On Maps: On Conformal Maps

"Two Riemann surfaces which can be mapped conformally onto each other are (conformally) equivalent and are to be regarded as different representations of one and the same ideal Riemann surface. The intrinsic properties of a Riemann surface will include only those properties which are invariant under conformal maps; that is, those properties which, if possessed by one Riemann surface are possessed by every equivalent surface. Obviously all topological properties are intrinsic properties of a Riemann surface; similarly with those properties belonging to the surface by virtue of its smoothness." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"Conformal mapping as it is often called is the representation of a bounded area in the plane of a complex variable by an area in the plane of another complex variable. Thus the method is a branch of mathematics based on the theory of functions of a complex variable." (William J Gibbs, "Conformal Transformations in Electrical Engineering", 1958)

"Conformal mappings are characterized by the fact that they infinitesimally (i) preserve angles, and (ii) preserve length (up to a scalar factor)." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"It is a truism that the Riemann mapping theorem allows us to transfer the complex function theory of any simply connected domain (except the plane itself) back to the unit disk, or vice versa. But many of the more delicate questions require something more. If we wish to studybehavior of functions at the boundary, or growth or regularity conditions, then we must know something about the boundary behavior of the conformal mapping." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"One of the most striking facts about the Poincaré metric on the disk is that it turns the disk into a complete metric space. How could this be? The boundary is missing! The reason that the disk is complete in the Poincaré metric is the same as the reason that the plane is complete in the Euclidean metric: the boundary is infinitely far away. [...] One of the important facts about the Poincaré metric is that it can be used to study not just conformal maps but all holomorphic maps of the disk. The key to this assertion is the classical Schwarz lemma." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)