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Complex Numbers

\begin{displaymath}
\begin{array}{rclrcl}
\mrr {j}{\isdef }{\sqrt{-1}}{z}{\isdef }{x+jy \; \mathrel{\isdef } \; re^{j\theta}}
\mr {x}{r \cos(\theta)}{y}{r \sin(\theta)}
\mr {r}{\left\vert z\right\vert \;=\; \sqrt{x^2+y^2}}{\theta}{\angle z \;=\; \tan^{-1}(y/x)}
\mr {\left\vert z_1 z_2\right\vert}{\left\vert z_1\right\vert\left\vert z_2\right\vert}{\left\vert z_1/z_2\right\vert}{\left\vert z_1\right\vert/\left\vert z_2\right\vert}
\mr {\vert e^{j\theta}\vert}{1}{\angle r}{0}
\mr {z_1 z_2}{(x_1x_2-y_1y_2) + j(x_1y_2+x_2y_1)}{z_1z_2}{r_1r_2e^{j(\theta_1+\theta_2)}}
\mrr {\overline{z}}{\isdef }{x-jy \;=\;re^{-j\theta}}{z\overline{z}}{=}{\left\vert z\right\vert^2 \;=\; x^2+y^2=r^2}
\end{array}\end{displaymath}


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA