ABSTRACT
Part 1 has been concerned with the representation of numbers on the complex plane (Chapters 1 and 3) that is the domain of a function (Chapters 3 and 5), and also with the range of the function (Chapters 5 and 7), in particular when it is multivalued (Chapters 7 and 9). The representation of a multivalued function on plane sheets is limited (Chapter 7), and an alternative is a one-to-one map between the plane and a sphere, viz. using the Riemann sphere as a Riemann surface. Projecting a sphere from one pole onto a plane passing through the equator, a one-to-one correspondence (Section 9.1) is obtained between the points on the sphere and the points on the plane (Section 9.2); the pole itself is projected to infinity, implying that there is only one “point at infinity” (Section 9.2). The stereographic projection (Section 9.3) has been used since the times of Ptolemy to make plane maps of the spherical surface of the earth; these maps are reasonably accurate near the equator, but become significantly distorted near the pole of projection. The section of the sphere by a plane is always a circle, and if the plane passes (does not pass) through the pole of projection, the circle is projected onto the plane as a straight line (Section 9.4) [another circle (Section 9.5)]. A particular case of the stereographic projection is obtained by taking a perpendicular plane through the pole of projection that makes an ordered projection of the circle onto a straight line (Section 9.6). A generalization of the stereographic projection is obtained by considering a sphere with branch-cuts (Section 9.7), for example, for a multivalued function. In the case of multivalued functions, taking two or more spheres joined by tubes along the branch-cuts leads, by continuous deformation, to spheres with one or more handles (Section 9.8) that can also be deformed into a torus with one or more holes (Section 9.9). Although the mapping into a plane is the simplest, it can be misleading if there are several sheets or the point-at-infinity is part of the domain or range; in such situations mapping into a closed surface may be more illuminating.
