% Format: latex % Copyright (C) 2006 Nick Urbanik % This program is free software; you can redistribute it and/or % modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % You should have received a copy of the GNU General Public License % along with this program; if not, write to the Free Software % Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. % 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 % Make sure duplexing is set to Short Edge (flip) % lpr fbook.ps % % or: % dvips -T148.5mm,210mm -f formula-book | psbook | % psnup -2 -W148.5mm -H210mm -pa4 | lpr % dvips -T148.5mm,210mm -f formula-book | psbook | psnup -2 -W148.5mm -H210mm -pa4 | lpr % To produce ps output suitable for the Adobe Acrobat Distiller: % dvips -Pcmz -Pamz -T148.5mm,210mm -o formula-book.ps formula-book % $Header$ % $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 % Removed an allowdisplaybreak that was causing an ugly looking page % break % Added more info in the pdf header % Added instructions on how to use with the Adobe Distiller. % % Revision 1.3 2001/06/16 14:08:52 nicku % Changed figures to using xfig output, together with psfrag. % Changed to use geometry instead of vmargin. % This makes pdf files come out at the right size. % With vmargin, had a5 pages printed in top left corner of a4 pages. % % Revision 1.2 2001/06/16 09:46:53 nicku % Fixed one spelling typo. % Indented using auctex. % % 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. % % 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 % \documentclass[% %dvips,% to use distiller, uncomment this. a5paper,% titlepage,% twoside,% %11pt% %12pt% ]{article} \usepackage{% %vmargin,% geometry,% varioref,% fancyhdr,% %emlines2,% graphicx,% array,% amssymb,% amsmath,% theorem,% ifthen,% psfrag,% booktabs,% xspace } \usepackage[% %pdfpagemode=Bookmarks,% %bookmarks,% pdfauthor={Nick Urbanik},% %pdfstartview=FitH,% %pdfpagemode=FitWidth,% pdfkeywords={For Gunter Beck},% pdftitle={Gunter Beck's Mathematical Formula Book},% pdfsubject={Mathematical Formulas}% ]{hyperref} \newboolean{smallBook} \setboolean{smallBook}{true} \ifthenelse{\boolean{smallBook}}% {% % Half A4 size: \geometry{a5paper, left=15mm, right=15mm, top=9mm, bottom=10mm, headsep=4mm, footskip=8mm, twosideshift=0pt} % \setpapersize{A5}% % \setmarginsrb{15mm}% left % {9mm}% top % {15mm}% right % {10mm}% bottom % {12pt}% headheight---increase to stop fancyhead warn % {4mm}% headsep % {0pt}% footheight % {8mm}% footskip }% %else {% \geometry{a4paper, left=30mm, right=30mm, top=18mm, bottom=10mm, headsep=8mm, footskip=11mm, twosideshift=0pt} % \setmarginsrb{30mm}% left % {18mm}% top % {30mm}% right % {10mm}% bottom % {15pt}% headheight---increase to stop fancyhead warn % {8mm}% headsep % {0pt}% footheight % {11mm}% footskip } % End of if small book ... else. \pagestyle{fancy} \lhead% [\fancyplain{}{}\textbf{\upshape Maths formula book}] {\fancyplain{}{}\textbf{\thepage}} \rhead% [\fancyplain{}{}\textbf{\upshape \thepage}] {\fancyplain{}{}\textbf{\upshape Maths formula book}} \lfoot[]{\fancyplain{}{}\tiny For Gunter Beck} \rfoot[\fancyplain{}{}\tiny For Gunter Beck]{} \cfoot{} \title{Maths formula book} \author{Nick Urbanik} \date{\normalsize In memory of Gunter Beck, a great teacher} \theorembodyfont{\rmfamily} \newtheorem{Def}{Definition}[section] \theorembodyfont{\slshape} \newtheorem{theorem}{Theorem}[section] \theoremheaderfont{\scshape} \newcommand{\Deg}{\ensuremath{^\circ}} \newcommand{\F}{\ensuremath{\mathcal{F}}\xspace} \numberwithin{equation}{section} \DeclareMathOperator{\cis}{cis} \begin{document} \maketitle % \thispagestyle{empty} \setcounter{page}{1} \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 $\cis\theta$. %This is from page 26 of %k:/emtexNew/packages/amslatex/math/amsldoc.tex: % It is defined in the amsthm package, which is automatically loaded % by amsmath.sty. See the file thmtest.tex. %\begin{proof} %... %\begin{equation} %g(t) = L\gamma!\,t^{-\gamma}+t^{-\delta}\eta(t) \qed %\end{equation} %\renewcommand{\qed}{}\end{proof} 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}