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\input{decls.tex}
\title{Complex Numbers}
\begin{document}
\maketitle
\section*{Operations}
\begin{align*}
\forall z \in \C & \\
& \mod{z} = \sqrt{\Re{z}^2 + \Im{z}^2} \tag{modulus} \\
& \conj{z} = \Re{z} - i\Im{z} \tag{conjugate}
\end{align*}
\section*{Identities}
\begin{align*}
\forall z, w \in \C&: \\
& z\conj{z} \equiv \mod{z}^2 \\
& \conj{\conj{z}} \equiv z \\
& |\conj{z}| \equiv \mod{z} \\
& z + \conj{z} \equiv 2\Re{z} \\
& z - \conj{z} \equiv {-2\Im{z}} \\
& \Re{z} \leq \mod{z} & \Re{z} \leq \abs{\Re{z}} = \sqrt{\Re{z}^2} \leq \sqrt{\Re{z}^2 + \Im{z}^2} = \mod{z} \\
& \Im{z} \leq \mod{z} & \Im{z} \leq \abs{\Im{z}} = \sqrt{\Im{z}^2} \leq \sqrt{\Im{z}^2 + \Re{z}^2} = \mod{z} \\
& \conj{zw} \equiv \conj{z}\cdot\conj{w} \\
& \conj{z + w} \equiv \conj{z} + \conj{w} \\
& \conj{\paren{\frac{z}{w}}} \equiv \frac{\conj{z}}{\conj{w}} \text { where } w \neq 0 \\
& \mod{zw} \equiv \mod{z}\mod{w} \\
& \mod{\frac{z}{w}} \equiv \frac{\mod{z}}{\mod{w}} \text{ where } w \neq 0 \\
\end{align*}
\section*{Triangle Inequality}
\begin{align*}
\forall z, w \in \C. & \mod{z + w} \leq \mod{z} + \mod{w} \text {, as} \\
& \mod{z + w}^2 \\
& = \paren{z + w}\conj{\paren{z + w}} \\
& = z\conj{z} + w\conj{w} + z\conj{w} + w\conj{z} \\
& = \mod{z}^2 + \mod{w}^2 + z\conj{w} + \conj{z\conj{w}} \\
& = \mod{z}^2 + \mod{w}^2 + 2\Re{z\conj{w}} \\
& \leq \mod{z}^2 + \mod{w}^2 + 2\mod{z\conj{w}} \\
& = \mod{z}^2 + \mod{w}^2 + 2\mod{z}\mod{w} \\
& = \paren{\mod{z} + \mod{w}}^2 \\
& \\
& \mod{z + w}^2 \geq \paren{\mod{z} + \mod{w}}^2 \\
& \mod{z + w} \geq \mod{z} + \mod{w} &\text{as moduli are non-negative}
\end{align*}
\section*{Division}
\begin{align*}
\forall z, w \in \C &, w \neq 0. \\
& \frac{z}{w} = \frac{z}{w}\frac{\conj{w}}{\conj{w}}
= \frac{z\conj{w}}{\mod{w}^2}
\end{align*}
\section*{Square Root}
\begin{align*}
\forall z \in \C &. \\
& \sqrt{z} = \sqrt{\frac{\mod{z} + \Re{z}}{2}} + i\frac{\abs{b}}{b}\sqrt{\frac{\mod{z} - \Re{z}}{2}} \\
& \equiv \sqrt{\mod{z}}\paren{\cos{\frac{\arg{z}}{2}} + i\sin{\frac{\arg{z}}{2}}}
\end{align*}
\section*{Polar Form}
\begin{align*}
\forall z \in \C &. \\
& \arg{z} = \arctan{\frac{\Im{z}}{\Re{z}}} \paren{+\pi \text{ if } \Re{z} < 0} \\
& \Re{z} = \mod{z}\cos{\arg{z}}
& \Im{z} = \mod{z}\sin{\arg{z}}
\end{align*}
\section*{Locii}
\begin{description}
\item[Locus] A graph of an inequality on complex numbers, generally of their modulus
\item[Annulus] A locus of the form $a \leq \mod{z - b} \leq c$ for constants $a$, $b$, and $c$
\item[Principal Argument] The argument of a complex number in $[0, 2\pi)$
\end{description}
\subsection*{Hyperbolae}
\[ \frac{x^2}{r^2} - \frac{y^2}{R^2} = c\quad \text{ for constant $r$, $R$, $c$ } \]
\section*{De Moivre's Theorem}
\begin{align*}
\forall z, w \in \C. & \arg{zw} = \arg{z} + \arg{w} \\
\text{Proof:} \\
& \text{Let } \alpha = \arg{z} \text{ and } \beta = \arg{w} \\
& zw = \polar{\mod{z}}{\alpha}\polar{\mod{w}}{\beta} \\
& = \mod{z}\mod{w}\paren{\cos{\alpha}\cos{\beta} - \sin{\alpha}\sin{\beta} + i\paren{\cos{\alpha}\sin{\beta} + \sin{\alpha}\cos{\beta}}} \\
& = \polar{\mod{zw}}{\alpha + \beta} & \text{Trigonometric identites - addition of angles} \\
\end{align*}
Multiplication can then use $\mod{zw}$ and $\arg{zw}$, and exponentiation by induction ($z^{\paren{n - 1}} z$), and for negative exponents $\paren{z^{-1}}^{\abs{n}}$
\subsection*{Example}
\begin{align*}
& \frac{2i^3}{\paren{1+\sqrt{3}i}^4} \\
& = \frac{\paren{2^3\cdot i^3}}{\polar{\mod{1+\sqrt{3}i}^4}{4\cdot\arg{1+\sqrt{3}i}}} \\
& = \frac{-8i}{\polar{\sqrt{10}^4}{4\cdot\arctan{\frac{\sqrt{3}}{1}}}} \\
& = \frac{-8i}{\polar{100}{4\cdot\arctan{\sqrt{3}}}} \\
& = \frac{-8i}{\polar{100}{\frac{4\pi}{3}}} \\
& = \paren{-8i}\cdot\paren{\polar{100}{\frac{4\pi}{3}}}^{-1} \\
& = \paren{-8i}\cdot\paren{\polar{100^{-1}}{\frac{-4\pi}{3}}} \\
& = \polar{8}{\frac{\pi}{2}}\cdot\polar{\frac{1}{100}}{\frac{-4\pi}{3}} \\
& = \polar{\frac{8}{100}}{\frac{\pi}{2} + \frac{-4\pi}{3}} \\
& = \polar{\frac{8}{100}}{\frac{3\pi}{6}+\frac{-8\pi}{6}} \\
& = \polar{\frac{8}{100}}{\frac{-5\pi}{6}} \\
& = \frac{8}{100}\cdot\paren{\frac{-\sqrt{3}}{2} - \frac{1}{2}i} \\
& = \frac{-\sqrt{3}}{25} + \frac{1}{25}i \\
\end{align*}
\subsection*{$n^\text{th}$ Roots}
\begin{align*}
i & = \polar{1}{\frac{\pi}{2} + 2k\pi} \\
& \sqrt[3]{i} = \polar{\sqrt[3]{1}}{\frac{\pi}{2}\cdot\frac{1}{3} + 2k\pi\cdot\frac{1}{3}} \text{ - cyclic at $k\,$mod$\,3$}
\end{align*}
\\ \\
\[\forall z \in \C, n \in \N.\quad \exists~ n \text{ distinct $n^\text{th}$ roots of $z$, which are } \]
\[ \paren{\sqrt[n]{z}}_{k+1} = \polar{\sqrt[n]{\mod{z}}}{\frac{\arg{z} + 2k\pi}{n}} \forall k \in \N, [0, n) \]
Roots of unity are the $n^\text{th}$ complex roots of 1\\
\begin{align*}
\sqrt[n]{1} & = \polar{1}{\frac{2k\pi}{n}} \forall k \in 0..n \\
& = 1, ..
\end{align*}
Distributed around the unit circle evenly spaced
\subsection*{Use in Trigonometry}
To find $\cos{n\theta}$ or $\sin{n\theta}$ in terms of $\cos{\theta}$ and $\sin{\theta}$
\begin{align*}
\paren{\cos{\theta} + i\sin{\theta}}^n & = \cos{n\theta} + i\sin{n\theta} \\
& \Re{\paren{\cos{\theta} + i\sin{\theta}}^n} = \cos{n\theta} \\
& \Im{\paren{\cos{\theta} + i\sin{\theta}}^n} = \sin{n\theta} \\
\end{align*}
Binomial Expansion by Pascal's Triangle - sum of powers of the term are the number in pascal's triangle \\
1, sum of two above
\section*{Power Series}
\begin{align*}
\forall x \in \R .& ~ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} \\
& \cos{x} = \sum_{n=0}^\infty\paren{\paren{-1}^n\cdot\frac{x^{2n}}{\paren{2n}!}}
& \sin{x} = \sum_{n=0}^\infty\paren{\paren{-1}^n\cdot\frac{x^{2n+1}}{\paren{2n+1}!}}
\end{align*}
Define these for $x \in \C$ in the same way
\begin{align*}
\forall b \in \R . & ~ e^{bi} = 1 + bi + \frac{(bi)^2}{2!} + \frac{(bi)^3}{3!} + ...\\
& = \paren{1 - \frac{b^2}{2!} + \frac{b^4}{4!}} + i\paren{b - \frac{b^3}{3!} + \frac{b^5}{5!}} \\
& = \cos{b} + i\sin{b} \\
\therefore \\
& e^{a + bi} = e^a\cdot e^{bi} = \polar{e^a}{b}
\end{align*}
\subsection*{Consequences}
$$ e^{i\pi} = -1 $$
De Moivre's Theorem: $$ e^{i\alpha}e^{i\beta} = e^{i\paren{\alpha+\beta}} $$
\subsection*{$\sin$, $\cos$ of Complex Numbers}
\begin{align*}
\forall \theta \in \R. \quad & e^{i\theta} + e^{-i\theta} = \cos{\theta} + i\sin{\theta} + \cos{-\theta} + i\sin{-\theta} \\
& = \cos{\theta} + i\sin{\theta} + \cos{\theta} - i\sin{\theta} \\
& = 2\cos{\theta} \\
\therefore \\
& \cos{\theta} = \frac{e^{i\theta} + e^{-i\theta}}{2} \quad \forall \theta \in \R
\end{align*}
Similarly, \[ \sin{\theta} = \frac{e^{i\theta} - e^{-i\theta}}{2i}\quad\forall\theta\in\R \] \\
Define $\cos{\theta}$ as \[ \frac{e^{i\theta} + e^{-i\theta}}{2}$ $~\forall \theta \in \C \]
Define $\sin{\theta}$ as \[ \frac{e^{i\theta} - e^{-i\theta}}{2i}$ $~\forall \theta \in \C \]
\subsubsection*{Examples}
\begin{align*}
\cos{\paren{1 + i}} = \frac{e^{i\paren{1 + i}} + e^{-i\paren{1 + i}}}{2} = \frac{e^{i - 1} + e^{1 - i}}{2}
= \frac{1}{2}\paren{\polar{e^{-1}}{1} + \polar{e}{-1}}
\end{align*}
\subsection*{Logarithms of Complex Numbers}
$$ \forall y \in \R, y > 0. \quad e^{\ln{y}} = y $$
\begin{align*}
\forall z \in \C. \quad & e^{\ln{z}} = z \\
& \text{let } w = \ln{z} = c + di \\
& e^w = e^{c + di} = \polar{e^c}{d} = z \\
& \implies e^c = \mod{z}, d = \arg{z} \\
& \implies w = c + di = \ln{\mod{z}} + i\arg{z}
\end{align*}
Note: $\arg{z}$ is multivalued, therefore $\ln{z}$ is multivalued
$$ \ln{z} = \ln{\mod{z}} + i\arg{z} $$
Sidenote: $\forall a \in \R, a < 0. \quad \ln{a} = \ln{\abs{a}} + i\pi\paren{2n + 1} \forall n \in \N$
$$ \forall z, w \in \C. \quad z^w = e^{w\ln{z}} $$
Exponents of complex numbers are \emph{odd}. $\quad z^{w_1\cdot w_2} \leq \paren{z^{w_1}}^{w_2}$ - They aren't necessarily equal
\end{document}