(6.1) |

where represents the square root of minus one. Here, and are both real, and are identified with the corresponding Cartesian coordinates. (Incidentally, should not be confused with a -coordinate: this is a strictly two-dimensional discussion.) We can also write

(6.2) |

where and are the modulus and argument of , respectively, but can also be identified with the corresponding plane polar coordinates. Finally,

(6.3) |

implies that

(6.4) | ||

(6.5) |

We can define functions of the complex variable, , in the same way that we define functions of a real variable. For instance,

(6.6) | ||

(6.7) |

For a given function, , we can substitute and write

where and are real two-dimensional functions. Thus, if

(6.9) |

then

(6.10) |

giving

(6.11) | ||

(6.12) |