ABSTRACT
The preceding applications of the conformal mapping have shown on several instances two properties: (i) a variety of domains, for example, a half-plane, a circle, or an arbitrary polygon can be mapped into each other; (ii) if the boundary of a region is mapped into a curve, the interior (exterior) of the region is mapped into the interior (exterior) of the curve. Since two regions can be mapped into each other through the unit disk, it is sufficient to address (i) and (ii) for mappings from the unit disc to an arbitrary region. The mappings besides being conformal or analytic should be univalent, that is, take each value in the range only once; in this way the range does not overlap with itself. The question thus arises of the most general conditions in that these properties hold. A first approach is to find necessary conditions, for example, conditions that if not met cause the property to fail; it is then proved that those necessary conditions are also sufficient. For example, the simple, that is, conformal and univalent, mapping into the unit disk does not exist for (a) multiply connected regions, for example, a doubly connected region may be mapped into an annulus rather than a disk; (b) a region with a single boundary point, for example, the punctured plane, that is, the plane less a point, has no simple mapping into a unit disk. Thus two necessary conditions for the existence of simple mapping of a region into a unit disk (Section 37.3) are that it be simply connected and the boundary consists of at least two points; these conditions are also sufficient (Section 37.5). The map is unique if a point is assigned as well as direction through it, that is, the conditions of unicity are the same as for bilinear mappings into the unit disk. Concerning the correspondence of boundaries and interiors (Section 37.7), the property (ii) above holds: (α) for a domain without singularities of the mapping function; (β) if the mapping function has one singularity in the interior that is a simple pole, viz. if it were a multiple pole or essential singularity it would fail to be univalent; (γ) if the mapping function has a singularity on the boundary, then the image region extends to infinity, and some asymptotic conditions have to be imposed for the theorem to hold. The mapping between multiply connected regions cannot be simple, and requires the use of multivalued functions (Section 37.4). The proof of existence of conformal simple (nonsimple) mapping between simply connected (multiply connected) regions involves the reference and minimax functions (Section 37.7) [the complete collection of fundamental regions of an automorphishm group (Section 37.6)]. The proof of existence of a conformal mapping from the unit disk to an arbitrary compact region is based on a sequence of holomorphic functions that converges uniformly, implying that the limit function is also holomorphic (Section 37.2); the uniform convergence of the sequence of functions holds if its elements are bounded (Section 37.1). Thus the starting point for all these proofs, as usual with existence and unicity theorems, is two sets of results concerning: (i) some bounds on the modulus, and real or imaginary parts of holomorphic functions and their derivates in disks (Section 37.1); (ii) the existence of uniformly convergent subsequences of sequences of functions in compact domains (Section 37.2). A set of results complementary to the existence and unicity theorems, is the construction of complex functions in a disk (in the complex half-plane) from the values of their real (or imaginary) parts on the boundary circle (Section 37.8) [straight line (Section 37.9)]; the latter specify the potential field inside or outside a cylinder (in the upper or lower half-plane) from the values of the potential or
T&F Cat#71181,
with Applications to
stream function on the boundary circle (real axis). These results (Sections 37.8 and 37.9) are based on an extension of the Cauchy theorems (Chapters 13, 15, and 31) to the exterior (Section 37.8) as well as the interior (Sections 13.6, 15.4, and 31.4) of a boundary. If the main interest is explicit solutions of problems rather than existence and unicity theorems it is possible to proceed directly to Sections 37.8 and 37.9. The latter concerns the field created by a distribution of sources or by a given potential on a boundary; both processes occur in nature and are used in engineering devices.
