^{+\infty} e^{-\frac{1}{2}at^2}dt
\]
\item
Find the Fourier transform for the function
\[
f(x) = \left\{ \begin{array}{ll}
		1 & -1 < t < 1\\
	     0 & \mbox{otherwise} \end{array} \right. \]
\item
Suppose we have a function with a certain width, what will the Fourier transform look like?
\item
Which of the following functions are or aren't Schwartz functions and why?
\[f(t) = e^{-t} \\
f(t) = e^{-|t|} \]
\item
Find $\tilde{f}(k)$ for the Schwartz function above.
\item
Suppose we have $f(t)$ and $f'(t)$, how do their respective Fourier transforms relate?
\item
What can we say about differentiation in general with regard to Fourier transforms?
\item
Find the solution to the equation $\displaystyle\frac{\partial^2{f}}{\partial{t^2}} + af = g$ in Fourier Space.
\item
Suppose we have $f{t}$ and $f(at)$, how do their respective Fourier transforms relate?
\item
State \emph{Plancheral's Theorem}.
\item
Find a value for $\displaystyle\int\limits_{-\infty}^{+\infty}e^{-2|t|}$ TODO FINISH THIS (WHAT@S GOING ON?)
\end{enumerate}
\subsection*{Week 4}
\begin{enumerate}
\item
Why do we require Schwartz functions for the Fourier Transform? Show why they meet the requirements.
\item
Show that the function $\displaystyle\tilde{f}(k) = \frac{1}{2\pi}$ is not a Schwartz function.
\item
Ignoring that it isn't a Schwartz function, find $\displaystyle\delta(t) = \int\limits_{-\infty}^{+\infty}\frac{1}{2\pi}e^{ikt}dk$



\end{enumerate}

\end{document}
