## Appendix III – A Brief Review of Complex Analysis

III.1. The Complex Number System

Complex numbers have many applications. The complex number system allows computations to be conducted that would be impossible to perform in the real world. In probability and statistics we mainly use complex numbers in dealing with characteristic functions, but in time series analysis complex analysis plays a key role. See for example Fuller (1996).

Complex numbers are actually two-dimensional vectors endowed with arithmetic operations that make them act as numbers. Therefore, complex numbers are introduced here in their “real” form as vectors in K2.

In addition to the usual addition and scalar multiplication operators on the elements of K2 (see Appendix I), we define the vector multiplication operator

(III1)

Observe that

Moreover,...

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