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% Printing (14 June 2001):
% dvips -T148.5mm,210mm -o formula-book.ps formula-book
% psbook formula-book.ps | psnup -2 -W148.5mm -H210mm -pa4 > fbook.ps
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% $Header$

% $Log: formula-book.tex,v $
% Revision 1.5  2005/07/08 21:38:03  nicku
% Added sin^{-1} and cos^{-1} in terms of tan^{-1}
% Useful with bc, which only provides atan2()
%
% Revision 1.4  2001/06/19 06:33:34  nicku
% Added use of booktabs style for the table
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% Revision 1.3  2001/06/16 14:08:52  nicku
% Changed figures to using xfig output, together with psfrag.
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% Revision 1.2  2001/06/16 09:46:53  nicku
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% Revision 1.1  2001/06/16 08:27:53  nicku
% Initial revision
%
% Revision 1.5  1996/01/09  11:32:25  Nick
% Now ready to print as an A5 booklet.
% Have fixed cases where the output was way too wide; some other cases remain,
% but perhaps not too much of a problem.
% Also fixed one more typo: in derivation of sum and product formulas.
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% Revision 1.4  1996/01/08  12:45:02  Nick
% Added common representation for binomial coefficients.
%
% Revision 1.3  1996/01/08  12:39:05  Nick
% Corrected five typos.
%
% Revision 1.2  1996/01/07  19:51:38  Nick
% This is a major revision.
% I've included most of the formula book here,
% Still needs to be checked more thoroughly---there are mistakes.
%
% Revision 1.1  1996/01/06  23:43:33  Nick
% Initial revision
%

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{\fancyplain{}{}\textbf{\upshape Maths formula book}}

\lfoot[]{\fancyplain{}{}\tiny For Gunter Beck}
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\title{Maths formula book}
\author{Nick Urbanik}
\date{\normalsize In memory of Gunter Beck, a great teacher}

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\begin{document}

\maketitle
% \thispagestyle{empty}
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\tableofcontents

\subsection*{Introduction}

This is a collection of formulas given by my mathematics teacher,
Gunter Beck, at Meadowbank Technical College, in Australia, during his
teaching of four unit maths (Higher School Certificate)\@.  As he
introduced each topic, he wrote the main formulas for that topic on
the board, and encouraged us to write them into our own formula books.
He died rather quickly from a cancer.  He was my favourite teacher.
After I learned of his death, and noticed that my old formula book was
getting very dog eared from so many years of constant use, I decided
to write this in fond memory of Gunter, using \AmS-\LaTeX{}.  There
are many who miss you, Gunter.

\newpage
\section{Algebraic Results}

\[ \text{if } \quad \frac{a}{b} = \frac{c}{d} \quad \text{ then:} \]
\begin{xalignat}{2}
  ad &= bc &\qquad\text{
    diagonal product} \\
  \frac{a}{c} &= \frac{b}{d} &\qquad\text{ diagonal exchange}\\
  \frac{b}{a} &= \frac{d}{c} &\qquad\text{ inverse}\\
  \frac{a+c}{b} &= \frac{c+d}{d} &\qquad\text{ addend}
\end{xalignat}

\paragraph{Factors of $x^n - a^n$}
\begin{equation}
  x^n - a^n = (x - a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 + \ldots
  + xa^{n-2} + a^{n-1})
\end{equation}

\paragraph{Sum and difference of two cubes}
\begin{align}
  \label{eqn:sumdiffcubes}
  a^3 + b^3 &= (a+b)(a^2 -ab +b^2)\\
  a^3-b^3 &= (a-b)(a^2 + ab + b^2)\\
  a^2 - b^2 &= (a + b)(a - b)
\end{align}


\paragraph{Expansions of various squares and cubes}
\begin{multline}
  (a+b+c+\dots+n)^2 = a^2 + b^2 + c^2 + \dots + n^2 + 2\sum ab \\
  \text{ (where $\sum ab$ represents all possible pairs of
    $a, b, c, \dots, n$)}
\end{multline} 
\begin{gather}
  (a + b+c+d)^2 = a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd +
  2cd\\
  (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc\\
  (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\\
  (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\end{gather}

\section{Absolute value}
\begin{Def}
  The definition of the absolute value of $a$ is
  \begin{equation}
    \label{eqn:absDef}
    |a| = \begin{cases}
      a & \text{ if $a > 0$}\\
      0 & \text{ if $a = 0$}\\
      -a & \text{ if $a < 0$}
    \end{cases}
  \end{equation}
\end{Def}

\paragraph{Some properties}

\begin{align}
  \text{if } |x| &= a \qquad \text{ then } \quad x = \pm a\\
  \text{if } |x| &< a \qquad \text{ then } \quad -a < x < a\\
  \text{if } |x| &> a \qquad \text{ then } \quad x < a \text{ or } x >
  a
\end{align}

\section{Inequalities}
If $a> b$, $c> d$, then:
\begin{xalignat}{2}
  a \pm c &> b \pm c & \\
  ac &> bc &\qquad (c > 0)\\
  ac &< bc &\qquad (c < 0)\\
  ac &> bd &\qquad (a, b, c, d \text{ all } > 0)\\
  a^2 &> b^2 &\qquad (a, b \text{ both } > 0) \\
  \frac{1}{a} &< \frac{1}{b} &\qquad (a, b \text{ both } > 0)
\end{xalignat}

\section{Trigonometry}
``All Stations To Central'': \parbox[c]{0.3\linewidth}{\includegraphics[width=0.5\linewidth]{all-stations-to-central}}
\paragraph{Reciprocal ratios}
\begin{align}
  \csc \theta &= \frac{1}{\sin \theta}\\
  \sec \theta &= \frac{1}{\cos \theta}\\
  \cot \theta &= \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\\
  \tan \theta &= \frac{\sin \theta}{\cos \theta}
\end{align}

\paragraph{Pythagorean identities}
\begin{align}
  \sin^2 \theta + \cos^2 \theta &= 1\\
  \tan^2 \theta + 1 &= \sec^2 \theta\\
  1 + \cot^2 \theta &= \csc^2 \theta
\end{align}

\paragraph{Inverse formulas}
\begin{align}
% $ perldoc -f sin
% For the inverse sine operation, you may use the
%          "Math::Trig::asin" function, or use this relation:
%              sub asin { atan2($_[0], sqrt(1 - $_[0] * $_[0])) }
  \label{eq:perlfunc-atan2-to-asin}
  \sin^{-1}x &= \tan^{-1}\frac{x}{\sqrt{1 - x^2}} & x \ne 1\\
% $ perldoc -f cos
% For the inverse cosine operation, you may use the
%          "Math::Trig::acos()" function, or use this relation:
%              sub acos { atan2( sqrt(1 - $_[0] * $_[0]), $_[0] ) }
  \cos^{-1}x &= \tan^{-1}\frac{\sqrt{1 - x^2}}{x} & x \ne 0
\end{align}

\paragraph{Co-ratios---(complementary angles)}
\begin{xalignat}{2}
  \sin(90\Deg - \theta) &= \cos \theta &
  \cos(90\Deg - \theta) &= \sin \theta \\
  \tan(90\Deg - \theta) &= \cot \theta &
  \cot(90\Deg - \theta) &= \tan \theta \\
  \sec(90\Deg - \theta) &= \csc \theta & \csc(90\Deg - \theta) &= \sec
  \theta
\end{xalignat}

\begin{xalignat}{2}
  \sin(180\Deg - \theta) &= \sin \theta &
  \sin(180\Deg + \theta) &= -\sin \theta \\
  \cos(180\Deg - \theta) &= -\cos \theta &
  \cos(180\Deg + \theta) &= -\cos \theta \\
  \tan(180\Deg - \theta) &= -\tan \theta & \tan(180\Deg + \theta) &=
  \tan \theta
\end{xalignat}
\begin{xalignat}{2}
  \sin(360\Deg - \theta) &= \sin -\theta = -\sin \theta &&\quad \text{(odd)}\\
  \cos(360\Deg - \theta) &= \cos -\theta = \cos \theta &&\quad \text{(even)}\\
  \tan(360\Deg - \theta) &= \tan -\theta = -\tan \theta &&\quad
  \text{(odd)}
\end{xalignat}

\paragraph{Exact values}
\mbox{}\\[2ex]
{\scriptsize%
\psfrag{1}{1}%
\psfrag{45}{$45\Deg$}%
\psfrag{r2}{$\sqrt{2}$}%
\includegraphics{forty-five-triangle}%
\hspace{3em}%
\psfrag{2}{2}%
\psfrag{30}{$30\Deg$}%
\psfrag{60}{$60\Deg$}%
\psfrag{r3}{$\sqrt{3}$}%
\includegraphics[scale=0.5]{sixty-thirty-triangle}%
}

%{\allowdisplaybreaks
\begin{align}
  \sin 45\Deg = \cos 45\Deg &= \frac{1}{\sqrt{2}}\\
  \sin 30\Deg = \cos 60\Deg &= \frac{1}{2}\\
  \sin 60\Deg = \cos 30\Deg &= \frac{\sqrt{3}}{2}\\
  \tan 45\Deg = \cot 45\Deg &= 1\\
  \tan 30\Deg = \cot 60\Deg &= \frac{1}{\sqrt{3}}\\
  \tan 60\Deg = \cot 30\Deg &= \sqrt{3}\\
  \sin 15\Deg = \cos 75\Deg &= \frac{\sqrt{6} - \sqrt{2}}{4}\\
  \sin 75\Deg = \cos 15\Deg &= \frac{\sqrt{6} + \sqrt{2}}{4}\\
  \tan 15\Deg = \cot 75\Deg &= 2 - \sqrt{3}\\
  \tan 75\Deg = \cot 15\Deg &= 2 + \sqrt{3}
\end{align}
%} % End of allowdisplaybreaks

\paragraph{Addition formulas}
\begin{align}
  \sin(\theta \pm \phi) &= \sin \theta\cos\phi \pm \cos\theta\sin\phi\\
  \cos(\theta \pm \phi) &= \cos\theta\cos\phi \mp \sin\theta\sin\phi\\
  \tan(\theta \pm \phi) &= \frac{\tan\theta \pm \tan\phi}%
                                {1 \mp \tan\theta\tan\phi}\\
   \cot(\theta \pm \phi) &= \frac{\cot\theta\cot\phi \mp 1}%
                                 {\cot\theta\pm  \cot\phi}
\end{align}
\begin{align}
  \sin 2\theta &= 2 \sin \theta\cos\theta\\
  \cos 2\theta &= \cos^2\theta - \sin^2 \theta\\
               &= 1 - 2\sin^2 \theta \\
               & = 2\cos^2 \theta - 1\\
  \tan 2\theta &= \frac{2\tan\theta}{1 - \tan^2\theta}\\
  \sin^2\theta &= \frac{1 -\cos2\theta}{2}\\
  \cos^2\theta &= \frac{1 + \cos 2\theta}{2}
\end{align}


\paragraph{Sine rule}

\begin{equation}
  \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\end{equation}

\paragraph{Area of a triangle}

\begin{equation}
  \text{Area of any triangle } = \frac{ab\sin C}{2}
\end{equation}

\paragraph{Cosine rule}
\begin{xalignat}{2}
  a^2 &= b^2 + c^2 - 2bc\cos A &\qquad \text{(side formula)}\\
  \cos A &= \frac{b^2 + c^2 - a^2}{2bc} &\qquad\text{(angle formula)}
\end{xalignat}

\paragraph{``Little $t$'' formula}

\begin{center}
  %\input{littlet.pic}
  \psfrag{x}{$x$}%
  \psfrag{2t}{$2t$}%
  \psfrag{1+t2}{$1+t^2$}%
  \psfrag{1-t2}{$1-t^2$}%
  \includegraphics[width=0.4\linewidth]{littlet}
\end{center}
\begin{align}
  t &= \tan\frac{x}{2}\\
  \tan x &= \frac{2\tan \frac{x}{2}}{1 - \tan^2 \frac{x}{2}}\\
  &= \frac{2t}{1 - t^2}\\
  \sin x &= \frac{2t}{1 + t^2}\\
  \cos x &= \frac{1 - t^2}{1 + t^2}\\
  \text{if $t = \tan\frac{x}{2}$ then } dx &= \frac{2\,dt}{1 + t^2}
\end{align}

\paragraph{Period and amplitude}

\begin{alignat}{2}
  y &= a \sin(\omega t + \phi) \qquad
  & \text{amplitude} &= a\text{, period} = \frac{2\pi}{\omega} \\
  y &= a \cos(\omega t + \phi) \qquad
  & \text{amplitude} &= a\text{, period} = \frac{2\pi}{\omega} \\
  y &= a \tan(\omega t + \phi) \qquad
  & \text{amplitude} &= \infty\text{, period} = \frac{\pi}{\omega}
\end{alignat}

\paragraph{Special limits}
\begin{align}
  \lim_{\theta \to 0} \frac{\sin \theta}{\theta} &= 1\\
  \lim_{\theta \to 0} \frac{\tan \theta}{\theta} &= 1
\end{align}

\paragraph{The $3\theta$ results}

\begin{align}
  \sin 3\theta &= 3 \sin \theta - 4\sin^3 \theta \\
  \cos 3\theta &= 4 \cos^3 \theta - 3\cos \theta \\
  \tan 3\theta &= \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2
    \theta}
\end{align}

\subsection{Sum and product formulas}

\begin{align}
  \sin(\theta + \phi) + \sin( \theta - \phi) &= 2\sin\theta \cos\phi\\
  \sin(\theta + \phi) - \sin(\theta - \phi) &= 2 \sin\phi\cos\theta\\
  \cos(\theta + \phi) + \cos(\theta - \phi) &= 2\cos\theta\cos\phi\\
  \cos(\theta + \phi) - \cos(\theta - \phi) &= -2\sin\theta \sin\phi
\end{align}
\begin{equation}
  \text{if }\begin{cases}
    \angle_1 = \theta + \phi\\
    \angle_2 = \theta - \phi\\
  \end{cases}%
  \text{ then }%
  \begin{aligned}
    \theta &= \tfrac{1}{2}(\angle_1 + \angle_2)\\
    \phi &= \tfrac{1}{2}(\angle_1 - \angle_2)\\
  \end{aligned}
  \qquad \text{use } \theta > \phi
\end{equation}

Using these formulas:
\begin{align}
  \sin a &= \frac{e^{ia} - e^{-ia}}{2i}\\
  \cos a &= \frac{e^{ia} + e^{-ia}}{2}
\end{align}
it is not too hard to derive the sum and product formulas, e.g.,
\begin{equation}
  \begin{split}
    \cos a \cos b
    &= \tfrac{1}{4}\left(e^{ia} + e^{-ia}\right)
                   \left(e^{ib} + e^{-ib}\right) \\
%
    &= \tfrac{1}{4}\left(e^{i(a+b)} + e^{i(a-b)}
                       + e^{-i(a-b)} + e^{-i(a+b)}\right) \\
%
    &= \tfrac{1}{4}\left(e^{i(a+b)} + e^{-i(a+b)}\right)
     + \tfrac{1}{4}\left(e^{i(a-b)} + e^{-i(a-b)}\right) \\
%
    &= \tfrac{1}{2}\cos(a+b) + \tfrac{1}{2}\cos(a-b).
  \end{split}
\end{equation}

\subsection{Sum of two waves}

\begin{equation}
  a \cos \omega t + b \sin \omega t = \sqrt{a^2 + b^2}\,\cos\left(\omega t -
    \tan^{-1} \frac{b}{a}\right)
\end{equation}
Here we use the fact that
\begin{align}
  a+jb &= \sqrt{a^2 + b^2}\,e^{j\tan^{-1}\frac{b}{a}} \label{eqn:a+jb}\\
  \intertext{and}
  a - jb &= \overline{a + jb} = \sqrt{a^2 +
    b^2}\,e^{-j\tan^{-1}\frac{b}{a}} \label{eqn:a-jb}\\
\end{align}
\begin{align}
  a \cos \omega t + b \sin \omega t
  &= \frac{a}{2}\left(e^{j\omega t} + e^{-j\omega t}\right)
  + \frac{b}{2j}\left(e^{j\omega t} - e^{-j\omega t}\right) \\
  &= \frac{a}{2}\left(e^{j\omega t} + e^{-j\omega t}\right)
  - \frac{jb}{2}\left(e^{j\omega t} - e^{-j\omega t}\right) \\
  &= \frac{1}{2}\left(e^{j\omega t}(a - jb) + e^{-j\omega t}(a +
    jb)\right) \quad \text{(from \eqref{eqn:a+jb} and \eqref{eqn:a-jb})} \\
  &= \frac{1}{2}\sqrt{a^2 +  b^2}
  \left(e^{j\omega t}e^{-j\tan^{-1}\frac{b}{a}} + e^{-j\omega
      t}e^{j\tan^{-1}\frac{b}{a}}\right)
  \\
  &= \frac{\sqrt{a^2 + b^2}}{2}
  \left(e^{j(\omega t - \tan^{-1}\frac{b}{a})} +
    e^{-j(\omega t - \tan^{-1}\frac{b}{a})}\right) \\
  &= \sqrt{a^2 + b^2} \,\cos\left(\omega t - \tan^{-1}\frac{b}{a}\right)
\end{align}

\paragraph{Period of sum of two sine waves:}

If $\cos \omega_1 t + \cos \omega_2 t$ is periodic with period $T$,
then $\exists m,n \in \mathbb{Z} : \omega_1 T = 2\pi m, \omega_2 T =
2\pi n$, i.e., $\dfrac{\omega_1}{\omega_2} = \dfrac{m}{n}$ is
rational.

Also, to find $T$:

$\omega_2 = \dfrac{n}{m}\omega_1$, \quad $T = \dfrac{2\pi}{\omega_1} m
= \dfrac{2 \pi}{\omega_2} n$.

The same is true of $\cos(\omega_1 t + \theta), \cos(\omega_2 t +
\phi), \quad \theta, \phi \in \mathbb{R}$, i.e., phase doesn't affect
the value of $T$ here.

\section{Coordinate geometry: straight lines}

\paragraph{Equations of straight lines}

{\allowdisplaybreaks

%\begin{xalignat}{2}
%y &= b & \text{line $\parallel$ $y$ axis through $(a,b)$}\\
%x &= a & \text{line $\parallel$ $x$ axis through $(a,b)$}\\
%y &= mx + b & \text{Slope intercept form}\\
%y - y_1 &= m(x - x_1) & \text{point slope form: through $(x_1, y_1)$
%  and slope $m$}\\
%y - y_1 &= \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) & \text{Two point
%  form: through $(x_1, y_1)$, $(x_2, y_2)$}\\
%\frac{x}{a} + \frac{y}{b} &= 1 & \text{2 intercept form: $x$ intercept
%  $a$, $y$ intercept $b$}\\
%Ax + By + C &= 0 & \text{slope = $\frac{-A}{B}$, $x$ intercept =
%  $\frac{-C}{A}$, $y$ intercept = $\frac{-C}{B}$}
%\end{xalignat}

The equation of a line $\parallel$ $y$ axis through $(a,b)$ is
\begin{equation}
  y = b
\end{equation}
The equation of a line $\parallel$ $x$ axis through $(a,b)$ is
\begin{equation}
  x = a 
\end{equation}
The equation of a line with slope $m$ and $y$ intercept $b$ is
\begin{equation}
  y = mx + b
\end{equation}
The equation of a line with slope $m$ through $(x_1, y_1)$ is
\begin{equation}
  y - y_1 = m(x - x_1)
\end{equation}
The equation of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is
\begin{equation}
  y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)
\end{equation}
The equation with $x$ intercept $a$ and $y$ intercept $b$ is
\begin{equation}
  \frac{x}{a} + \frac{y}{b} = 1
\end{equation}
This ``general form'' of equation for a line has slope =
$\frac{-A}{B}$, $x$ intercept = $\frac{-C}{A}$, $y$ intercept =
$\frac{-C}{B}$:
\begin{equation}
  Ax + By + C = 0
\end{equation}


} % End of allowdisplaybreaks

\paragraph{Distance and point formulas}
\mbox{}

The distance between $(x_1, x_2)$ and $(x_2, y_2)$ is
\begin{equation}
  \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\end{equation}
The distance of $(x_1, y_1)$ from $Ax + Bx + C = 0$ is
\begin{equation}
  d = \left|\frac{Ax_1 + By_1 + C}{\sqrt{A^2 + B^2}}\right|
\end{equation}
The coordinates of the point which divides the join of $Q =
(x_1, y_1)$ and $R = (x_2, y_2)$ in the ratio $m_i : m_2$ is
\begin{equation}
  P = \left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 +
      m_2y_1}{m_1 + m_2}\right)
\end{equation}
From $Q$ to $P$ to $R$.  If $P$ is
outside of the line $QR$, then $QP$, $RP$ are measured in opposite
senses, and $\frac{m_1}{m_2}$ is negative.

The coordinates of the midpoint of the join of $(x_1, y_1)$ and $(x_2,
y_2)$ is
\begin{equation}
  \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\end{equation}
The angle $\alpha$ between two lines with slopes $m_1$, $m_2$ is given
by
\begin{xalignat}{2}
  \tan \alpha &= \left|\frac{m_2 - m_1}{1 + m_2m_1}\right| & \text{ for
    $\alpha$ acute} \\
  \tan \alpha &= -\left|\frac{m_2 - m_1}{1 + m_2m_1}\right| & \text{ for
    $\alpha$ obtuse}
\end{xalignat}

\section{Logarithms and Indexes}

\paragraph{Basic index laws}
\begin{align}
  a^m \times a^n & =  a^{m + n} \\
  a^m \div a^n & =  a^{m - n} \\
  (a^m)^n & = a^{mn} \\
  a^0 &= 1\\
  a^{-1} & = \frac{1}{a^n}\\
  a^{\frac{m}{n}} & = \sqrt[n]{a^m}\qquad\mbox{(if $a^m > 0$)}\\
  (ab)^n & = a^nb^n\\
  \sqrt[n]{\frac{a}{b}} & = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
\end{align}

\begin{Def}[Definition of a logarithm]
  
  $\mbox{if }\quad \log_a x = y \quad\mbox{ then }\quad x = a^y.$

  The number $a$ is called the {\em base\/} of the logarithm.
\end{Def}

\paragraph{Change of base of logarithms}

\begin{equation}
  \log_a x = \frac{\log_b x}{\log_b a}
\end{equation}

\paragraph{Basic laws of logarithms}
\begin{eqnarray}
  \log_a mn & = & \log_a m + \log_a n\\
  \log_a \frac{m}{n} & = & \log_a m - \log_a n\\
  \log_a m^x &= & x\log_a m
\end{eqnarray}


\begin{align}
  a^x &= e^{x\ln a}\\
  x^e &= e^{e\log x}\\
  e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \qquad
  \forall x.
\end{align}

\section{Series and Sequences}

\paragraph{Arithmetic progression (AP):} is of the form:
\[ a, a+d, a+2d,\dots,u_n \]
\begin{equation}
  u_n = a + (a-1)d
\end{equation}
\paragraph{Tests for arithmetic progression}
\[ u_2 - u_1 = u_3 - u_2 = \dots = u_n - u_{n-1} \]
\[\text{if $a, b, c$ are sequential terms in AP, then } b =
\frac{a+c}{2} \]
\paragraph{Sum of an AP}
\begin{align}
  S_n &= \sum^n_{k=1} \big(a + (k-1)d\big)\\
  u_n &= S_n - S_{n-1}
\end{align}
\begin{xalignat}{2}
  S_n &= \frac{n(a + u_n)}{2} & \quad \text{ if given last term
    $u_n$}\\
  S_n &= \frac{n\big(2a + (n-1)d\big)}{2} & \quad \text{ if last term
    not given}
\end{xalignat}
\paragraph{Geometric progression (GP):}

is of the form:
\[a, ar, ar^2, \ldots, ar^{n-1} \]
\begin{equation}
  u_n = ar^{n-1}
\end{equation}

\paragraph{Test for GP}
\[ \frac{u_2}{u_1} = \frac{u_3}{u_2} = \dots = \frac{u_n}{u_{n-1}} \]
\[ u_2^2 = u_1 u_3, \dots, u_{n-1}^2 = u_{n-2}u_n \]

\paragraph{Sum of GP}
\[ S_n = a\sum^n_{k=1} r^{k - 1} \]
\begin{xalignat}{2}
  S_n &= \frac{a(r^n - 1)}{r-1} & \text{ use if $|r| > 1$}\\
  S_n &= \frac{a(1 - r^n)}{1 - r} & \text{ use if $|r| < 1$}
\end{xalignat}

\paragraph{Sum to infinity}
\begin{equation}
  S_\infty = \frac{a}{1 - r} \qquad \text{ \hfill where $|r| < 1$.}
\end{equation}

\begin{equation}
  \begin{split}
    \sum^n_{k=1} r^k &= r\sum^n_{k=1} r^{n-1}\\
    &= \frac{r(r^n - 1)}{r-1}\\
    &= \frac{r^{n+1} - r}{r-1}
%             \\ &= r^{n+1} % THIS IS NOT FINISHED!
  \end{split}
\end{equation}

\paragraph{Some other useful sums}
\begin{align}
  \sum^n_{k=0} k^2 &= \frac{n(n+1)(2n+1)}{6}\\
  \sum^n_{i=0} k^3 &= \frac{n^2(n+1)^2}{4}\\
  \sum^n_{k=0} k &= \frac{n(n+ 1)}{2}
\end{align}
\begin{align}
  e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots\\
  a^x = e^{x\ln a} &= 1 + x\ln a + \frac{(x\ln a)^2}{2!} + \frac{(x\ln
    a)^3}{3!} + \dots\\
  \ln(1 + x) &= x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} +
  \dots, \qquad -1 < x \le 1.
\end{align}

\paragraph{Series relevant to $\mathcal{Z}$ transforms}

\begin{xalignat}{2}
  \sum_{n=0}^M z^n &= 1 + z + z^2 + z^3 + \dots + z^{M-1} + z^M
  & \label{eqn:zSum} \\
%
  z\sum_{n=0}^M z^n &= z + z^2 + z^3 + \dots + z^{M-1} + z^{M+1}
  &
  \text{multiplying b.s. by $z$} \label{eqn:zzSum} \\
%
  (1-z)\sum_{n=0}^M z^n &= 1 - z^{M+1} & \text{ subtracting
    \eqref{eqn:zSum} $-$ \eqref{eqn:zzSum}} \\
%
  \therefore \sum_{n=0}^M z^n &= \frac{1-z^{M+1}}{1 - z} & \\
%
  \intertext{Now}
%
  \sum_{n=0}^\infty z^n &= \lim_{M \to \infty} \sum_{n=0}^M z^n & \\
%
  &= \lim_{M\to\infty} \frac{1 - z^{M+1}}{1 - z} & \\
%
  &= \frac{1}{1-z} \quad \forall z : |z| < 1 &
\end{xalignat}
This is from Thomas \& Finney, pages 610--611.
\section{Finance}

\paragraph{Simple interest}
\begin{equation}
  I = Ptr
\end{equation}
where $P$ = principal, $t$ = time, $r$ = rate (i.e., $6\%
\longrightarrow
\frac{6}{100}$)

\paragraph{Compound interest}
\begin{equation}
  A = P(1 + r)^n
\end{equation}
where $P$ = principal, $n$ = time, $r$ = rate as above, $A$ = amount
you get.

\paragraph{Reducible interest:}
This uses $S_n = \frac{a(r^n - 1)}{r - 1}$ where $a$ is $M$, $r = 1 +
\frac{R}{100}$.
\begin{align}
  \frac{M\big((1+\frac{R}{100})^n - 1\big)}{\frac{R}{100}} &= P\left(1 +
    \frac{R}{100}\right)^n\\
  \text{i.e., } M &= \frac{P(1 + \frac{R}{100})^n\cdot \frac{R}{100}}{(1
    + \frac{R}{100})^n - 1}
\end{align}
where $M$ = equal installment paid per unit time\\
$R$ = \% rate of interest per unit time (Note: same unit of time as
for $M$)\\
$n$ = number of installments.

\paragraph{Superannuation}

\begin{equation}
  S_n = \frac{A\Big(1 + \frac{R}{100}\big((1 + \frac{R}{100})^n - 1
    \big)\Big)}{\frac{R}{100}}
\end{equation}
where $A$ is an equal installment per unit time.  $n$, $R$ are as above.

\section{Quadratic equations}
The solution of $ax^2 + bx + c = 0$ is
\begin{equation}
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}

\section{Differentiation}

\paragraph{From first principles:}
\begin{align}
  f'(x) &= \lim_{h\to 0}\frac{f(x + h) - f(x)}{h}\\
  \text{or } f'(c) &= \lim_{x \to c} \frac{f(x) - f(c)}{x - c}
\end{align}
\begin{align}
  \frac{d}{dx} x^n &= nx^{n - 1}\\
  \frac{d}{dx} c &= 0\\
  \frac{d}{dx} c\,f(x) &= c\,f'(x)\\
  \frac{d}{dx}\big(f(x) \pm g(x)\big) &= f'(x) \pm g'(x)
\end{align}

\paragraph{Product rule}
\begin{align}
  \frac{d}{dx}(uv) &= u\frac{dv}{dx} + v\frac{du}{dx}\\
  \frac{d}{dx}(uvw) &= uv\frac{dw}{dx} + uw\frac{dv}{dx} +
  vw\frac{du}{dx}
\end{align}

\paragraph{Quotient rule}
\begin{equation}
  \frac{d}{dx}\frac{u}{v} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}
\end{equation}

\paragraph{Chain rule (Function of a function rule, composite function
  rule, substitution rule)}
\begin{equation}
  \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\end{equation}

\subsection{Derivatives of trig functions}

{\allowdisplaybreaks
\begin{align}
  \frac{d}{dx} \sin x &= \cos x\\
  \frac{d}{dx} \cos x &= -\sin x \\
  \frac{d}{dx} \tan x &= \sec^2 x\\
% CHECK THIS!!!!!!!!!!!!!!!
  \frac{d}{dx} \sin^{-1} x &= \frac{1}{\sqrt{1 - x^2}} \qquad
  -\frac{\pi}{2} < \sin^{-1} x < \frac{\pi}{2}\\
  \frac{d}{dx}  \cos^{-1} x &= \frac{- 1}{\sqrt{1 - x^2}} \qquad
  -\frac{\pi}{2} < \cos^{-1} x < \frac{\pi}{2}\\
  \frac{d}{dx} \tan^{-1} x &= \frac{1}{1 + x^2} \qquad
  -\frac{\pi}{2} < \tan^{-1} x < \frac{\pi}{2}\\ 
  \frac{d}{dx} \sec x &= \sec x \tan x \\
  \frac{d}{dx} \csc x &= -\csc x \cot x\\
  \frac{d}{dx} \cot x &= -\csc^2 x \\
  \frac{d}{dx} \sinh x &= \cosh x \\
  \frac{d}{dx} \cosh x &= \sinh x
\end{align}
} % end of allowdisplaybreaks.

\subsection{Derivatives of exponential and log functions}

\begin{align}
  \frac{d}{dx} \ln f(x) &= \frac{f'(x)}{f(x)} \\
  \frac{d}{dx} e^{f(x)} &= f'(x) e^{f(x)} \\
  \frac{d}{dx} e^x &= e^x \\
  \frac{d}{dx} a^x &= \ln a\cdot a^x \qquad \text{ since } a^k = e^{k\ln
    a}\\
  \frac{d}{dx} \ln y &= \frac{d}{dx} \ln y \,\frac{dy}{dx}\\
  \frac{d}{dx} \ln y &= \frac{1}{y} \frac{dy}{dx}
\end{align}

\section{Integration}

\subsection{Integration by parts}

\begin{equation}
  \int u \frac{dv}{dx}\,dx = uv - \int v\frac{du}{dx}\,dx
\end{equation}

\subsection{Numerical integration}

\paragraph{Trapezoidal rule}
\begin{equation}
  \int_a^b f(x)\,dx \approx \frac{b - a}{2n}\big(y_0 + y_n + 2( y_1 +
  y_2 + y_3 + \dots + y_{n-1})\big)
\end{equation}

\paragraph{Simpson's rule}

\begin{equation}
  \int_a^b f(x)\,dx \approx \frac{b-a}{3n}(y_0 + 4y_1 + 2y_2 + 4y_3 +
  2y_4 + \dots + 4y_{n-1} + y_n) \qquad \text{ $n$ is even}
\end{equation}

\subsection{Indefinite integrals}

{\allowdisplaybreaks

\begin{align}
  \int K\,dx &= Kx + c\\
  \int x^n\,dx &= \frac{x^{n + 1}}{n + 1} + c\\
  \int (ax + b)^n\,dx &= \frac{(ax + b)^{n + 1}}{a(n + 1)} + c\\
  \int \sin ax\,dx &= \frac{-1}{a}\cos ax + c\\
  \int \cos ax \, dx &= \frac{1}{a}\sin ax + c\\
  \int \sec^2 ax \,dx &= \frac{1}{a}\tan ax + c\\
  \int e^{ax}\,dx &= \frac{1}{a}e^{ax} + c\\
  \int \frac{dx}{x} &= \ln x + c\\
  \int \frac{f'(x)}{f(x)}\,dx &= \ln f(x) + c\\
  \int \frac{dx}{\sqrt{1 - x^2}}\, dx &= \sin^{-1}x + c \\
  \int \frac{-1}{\sqrt{1 - x^2}}\,dx &= \cos^{-1}x + c \\
  \int \frac{dx}{1 + x^2} &= \tan^{-1} x + c \\
  \int \frac{dx}{\sqrt{a^2 - x^2}} &= \sin^{-1}\frac{x}{a} + c\\
  &= \cos^{-1}\frac{x}{a} + c_2, \qquad
  -a < x < a\\
  \int \frac{dx}{a^2 + x^2} &= \frac{1}{a}\tan^{-1}\frac{x}{a} + c\\
  \int e^u\,du &= e^u + c, \quad \text{ i.e., } \int e^{f(x)}f'(x)\,dx
  =
  e^{f(x)} + c\\
  \int e^{ax}\,dx &= \frac{e^{ax}}{a} + c\\
  \int \frac{1}{x^2 - a^2}\,dx &= \frac{1}{2a}\log\frac{x-a}{a+a} + c\\
  \int \frac{dx}{a^2 - x^2} &= \frac{1}{2a} \log \frac{a + x}{a - x} +
  c \\
  \int \ln x\,dx &= x\ln x - x + c\\
  \int \frac{dx}{\sqrt{a^2 + x^2}} &= \log(x + \sqrt{a^2 + x^2}) + c\\
  \int \frac{dx}{\sqrt{x^2 - a^2}} &= \log(x + \sqrt{x^2 - a^2}) + c
  \quad \forall x : |x| > a \\
  \int \tan x\,dx &= \log\sec x + c \\
  \int \sec x\,dx &= \ln(\sec x + \tan x) + c\\
  &= -\ln(\sec x - \tan x) + c\\
  &= \ln \tan\left(\frac{\pi}{4} + \frac{x}{2}\right) + c\\
  \int \cot x\,dx &= \ln \sin x + c \\
  \int \csc^2 (ax)\,dx &= -\frac{1}{a}\cot( ax) + c \\
  \int a^x\,dx = \int e^{x\ln a}\,dx &= \frac{e^{x\ln a}}{\ln a} + c
  = \frac{a^x}{\ln a} + c_2\\
  &\phantom{=}\text{(let $y=a^x$, then $\ln y = x\ln a$, so $y =
    e^{x\ln
      a}$).}\\
  \int \cosh ax\,dx &= \frac{1}{a} \sinh ax + c \\
  \int \sinh ax\,dx &= \frac{1}{a} \cosh ax + c
\end{align}
} % End of allowdisplaybreaks

\paragraph{Trig substitutions}
\begin{align}
  \text{for } \quad a^2 &+   x^2 \qquad \text{ try } \qquad x = a\tan
  \theta \\
  \quad a^2 &-   x^2 \qquad \text{ try } \qquad x = a\sin
  \theta \\
  \quad x^2 &+   a^2 \qquad \text{ try } \qquad x = a\sec
  \theta
\end{align}

\begin{equation}
  \text{if } t = \tan \frac{x}{2} \text{ then } dx = \frac{2\,dt}{1 +
   t^2}
\end{equation}

\subsection{Definite integrals}

\begin{align}
  \frac{d}{dx}\left\{\int_a^0 f(t)\,dt \right\} &= f(x)\\
  \int_0^a f(x)\,dx &= \int_0^a f(a - x)\,dx
\end{align}

\section{Areas and volumes of geometric shapes}

Length of an arc with angle $\theta$ and radius $r$ is $\ell =
r\theta$.

Area of a sector angle $\theta$ and radius $r$ is $A =
\frac{1}{2}r^2\theta$

Area of a segment of a circle with angle $\theta$ and radius $r$ is $A
= \frac{1}{2}r^2(\theta - \sin \theta)$
\begin{align*}
  \text{Area of triangle} &= \frac{1}{2}r^2\sin\theta\\
  \therefore \text{area of segment} &= \frac{1}{2}r^2\theta -
  \frac{1}{2}r^2\sin\theta
\end{align*}

Area of a triangle with height $h$ and base $b$ is $A = \frac{1}{2}hb$

Area of parallelogram with height $h$ and base $b$ is $A = bh$

Area of a trapezium with parallel sides of length $a, b$ and height
$h$ is $A = \frac{1}{2}h(a + b)$

Area of a rhombus with diagonals of length $a, b$ is $A =
\frac{1}{2}ab$

Area of a circle is $A = \pi r^2$, circumference is $C = 2\pi r$

Area of an ellipse with semi-major axis length $a$, semi-minor axis
length $b$ is $A = \pi ab$.  (Note that $a$, $b$ become the radius as
the ellipse approaches the shape of a circle.)

\subsection{Volumes}

In the following, $h$ is the height of the shape, $r$ is a radius.

Right rectangular prism or oblique rectangular prism with base lengths
$a$ and $b$: $V = abh$

Cylinder with circular base, both upright and oblique, $V = \pi r^2 h$

Pyramids with rectangular bases of side length $a$, $b$, both right
pyramids and oblique pyramids: $V = \frac{1}{3}abh$

Cones: $V = \frac{1}{3}\pi r^2 h$

The curved surface area of an upright cone with length from apex to
edge of the base $s$ ($s$ is not the height, but the length of the side
of the cone): $S = \pi rs$.

Volume of a sphere $V = \frac{4}{3}\pi r^3$

Surface area of a sphere is $S = 4\pi r^2$

\section{Simple harmonic motion}

\begin{Def}
  \begin{equation}
    \ddot{x} = -n^2x \qquad \qquad \text{ $n$ is any constant}
  \end{equation}
\end{Def}

\paragraph{Other properties}

\begin{xalignat}{2}
  v^2 &= n^2(a^2 - x^2) & \text{$a$ = amplitude}\\
  x &= a \cos nt & \\
  T &= \frac{2\pi}{n} & \text{$T$ is the period}\\
  v_{\text{max}} &= |na| & \\
  \ddot{x}_{\text{max}} &= -n^2a &
\end{xalignat}

\section{Newton's method}

If $x_1$ is a good approximation to $f(x) = 0$ then
\begin{equation}
  x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}
\end{equation}
is a better approximation.

\section{Binomial}

The coefficients in the expansion of $(a + b)^n$ where $n = 0, 1, 2,
3,\dots$ are given by Pascal's triangle.

\subsection{Binomial theorem}

\begin{align}
  (a + b)^n &= \sum_{r=0}^n T_{r+1}\\
  &= \sum_{r=0}^n \frac{n!}{r!(n-r)!} \, a^{n-r}b^r\\
  T_{r+1} &= \mbox{}^nC_r a^{n-r}b^r\\
  \text{where } \mbox{}^nC_r = \binom{n}{r} &= \frac{n!}{r!(n-r)!}
\end{align}

\paragraph{Pascal's triangle relationship}
\begin{equation}
  ^{n+1}C_r = \text{}^nC_{r-1} + \text{}^nC_{r-n}
\end{equation}

\paragraph{Sum of coefficients}
\begin{equation}
  \sum_{r=0}^n \text{}^nC_r = 2^n
\end{equation}

\paragraph{Symmetrical relationship}

\begin{equation}
^nC_r = \text{}^nC_{n-r}
\end{equation}


\section{Hyperbolic functions}

\begin{xalignat}{2}
  \cosh x &= \frac{e^x + e^{-x}}{2} & \text{even: catenary}\\
  \sinh x &= \frac{e^x - e^{-x}}{2} & \text{odd: $1\to1$, onto.} \\
  \frac{d}{dx} \cosh x &= \sinh x & \\
  \frac{d}{dx} \sinh x &= \cosh x &
\end{xalignat}

\paragraph{Identities}

\begin{align}
  \cosh^2 x - \sinh^2 x &= 1 \\
  \cosh 2x &= \cosh^2 x + \sinh^2 x \\
  \sinh( x \pm y) &= \sinh x \cosh y \pm \cosh x \sinh y
\end{align}

\section{Sets of numbers}

\begin{tabular}{@{}l>{$}c<{$}>{$}l<{$}@{}}
  \toprule%
  \textbf{subset} & \makebox[2em]{\textbf{notation}}
  & \multicolumn{1}{c}{\textbf{elements}} \\
  \midrule
  Complex numbers & \mathbb{C} & \\
  Real numbers & \mathbb{R} & \\
  Natural numbers (or whole numbers) & \mathbb{N} & 1, 2, 3, 4, 5,
  \dots \\
  Integers & \mathbb{Z} & \dots, -2, -1, 0, 1, 2, 3, \dots \\
  Rational numbers (or fractions) & \mathbb{Q} & 0, 1, 2, -1,
  \frac{1}{2}, \frac{3}{4}, \frac{5}{3}, -\frac{1}{2}, -\frac{3}{7},
  \dots\\
  \bottomrule
\end{tabular}

\vspace{1ex}

Reference: K. G. Binmore, \emph{Mathematical Analysis: a
  straightforward approach}, Cambridge, 1983.

Note that Gunter called the set of integers $\mathbb{J}$.  He also
referred to a set $\mathbb{R}*$, \label{sec:R*}and called the set of rationals
$\mathbb{Q}$ or $\mathbb{R}$\@.  He included 0 in the set of Cardinals
or natural numbers, $\mathbb{N}$.


\section{Complex numbers}

\paragraph{Definition of $i$}

\begin{Def}
  \begin{equation}
    \sqrt{-1} = \pm i
  \end{equation}
\end{Def}

\paragraph{Equality}
\begin{Def}
  If $a + ib = c + id$, then $a = c$ and $b = d$.
\end{Def}

\paragraph{Mod-arg theorems}
\begin{theorem}
  \begin{align}
    |z_1z_2| &= |z_1| \cdot |z_2| \\
    \arg(z_1z_2) &= \arg z1 + \arg z_2
  \end{align}
\end{theorem}

\begin{theorem}
  \begin{align}
    \left|\frac{z_1}{z_2}\right| &= \frac{|z_1|}{|z_2|} \\
    \arg\frac{z_1}{z_2} &= \arg z1 - \arg z_2
  \end{align}
\end{theorem}

\paragraph{Euler's formula}
\begin{equation}
  \cos \theta + i \sin \theta = \cis \theta = e^{i \theta}
\end{equation}

Gunter called $\cos \theta + i \sin \theta \qquad \cis\theta$\label{sec:cis}.

\paragraph{de Moivre's theorem}
\begin{theorem}
  \begin{equation}
    \cos n \theta + i n \sin \theta = \cis n\theta = e^{i n\theta}
  \end{equation}
\end{theorem}

\paragraph{Cube roots of unity:}

if $z^3 = 1$ then $z = 1$ or $-\frac{1}{2} \pm \frac{i\sqrt{3}}{2}$.
Each root is the square of the other.

\paragraph{Using de Moivre's theorem:}
Here we use 
\begin{equation*}
  z^3 = 1 \implies |z^3| = 1 \implies   |z|^3 = 1 \implies |z| = 1.
\end{equation*}
\begin{alignat*}{2}
  \text{let } z &= r^3( \cos \theta + i \sin \theta)^3 & \text{Now $r =
    1$, as above}\\
%
  \text{further, } z &= \cos 3 \theta + i\sin 3 \theta & \text{ (by de
    Moivre's Theorem)}\\
%
  \therefore \cos 3\theta + i \sin 3\theta &= 1 + i0 &\\
%
  \therefore \cos 3\theta &= 1, \quad i\sin 3\theta = i0 &\\
%
  \therefore 3\theta &= 0, 2\pi, 4\pi,\dots &\\
%
  \therefore \theta &= 0, \frac{2\pi}{3}, \frac{-2\pi}{3} & \text{since
    $\theta < |\pi|$.} \\
%
  \text{Hence } z &= \cis 0, \cis \frac{2\pi}{3},
  \cis\frac{-2\pi}{3} & \\
%
  \text{Now } \left(\cis\frac{2\pi}{3}\right)^2 &=
  \cis\frac{4\pi}{3} & \text{ (de Moivre's Theorem)} \\
%
  &= \cis\frac{-2\pi}{3} & = \text{other cube root of unity.} \\
%
  \therefore \text{ one complex root is the }&\text{square of the
    other.}
  & \square
\end{alignat*}
This same method is used to find the $n$th root of unity.

See section~\vref{sec:cis} for the meaning of $\cis\theta$.

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Multiplying a complex number by $i$ rotates the number by
$\frac{\pi}{2}$ in a positive (anti-clockwise) sense:
\begin{align}
  z + \overline{z} &= 2 \Re(z) \\
  z - \overline{z} &= 2 i \Im(z) \\
  |z| &= \sqrt{z \cdot \overline{z}} \\
  \arg z^{-1} &= \arg\overline{z} \\
  \overline{i \omega} &= -i \overline{\omega}
\end{align}

\section{Polynomials}

\begin{Def}
  \emph{Monic} means the leading coefficient $a_n = 1$.
\end{Def}

\begin{theorem}
  Every polynomial over $\mathcal{F}$ can be factorised into monic
  factors of degree $\ge$ 1 and a constant
\end{theorem}

\paragraph{Remainder theorem:}

\begin{theorem}
  $A(x)$ is a polynomial over $\mathcal{F}$; $A(x) = A(a)$ when $A(x)$
  is divided by $(x - a)$.
\end{theorem}

\paragraph{The factor theorem}

\begin{theorem}
  $(x - a)$ is a factor of $A(x)$ iff $A(a) = 0$
\end{theorem}

\paragraph{Theorem of rational roots of polynomials}

\begin{theorem}
  If $\frac{r}{s}$ is a zero of $P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots
  + a_1x + a_0$ $(a_n \ne 0)$, $r$, $s$ are relatively prime and $a_0,
  a_1, \dots, a_n \in \mathbb{Z}$, then $s/a_n$ and $r/a_0$, i.e., $(sx
  - r)$ is a factor of $P(x)$.
\end{theorem}

\paragraph{Fundamental theorem of algebra}
\begin{theorem}
  Every $P(x)$ with coefficients over $\mathbb{R}*$, $\mathbb{R}$ or
  $\mathbb{C}$ has a root $(P(\alpha) = 0)$ for some complex number
  $\alpha$.
\end{theorem}
See section~\vref{sec:R*} for the particular meaning of 
``$\mathbb{R}*$, $\mathbb{R}$'' here.

\begin{theorem}
  A polynomial $P(x)$ of degree $n > 0$ has exactly $n$ zeros in field
  $\mathbb{C}$ and hence exactly $n$ linear factors over $\mathbb{C}$.
\end{theorem}

\begin{theorem}
  If $\alpha = a + ib$ is a root of $P(x)$ (whose coefficients are
  real), then its complex conjugate, $\overline{\alpha} = a - ib$ is also
  a root.
\end{theorem}

\begin{theorem}
  If $P(x)$ has degree $n$ with real coefficients, if $n$ is odd there
  is always at least one real root.
\end{theorem}

\subsection{Multiple roots \& derived polynomials}

\begin{theorem}
  If the factor $(x - \alpha)$ occurs $r$ times then we say it is an
  $r$-fold root, or $\alpha$ has a multiplicity of $r$.
\end{theorem}

\begin{theorem}
  If $P(x)$ has a root of multiplicity $m$ then $P'(x)$ has a root of
  multiplicity $(m - 1)$.
\end{theorem}

\begin{theorem}
  If $P(x)$ over $\mathcal{F}$ is irreducible over $\mathcal{F}$ and
  $\deg P(x) > 1$, then there are no roots of $P(x)$ over
  $\mathcal{F}$.
\end{theorem}

\begin{theorem}
  2 different polynomials $A(x)$ and $B(x)$ over a non-finite field \F
  cannot specify the same function in \F.
\end{theorem}

\begin{theorem}
  If two polynomials of the $n$th degree over a field \F specify the
  same function for more than $n$ elements of \F, then the two
  polynomials are equal.
\end{theorem}

\begin{theorem}
  If $P(x) = a_0 + a_1x + a_2x^2 + \dots + a_nx^n$ over \F where $n
  \ne 0$ and if $P(x)$ is completely reducible to $n$ linear factors
  over \F (i.e., if $P(x) = a_n(x - \alpha_1)(x - \alpha_2)\dots(x
  -\alpha_n)$ where $\alpha_1, \alpha_2, \dots, \alpha_n$ are the
  roots of $P(x)$), then
  \begin{align}
    \sum \alpha &= \frac{-\alpha_{n-1}}{a_n}\\
    \sum \alpha\beta &= \frac{\alpha_{n-2}}{a_n}\\
    \sum \alpha\beta\gamma &= \frac{-\alpha_{n-3}}{a_n}\\
    &\phantom{a} \vdots \\
    \sum \dots\alpha\beta\gamma\dots\omega &= (-1)^n\frac{\alpha_{0}}{a_n}\\
    \intertext{Also} \sum \alpha^2 &= \left(\sum\alpha\right)^2 - 2
    \sum \alpha\beta
  \end{align}
\end{theorem}

\paragraph{Definition of a rational function:}

\begin{Def}
  If $P(x)$, $Q(x)$ are 2 polynomials over \F, then
  $\frac{P(x)}{Q(x)}$ is a rational function over \F.
\end{Def}

\paragraph{Definition of sum and product of rational functions:}

\begin{Def}
  \begin{align}
    \frac{A(x)}{B(x)} + \frac{C(x)}{D(x)} &= \frac{A(x)D(x) +
      C(x)B(x)}{B(x)D(x)} \\
    \frac{A(x)}{B(x)} \cdot \frac{C(x)}{D(x)} &= \frac{A(x)C(x)}{B(x)D(x)}
  \end{align}
\end{Def}

\subsection{Partial fractions}

\begin{theorem}
  If $F, P, Q$ are relatively prime polynomials over \F and $\deg F <
  \deg PQ$ then we can find unique polynomials $A$, $B$ such that 
  \begin{equation}
    \frac{F}{PQ} = \frac{A}{P} + \frac{B}{Q}\qquad \qquad \deg A < \deg P,
    \quad \deg B < \deg Q.
  \end{equation}
\end{theorem}

\end{document}

