Complex Numbers 59
Expressions such as oo + oo, oo - oo, 0 · oo, or oo/oo are not defined since
they do not lead to meaningful results.
The significance of the ideal element oo can be shown geometrically
by means of the spherical representation of the complex plane due to
B. Riemann (1826-1866) and C. Neumann (1832-1925).
Consider a sphere S of unit diameter tangent to the complex plane at the
origin 0 (Fig. 1.14). Choosing a Cartesian rectangular frame of reference
OXYU, as shown in the figure, the equation of Sis
1
x2 + y2 + ( u - %)2 = -
4
or
x2 + y2 + u2 - u = (^0) (1.17-1)
The point N with coordinates (0, O, 1) will be referred to as the north pole
of the sphere. The point 0 with coordinates (0, O, 0) is then called the south
pole, and the great circle in the plane u = ~ is said to be the equator.
To associate an arbitrary point P of the complex plane with a point of
S we determine the point Q where the segment NP meets the sphere. This
construction establishes a one-to-one correspondence between the complex
plane and S-{N}, i.e., the sphere with the north pole deleted. The point Q
is called the stereographic projection of P upon S, and may also be regarded
as representing the complex number z corresponding to P. Hence the set
S - { N} is as appropriate as the complex plane for the representation of
the complex numbers. However, the presence of the extra point N on S
suggests the possibility of representing the improper complex number oo
by N, thus establishing a one-to-one correspondence between the extended
complex plane C* and S. With this convention adopted the sphere S is
called the Riemann sphere, or the sphere of complex numbers.
uFig. 1.14N(O, 0, 1)Q(cx, [), -y)y
(x, y, O)