1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 67

where A = a+ %bf3 and B = -^1 / 2 b,,;:::iS. It is easy to check that (1.18-3)

establishes an isomorphism between the complex numbers a + bi and the

complex numbers A+ Bj with j2 = -1. That is, if we let z = a+ bi,

Z = A+ Bj and, similarly, z' = a'+ b'i, Z' = A'+ B'j, the one-to-

one correspondence z <--> Z, z' <--> Z' is such that z = z' iff Z = Z', and

z + z^1 <--> Z + Z', zz' <--> ZZ'.

If 6. = O, the system is called parabolic or dual. A parabolic system

always contains an element

j = /3-2i

such that

j2 = /3^2 - 4(3i + 4i^2 = /3^2 + 4a .= 0

From (1.18-4) it follows that i = 1/ 2 /3 -^1 / 2 j, so

a+ bi= a+ b(1/ 2 (3-^1 M) =(a+^1 / 2 b/3) -^1 / 2 bj =A+ Bj

(1.18-4)

(1.18-5)

where A = a + ~ bf3 and B = - ~ b. Equation (1.18-5) establishes an
isomorphism between the complex numbers a+bi and the complex numbers

A+ Bj with j2 = 0.

If 6. > 0, the system is called hyperbolic, duo, or double. A hyperbolic

system contains an element

. f3 - 2i

J = VE

such that

(3^2 - 4(3i + 4i^2 (3^2 + 4a

j2 = = = +1

From (1.18-6) it follows that i = (/3 - VE.i)/2, so

a + bi = a + b(/3 - v'fij) = A + B j

2

(1.18-6)

(1.18-7)

where A = a+ %bf3 and B = -^1 / 2 b0S. Again, (1.18-7) establishes an
isomorphism between the complex numbers a + bi and those of the form
A + Bj with j2 = +l.
Summing up, we see that a general complex number system can be
reduced, by an appropriate change of the basic unit i, to either of the
following essentially different systems:

1. Ordinary (elliptic) complex numbers a+ bj (j^2 = -1)

2. Parabolic complex numbers a+ bj (j^2 = 0)


3. Hyperbolic complex numbers a + bj (j^2 = + 1)