The sphere

There are several advantages in using the set ℂ of complex numbers as the domain of definition of functions. The complex numbers form a field which is algebraically closed, that is, polynomials of degree n have n roots in ℂ, counting multiplicities. Geometrically, ℂ can be regarded as the Euclidean plane ℝ2, probably the most familiar geometric structure of all (hence we sometimes call ℂ the complex plane). As a domain of definition of functions, ℂ has the following remarkable property: if f is a function of a complex variable and is differentiable on some region R ⊆ ℂ (recall that a region is a non-empty, path-connected, open set), then f is infinitely differentiable on R, and for each a ∈ R we can expand f as a convergent power series in some sufficiently small disc containing a. (In contrast, there are functions of a real variable which are once but not twice differentiable, or which are infinitely differentiable but cannot be represented by power series.) When f is differentiate on a region R, we will say that f is analytic on R; in some books the words ‘holomorphic’ or ‘regular’ are used instead of ‘analytic’. A function whose only singularities in R are poles is called meromorphic in R.

There are, however, some disadvantages in using ℂ. Division by 0 is impossible, and so some standard functions are not defined everywhere; for example, z-1 is undefined at z = 0.