Complex Analysis and Topology

Since the square of a non-zero real number is always positive, there can be no real square root of −1. Inventing such a square root i and adjoining it to the real numbers, as in §IV.10, leads to extensive and important developments. On the one hand, the resulting complex numbers x + iy represent well the properties of the Euclidean x−y plane and derive part of their “reality” from the geometric reality of the plane. On the other hand, well behaved functions f of such a complex number z = x + iy are those functions f which have a complex derivative, and the properties of these functions are truly remarkable. The resulting study of “complex variables”, that is, of differentiable functions of a complex number z, leads to deep mathematical theorems with unexpected practical connections, for example to electrostatic potential and to the steady flow of fluids as well as to aerodynamics. This chapter will introduce these concepts of differentiation and the corresponding integrals and will indicate some of these connections, all with a view to seeing how the apparently simple algebraic device of inventing a “number” i with i2 = −1 has both geometric and analytic consequences—all a striking instance of the remarkable interconnections of formal ideas.