NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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More Notation and Terminology
It's already been mentioned that the rectilinear coordinates of a complex number in the complex plane are called the real part andimaginary part, respectively.
We also have special notation and various names for the radius and angle of a complex number expressed in polar coordinates :
The complex conjugate of is denoted and is defined by
where, of course, . Sometimes you'll see the notation in place of , but we won't use that here.
In general, you can always obtain the complex conjugate of any expression by simply replacing with . In the complex plane, this is a vertical flip about the real axis; in other words, complex conjugation replaces each point in the complex plane by its mirror image on the other side of the axis.