Analytic function

Abstract: X-ray scattering factors for neutral atoms from He to Lw and for most of the chemically significant ions through Lu3+ have been computed from numerical Hartree–Fock wave functions. The results are given in the form of coefficients for an analytic function.

Abstract: A method is developed for treating Einstein's field equations, applied to static spheres of fluid, in such a manner as to provide explicit solutions in terms of known analytic functions. A number of new solutions are thus obtained, and the properties of three of the new solutions are examined in detail. It is hoped that the investigation may be of some help in connection with studies of stellar structure. (See the accompanying article by Professor Oppenheimer and Mr. Volkoff.)

Abstract: I. Analytic Functions of One Complex Variable. II. Elementary Properties of Functions of Several Complex Variables. III. Applications to Commutative Banach Algebras. IV. L2 Estimates and Existence Theorems for the Operator. V. Stein Manifolds. VI. Local Properties of Analytic Functions. VII. Coherent Analytic Sheaves on Stein Manifolds. Bibliography. Index.

Abstract: The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2].1 While it is admitted that the main idea to be used is more in the nature of a lucky guess than of a method, it will be found that the gain in brevity and simplicity of the argument is considerable. As a by-product, we shall also obtain a number of hitherto unknown identities between various domain functions. The basic problem treated in the above-mentioned papers is the following generalization of the classical Schwarz lemma: Given a finite schlicht domain D of connectivity n (n> 1) in the complex zplane and a point r in D, to find a function F(z) with the following properties: (a) F(z) belongs to the family B of analytic functions f(z) which are single-valued and regular in D and satisfy there If(z) I _ 1; (b) I F'(r) I > lf'(r) J, where f(z) is any function in B. Evidently, it is sufficient to solve this problem for any domain D' which is conformally equivalent to D. In particular, we may therefore assume, without restricting the generality of what follows, that D is bounded by analytic curves. It was shown by Ahlfors that F(z) yields a (1, n) conformal mapping of D onto the interior of the unit circle and that n-I of the n zeros of F(z) coincide with the zeros of a single-valued function h(z) which is regular in D with the exception of a simple pole at z = and satisfies -'ih(z)dz>O on the boundary r of D; the nth zero of F(z) is located at zx=. It was subsequently noticed by Garabedian that the function h(z) can be written in the form h(z) = F(z)q(z) where q(z) (Z )-2 is regular in D and that the extremal property of F(z) can be deduced in a very elegant manner from the resulting inequality