We show that there exists a function f meromorphic in the plane C such that the family of all functions g holomorphic in the unit disk D for which f ◦ g has no fixed point in D is not normal. This answers a question of Hinchliffe who had shown that this family is normal if Ĉ\f(C) does not consist of exactly one point in D. We also investigate the normality of the family of all holomorphic funct...

We shall assume that reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions(see [11,14]). In addition, we will use the notation σ(f) to denote the order of growth of entire function f(z), σ2(f) to denote the hyper-order of f(z), λ(f)(λ2(f)) to denote the exponent(hyper-exponent) of convergence of the zero-s...

We give a complete classification and present new exotic phenomena of the meromorphic structure of ζ-functions associated to general selfadjoint extensions of Laplace-type operators over conic manifolds. We show that the meromorphic extensions of these ζ-functions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitraril...

This paper studies recurrence phenomena in iterative holomorphic dynamics of certain multi-valued maps. In particular, we prove an analogue the Poincare theorem for meromorphic correspondences with respect to dynamically interesting measures associated them. Meromorphic present a significant measure-theoretic obstacle: image Borel set under correspondence need not be Borel. We manage this issue...

of a Laurent series. If Op is the local ring of holomorphic functions around p. Take Mp the field of meromorphic functions around p, a principle part is just an element of the quotient groupMp/Op. The Mittag-Leffler question is, given a discrete set {pn} of points in S and a principle part at pn for each n, does there exist a meromorphic function f on S, holomorphic outside {pn}, whose principl...

A singular flat geometry may be canonically assigned to a real algebraic curve Γ; namely, via analytic continuation of unit speed parameterization of the real locus ΓR. Globally, the metric ρ = |Q| = |q(z)|dzdz̄ is given by the meromorphic quadratic differential Q on Γ induced by the standard complex form dx + dy on C = {(x, y)}. By considering basic properties of Q, we show that the condition f...

We describe the explicit construction of M. J. Ferreira, B. A. Simões and the author of all harmonic maps from the 2-sphere to the unitary group in terms of freely chosen meromorphic functions. The harmonic maps are displayed as the product of unitons with an explicit spanning set for each uniton given by an algebraic formula in terms of the meromorphic data. We then discuss an extension of thi...

Among the invariant factors g of a positive semidefinite analytic function f on R, those g whose zero set Y is a curve are called special. We show that if each special g is a sum of squares of global meromorphic functions on a neighbourhood of Y , then f is a sum of squares of global meromorphic functions. Here sums can be (convergent) infinite, but we also find some sufficient conditions to ge...

We study the normality of families of meromorphic functions related to a result of Drasin. We consider whether a family meromorphic functions F whose each function does not take zero is normal in D, if for every pair of functions f and g in F, f z and g z share ∞ or H f − 1 and H g − 1 share 0, whereH f : f k z ak−1f k−1 z · · · , a0f z . Some examples show that the conditions in our results ar...

We present the basics of two-body quantum-mechanical scattering theory and the theory of quantum resonances. The wave operators and S-matrix are constructed for smooth, compactly-supported potential perturbations of the Laplacian. The meromorphic continuation of the cut-off resolvent is proved for the same family of Schrödinger operators. Quantum resonances are defined as the poles of the merom...