diff options
| -rw-r--r-- | komfour_zf.tex | 152 |
1 files changed, 141 insertions, 11 deletions
diff --git a/komfour_zf.tex b/komfour_zf.tex index c1a0813..7d3786b 100644 --- a/komfour_zf.tex +++ b/komfour_zf.tex @@ -3,7 +3,7 @@ % !TeX root = komfour_zf.tex %% TODO: publish to CTAN -\documentclass[twocolumn, small]{tex/hsrzf} +\documentclass[twocolumn, margin=normal]{tex/hsrzf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages @@ -34,14 +34,14 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Metadata -\course{Elektrotechnik} +\course{Electrical Engineering} \module{KomFour} -\semester{Fr\"uhlingssemester 2020} +\semester{Spring Semester 2020} \authoremail{npross@hsr.ch} \author{Naoki Pross -- \texttt{\theauthoremail}} -\title{\texttt{\themodule} Zusammenfassung} +\title{Cheat sheets for \texttt{\themodule}} \date{\thesemester} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -52,6 +52,8 @@ %% number sets \newcommand\Nset{\mathbb{N}} +\newcommand\Zset{\mathbb{Z}} +\newcommand\Qset{\mathbb{Q}} \newcommand\Rset{\mathbb{R}} \newcommand\Cset{\mathbb{C}} @@ -148,11 +150,11 @@ \end{definition} \begin{definition}[Absolute value] - If \(z = a + jb\) we define the \emph{abolute value} \(\len{z} = \sqrt{a^2 + b^2}\) + If \(z = a + jb\) we define the \emph{absolute value} \(\len{z} = \sqrt{a^2 + b^2}\) \end{definition} \begin{lemma}[Properties of absolute value] - Let \(z,w\in\Cset\). We have \(\len{z} \in\Rset^+_0\), \(z\conj{z} = \len{z}^2\) and as a consequence \(\len{zw} = \len{z}\cdot\len{w}\) and \(\len{\conj{z}} = \len{z}\). In additon we have the inequalities + Let \(z,w\in\Cset\). We have \(\len{z} \in\Rset^+_0\), \(z\conj{z} = \len{z}^2\) and as a consequence \(\len{zw} = \len{z}\cdot\len{w}\) and \(\len{\conj{z}} = \len{z}\). In addition we have the inequalities \begin{align*} -\len{z} \leq &\Re(z) \leq \len{z} & \len{z} &\leq \len{\Re(z)} + \len{\Im(z)} \\ @@ -182,7 +184,7 @@ \end{lemma} \begin{definition}[Polar notation] - An alternative rapresentation of a complex number \(z = a + jb\) is its \emph{polar form} \(z = r \angle \phi\), where \(r = \len{z}\) and \(\phi = \arg{z} = r(\cos\phi + j\sin\phi)\). + An alternative representation of a complex number \(z = a + jb\) is its \emph{polar form} \(z = r \angle \phi\), where \(r = \len{z}\) and \(\phi = \arg{z} = r(\cos\phi + j\sin\phi)\). \begin{align*} a &= r\cos\phi & b &= r \sin\phi & @@ -217,6 +219,11 @@ \len{z/w} = \len{z}/\len{w} \quad \arg(z/w) = \arg z - \arg w \] + Lastly from the product we see that for \(k \in \Nset\) + \[ + \len{z^k} = \len{z}^k \quad + \arg{z^k} = k \arg{z} + \] \end{lemma} \begin{theorem}[De Moivre's formula] @@ -225,16 +232,139 @@ \left(\cos\phi + j\sin\phi\right)^n = \cos(n\phi) + j\sin(n\phi) \] As a consequence with the binomial formula - \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^n\) we have + \((a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^n\), recalling that \(\binom{n}{k} = n!/(k!(n-k)!)\) (Pascal's triangle), we have + \begin{align*} + \sin(nx)&=\sum _{k=0}^{n}{\binom {n}{k}}(\cos x)^{k}\,(\sin x)^{n-k}\,\sin {\frac {(n-k)\pi }{2}}\\ + \cos(nx)&=\sum _{k=0}^{n}{\binom {n}{k}}(\cos x)^{k}\,(\sin x)^{n-k}\,\cos {\frac {(n-k)\pi }{2}} + \end{align*} +\end{theorem} + +\section{Complex valued functions} + +\begin{definition}[Function in \(\Cset\)] + Let \(f: \mathbb{D} \to \mathbb{W}\) with both \(\mathbb{D}, \mathbb{W} \subseteq \Cset\) + that maps \(z = (a + jb) \mapsto w = (u + jv)\), + then \(u = \Re f(z)\) and \(v = \Im f(z)\). + If \(f\) is a bijection with inverse \(f^{-1}\), then \(a = \Re f^{-1}(w), b = \Im f^{-1}(w)\). +\end{definition} + +\begin{definition}[Differentiation in \(\Cset\)] + Let \(f\) be a function of \(z\) and \(h \in \Cset\). We have the limit + \[ + \lim_{\len{h} \to 0} \frac{f(z_0 + h) - f(z_0)}{h} = f'(z_0) + \] + to define the \emph{derivative} of \(f\) at the point \(z_0\). +\end{definition} + +\begin{lemma}[Local dilation and rotation] + Let \(f\) be a differentiable function in \(\Cset\). + If \(f'(z) \neq 0\) everywhere, then \(f\) is a conformal map (i.e. preserves angles) with local dilation of \(\len{f'(z)}\) and rotation of \(\arg f'(z)\) +\end{lemma} + +\begin{definition}[Linear function] + +\end{definition} + +\begin{definition}[Monomial and \(n\)-th root] + Let \(w = z^n\) be a monomial of degree \(n\in\Nset\). Using the polar notation we see that + \((r\angle \phi)^n = r^n \angle (n\phi)\). Because \(r\angle\phi = r\angle(\phi+2\pi)\) there cannot be a bijection between \(w\) and \(z\), if we want to define an inverse function \(z = \sqrt[n]{w}\) we get many values with the form \[ - \sin(n\phi) = + z_k = \sqrt[n]{r}\angle(\phi + k2\pi)/n \qquad 0 \leq k < n \] + This fact implies that in general for \(z,u \in\Cset\) \(\sqrt[n]{zu} \neq \sqrt[n]{z}\sqrt[n]{u}\), as the relationship holds only for \emph{some} values of \(\sqrt[n]{z} \text{ and } \sqrt[n]{u}\). +\end{definition} + +\begin{theorem}[Roots of a polynomial] + Every complex polynomial of degree \(n\) has always \(n\) roots in \(\Cset\). \end{theorem} -\section{Complex valued functions} +\begin{theorem} + Every complex polynomial of degree \(n\) with coefficients can be \emph{uniquely} rewritten in term of its roots. + \[ + P(z) = \sum_{k=0}^n a_k z^k = a_n \prod_{k=0}^{n} (z - z_k) + \] +\end{theorem} + +\begin{theorem}[Polynomal with real coefficients] + The roots of a polynomial with real coefficients of degree \(n\), always come in conjugate complex pairs of \(r\) and \(\conj{r}\). That is because + \[ + (z - r)(z - \conj{r}) = z^2 - 2\Re(r)z + \len{z}^2 + \] +\end{theorem} + +\begin{lemma} + From the previous theorem follows that a polynomial of \emph{odd} degree, has always at least one real solution because \(r \in\Rset \iff r = \conj{r}\). +\end{lemma} + +\begin{theorem} + All roots of a polynomial \(p(z) = \sum_{k=0}^n a_k z^k\) are inside of the open disk centered at the origin of radius \(\sum_{k=0}^n \len{a_k / a_n}\). +\end{theorem} + +\begin{theorem}[Cardano's cubic formula] + % TODO +\end{theorem} + +\begin{definition}[Exponential] + If \(z\) is a complex number we define the exponential function \(e^z\) by its convergent power series + \[ + e^z = \sum_{n=0}^\infty \frac{z^n}{n!} + \] +\end{definition} + +\begin{theorem}[Euler's formula] + By setting the argument of the exponential function to \(jt\) for some \(t \in\Rset\) we can reorder the power series to be a sum of the power series of \(j\sin\) and \(\cos\), and thus define + \[ + e^{jt} = \cos t + j\sin t = \cjs t = 1\angle t + \] +\end{theorem} + +\begin{lemma} Let \(a,b \in\Cset\) and \(k\in\Zset\) + \[ + e^a e^b = e^{a+b} \quad + e^a / e^b = e^{a-b} \quad + \left(e^a \right)^k = e^{ak} + \] +\end{lemma} + +\begin{definition}[Trigonometric functions] +\end{definition} + +\begin{definition}[Logarithm] + Because \(w = e^z\) defined from \(\Cset \to \Cset\) is not a bijection (\(e^{z + 2\pi j} = e^z\)), unless we restrict the imaginary part of the domain to \((\pi, \pi]\), we get only an equivalence relationship because + \[ + \ln\left[\len{w} e^{j(\phi + k2\pi)}\right] = \ln\len{w} + j(\phi + k2\pi) + \] + where \(k \in\Zset\). Similarly for \(z,w\in\Cset\) + \begin{align*} + \ln(w) &\equiv z &\pmod{2\pi j} \\ + \ln(w^k) &\equiv k\ln(w) &\pmod{2\pi j} \\ + \ln(zw) &\equiv \ln(z) + \ln(w) &\pmod{2\pi j} \\ + \ln(z/w) &\equiv \ln(z) - \ln(w) &\pmod{2\pi j} + \end{align*} +\end{definition} + +\begin{lemma}[General exponentiation] + So far we have only exponentiation for an exponent \(k\in\Zset\), by adding \(m \in\Nset\) we can define the quotient \(k/m \in\Qset\) that together with \(z\in\Cset\) gives + \begin{align*} + z^{k/m} &= e^{\ln(z) k/m} \\ + &= \exp\big((\ln\len{z} + j(\arg z + 2\pi n)) k/m\big) \\ + &= \exp\big(\ln\len{z}\cdot k/m)\exp((\arg z + 2\pi n)jk/m\big) \\ + &= \len{z}^{k/m}\exp\big((\arg z + 2\pi n)jk/m\big)= \sqrt[m]{z^k} + \end{align*} + like in the reals, except that we have \(m\) values instead of 1 or 2. If we let \(w \in\Cset\) the expression \(z^w\) cannot be equal to an unique value because + \begin{align*} + z^w = e^{w \ln z} &= \exp\big( w (\ln\len{z} + j \arg{z} + 2\pi nj)\big) \\ + &= e^{w(\ln\len{z} + j\arg z)} e^{w2\pi nj} + \end{align*} + instead it is said to be \emph{multivalued}. +\end{lemma} + +\section{Fourier Series} + +\section{Fourier Transform} \section{License} \doclicenseThis \end{document} -% vim: set et ts=2 sw=2 spelllang=us spell linebreak : +% vim: set et ts=2 sw=2 spelllang=en spell linebreak : |