Conformal mapping

Abstract

Abstract This chapter concerns angle-preserving mappings between regions in the complex plane. As we shall see, every holomorphic function whose derivative is non-zero defines such a mapping. These mappings are of intrinsic geometric interest and of importance in advanced complex analysis (Chapter 16 hints at this). They are also worth studying because of their usefulness in solving certain physical problems, for example, problems about two-dimensional fluid flow, the idea being to transform a given problem into an equivalent one which is easier to solve. So we wish to consider the problem of mapping a given region G onto a geometrically simpler region G’, for example the open unit disc or the open upper half-plane. We concentrate in this basic track chapter on presenting and illustrating the principles of conformal mapping. We therefore restrict attention to mapping a region whose boundary is a circline or is a pair of arcs (that is, lines, rays, circular arcs, line segments).