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Elementary Relationships

From the above definitions, one can quickly verify

\begin{eqnarray*}
z+\overline{z} &=& 2 \, \mbox{re}\left\{z\right\} \\
z-\overline{z} &=& 2\, j\,\, \mbox{im}\left\{z\right\} \\
z\overline{z} &=& \left\vert z\right\vert^2.
\end{eqnarray*}

Let's verify the third relationship which states that a complex number multiplied by its conjugate is equal to its magnitude squared:

$\displaystyle z \overline{z} \isdef (x+jy)(x-jy) = x^2-(jy)^2 = x^2 + y^2 \isdef \vert z\vert^2 \protect$ (2.4)


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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