The representation, beloved of engineers and physicists, of a Complex Number in terms of a Complex exponential

(1) |

(2) | |||

(3) |

Here, is the counterclockwise Angle from the Positive Real axis. In the degenerate case when ,

(4) |

It is trivially true that

(5) |

(6) |

Look at the time averages of each term,

(7) |

(8) |

(9) |

(10) |

Consider now two scalar functions

(11) | |||

(12) |

Then

(13) | |||

(14) |

In general,

(15) |

© 1996-9

1999-05-26