In this section I present some theory of complex variables, with physical insight but without rigorous proofs. No prior exposure to the theory of complex variables is assumed. One idea, used in about half of the chapter, links any complex function with an idealized fluid flow in the plane (the details are in section 11.3). With this compact idea some of the basic facts of the theory become intuitively obvious.
The first section on complex numbers requires little background. The rest of the chapter should be accessible to anyone who saw line integrals. The concepts of the divergence and curl are explained to the degree they are used.
Here are some of the highlights.
1. The Cauchy integral formula, ...