Lars V. Ahlfors

The boundary correspondence under quasiconformal mappings

We study boundary properties of quasiconformal self-mappings depending on complex dilatations. We give some new conditions for the corresponding quasisymmetric func- tion to be asymptotically symmetric and obtain an explicit asymptotical representation for the distortion ratio of boundary correspondence when the complex dilatation has directional limits.

Complex analysis: an introduction to the theory of analytic functions of one complex variable

Publisher: Publication Date: Number of Pages: Format: Edition: Price: ISBN: Category: BLL Rating: Lars Ahlfors McGraw-Hill 1979 345 Hardcover 3

Conformal invariants and function-theoretic null-sets

286.67 9780070006577 Textbook BLL*** The Basic Library List Committee considers this book essential for undergraduate mathematics libraries. Home » MAA Publications » MAA Reviews » Complex Analysis : An Introduction to The Theory of Analytic Functions of One Complex Variable

An extension of Schwarz’s lemma

The most useful conformal invar iants are obtained by solving conformMly invar ian t ex t remal problems. The i r usefulness derives f rom the fac t tha t they must automat ical ly satisfy a principle of majorizat ion. There is a r ich variety of such problems, and if we would aim at completeness this paper would assume forbidding proport ions. We shall therefore l imit ourselves to a few part icular ly simple i n v a r i a n t s and study thei r propert ies and in ter re la t ions in considerable detail. Each class of invar ian ts is connected wi th a category of null-sets, which by this very fac t en ter na tura l ly in funct iontheoret ic considerat ions. A null-set is the complement of a region for which a cer ta in conformal invar iant degenerates. Inequal i t ies between invar iants lead to inclusion relat ions between the corresponding classes of null-sets. Th roughou t this paper Y2 will denote an open region in the extended z plane, and Zo will be a dis t inguished point in t~. Most results will be formula ted for the case z 0 ~ c~, but the t rans i t ion to z o = c~ is always trivial. In some instances the la t te r case offers formal advantages. We shall consider classes of funct ions f(z) which are analyt ic and singlevalued in some region t). F o r a general class ~ the region t~ is al lowed to vary with f , but the subclass of funct ions in a fixed region t~ will be denoted by ~(t2). For ZoE ~ we in t roduce the quant i ty

Möbius transformations in expressed through 2×2 matrices of clifford numbers

and has the constant curvature -4. 2. Consider now an analytic function o =f(z) from the circle I zx < 1 to a Riemann surface W. The analyticity is expressed by the fact that every local parameter w is an analytic function of z. To a differential element dz corresponds an element dw whose length does not depend on the dlirection of dz. The corresponding value of ds =X j dw|N dz I is therefore uniquely de* Presented to the Society, September 8, 1937; received by the editors April 1, 1937. t For the definition of a Riemann surface see T. Rad6, Uber den Begriff der Riemannschen Fldche, Acta Szeged, vol. 2 (1925). 359

Cross-ratios and Schwarzian Derivatives in Rn

This is an expository paper. Its purpose is to show, in contemporary mathematical language, how to study Mobius transformations in several dimensions as linear fractions with Clifford numbers as coefficients. The method was introduced by K. Th. Vahlen in 1902, but was forgotten for a long time.

On Phragmén-Lindelöf’s principle

This paper was written several years ago, but no part of it has been published previously. A preprint was distributed to selected experts and seems to have been favorably received. For some time I had hoped to improve on the results of the paper, but as years went by my research took a different direction, and it became implausible that I would add anything significant to the paper as it stands.

Clifford Numbers and Möbius Transformations in Rn

In the first part of this paper we give a proof of Phragmen-Lindelkfs now classical principlet which is simpler and yields more detailed information than any of the proofs hitherto known. Our procedure consists in proving a theorem in finite terms, similar to Hadamards three circles theorem, from which the asymptotic statement is shown to follow by a very simple and transparent reasoning. A certain symmetry in the result is obtained by allowing the functions considered to have two possible singularities, one at 0 and one at oo. The ultimate theorem is the sharpest possible and contains all previous results, including those of the brothers Nevanlinna.t In Part II we generalize Phragmen-Lindel6fs principle to harmonic functions of n variables. The methods of Part I are seen to carry over without any difficulties. The result is particularly interesting in so far as the symmetry of the two-dimensional case is not maintained, the extremal functions corresponding to the two singularities being now essentially different.

Conditions for Quasiconformal Deformations in Several Variables

Mobius transformations in any dimension can be expressed through 2x2 matrices with Clifford numbers as entries. This technique is relatively unknown in spite of having been introduced as early as 1902. The present paper should be viewed as a strong endorsement for the use of Clifford algebras in this particular context. In addition to an expository introduction to Clifford numbers it features a discussion of the fixed points and classification of Mobius transformations.

Some remarks on clifford algebras

Publisher Summary If v(z) is a measurable complex-valued function in the complex plane. Subject only to growth conditions, the inhomogeneous Cauchy–Riemann equation has the explicit solution , and this solution satisfies . This observation is fundamental for many questions in the theory of holomorphic functions of one or several variables, for partial differential equations, and in the theory of quasiconformal mappings in two dimensions. In particular, Cauchy–Riemann equation plays an important role in the study of Kleinian groups, the right-hand side being a Beltrami differential. In that connection, the solution f has a natural interpretation as an infinitesimal deformation. Such deformations can be studied in the space of n real variables.