Introduction to the Mathematical and Statistical Foundations of Econometrics

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, define the inverse operator “—1” by

and thus

a/c

0

provided that c = 0. Therefore, all the basic arithmetic operations (addition, subtraction, multiplication, division) of the real number system К apply to K^ and vice versa.

In the subspace R = {(0, x)T, x є К} the multiplication operator “ x ” yields

(0) X (0)=(—Г )■

In particular, note that

01 X 01 = —01

Now let

and interpret a + i.0 as the mapping

a ■ c — b ■ d b ■ c + a ■ d = (a ■ c — b ■ d) + i ■ (b ■ c + a ■ d). (Ш.7)

However, the same result can be obtained by using standard arithmetic opera­tions and treating the identifier i as л/—I:

(a + i ■ b) x (c + i ■ d) = a ■ c + i2 ■ b ■ d + i ■ b ■ c + i ■ a ■ d

= (a ■ c — b ■ d) + i ■ (b ■ c + a ■ d). (III.8)

In particular, it follows from (Ш.4)-(Ш.6) that

which can also be obtained by standard arithmetic operations with treated i as V—I and i.0 as 0.

Similarly, we have

(a /^c^— 1 fa ■ c + b ■ d

b// d) = c2 + d2b ■ c — a ■ d)

a ■ c + b ■ d b ■ c — a ■ d c2 + d2 + i c2 + d2

provided that c2 + d2 > 0. Again, this result can also be obtained by standard arithmetic operations with i treated as */—I:

a + i ■ b c — i ■ d _

c + i ■ d c — i ■ d (a + i ■ b) x (c — i ■ d)

(c + i ■ d) x (c — i ■ d) a ■ c + b ■ d, b ■ c — a ■ d c2 + d2 + i c2 + d2

The Euclidean space K2 endowed with the arithmetic operations (III.1)— (Ш.3) resembles a number system except that the “numbers” involved cannot be ordered. However, it is possible to measure the distance between these “num­bers” using the Euclidean norm:

If the “numbers” in this system are denoted by (Ш.5) and standard arithmetic operations are applied with i treated as */—I and i.0 as 0, the results are the
same as for the arithmetic operations (Ш.1), (III.2), and (Ш.3) on the elements of K2. Therefore, we may interpret (III.5) as a number, bearing in mind that this number has two dimensions if b = 0.

From now on I will use the standard notation for multiplication, that is,

(a + i ■ b)(c + і ■ d) instead of (III.8).

The “a” of a + і ■ b is called the real part of the complex number involved, denoted by Re(a + і ■ b) = a, and b is called the imaginary part, denoted by Im(a + і ■ b) = b ■ Moreover, a — і ■ b is called the complex conjugate of a + і ■ b and vice versa. The complex conjugate of z = a + і ■ b is denoted by a bar: z = a — і ■ b. It follows from (III.7) that, for z = a + і ■ b and w = c + і ■ d, zw = z ■ w. Moreover, |z| = fZZ. Finally, the complex number system itself is denoted by C.