Normal families of meromorphic functions with multiple zeros and poles

LetF be a family of functions meromorphic in the plane domainD, all of whose zeros and poles are multiple. Leth be a continuous function onD. Suppose that, for eachf ≠F,f ¹(z) εh(z) forz εD. We show that ifh(z) ≠ 0 for allz εD, or ifh is holomorphic onD but not identically zero there and all zeros of functions inF have multiplicity at least 3, thenF is a normal family onD. Partially supported by the Shanghai Priority Academic Discipline and by the NNSF of China Approved No. 10271122. Research supported by the German-Israeli Foundation for Scientific Research and Development, G.I.F. Grant No. G-643-117.6/1999.     

Normal families of meromorphic functions with multiple zeros and poles

Let k be a positive integer with k?2 and let be a family of functions meromorphic on a domain D in , all of whose poles have multiplicity at least 3, and of whose zeros all have multiplicity at least k+1. Let a(z) be a function holomorphic on D, a(z)?0. Suppose that for each , f^(k)(z)≠a(z) for z∈D. Then is a normal family on D.     

Application of the generalized Siewert-Burniston method to locating zeros and poles of meromorphic functions

Summary The generalized Siewert-Burniston method for the determination of the zeros of an analytic function inside a simple closed contourC in the complex plane is modified to be applicable to the determination of both the zeros and the poles of a meromorphic function insideC. The method is based again on the solution of a homogeneous Riemann-Hilbert boundary value problem onC and the values of the meromorphic function onC are sufficient for the application of the method.

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Résumé La méthode de Siewert et Burniston généralisée pour la détermination des zéros d'une fonction analytique à l'intérieur d'un contourC simple fermé dans le plan complexe est modifiée pour être applicable à la détermination des zéros ainsi que des pôles d'une fonction méromorphe à l'intérieur deC. La méthode se base encore une fois à la résolution d'un problème de valeurs aux limites de Riemann-Hilbert surC; les valeurs de la fonction méromorphe surC étant suffisantes pour l'application de la méthode.

     

The logarithmic energy of zeros and poles of a rational function

On assuming that certain lemniscates of a rational function are connected, we establish some sharp inequality that involves the logarithmic energy of a discrete charge concentrated at the zeros and poles of this function and the absolute values of its derivatives at these points. The equality in this estimate is attained for specially arranged zeros and poles of a suitable Zolotarev fraction and for special distributions of charge.     

关于多极点的k-格林函数

研究了k-Hessian算子对应的多极点k-格林函数,得到了在k-超凸域上的连续性与边界行为,并通过Lelong-Jensen公式展示了k-Hessian边界测度与k-凸函数在极点处函数值之间的关系.     

Heegner zeros of theta functions

Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant . This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.

     

On the zeros and poles of Padé approximants toe z

Summary In this paper, we study the location of the zeros and poles of general Padé approximats toe ^z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Fla. Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT(11-1)-2075.     

Bank-Laine functions with sparse zeros

A Bank-Laine function is an entire function satisfying at every zero of . We construct a Bank-Laine function of finite order with arbitrarily sparse zero-sequence. On the other hand, we show that a real sequence of at most order 1, convergence class, cannot be the zero-sequence of a Bank-Laine function of finite order.

     

Determination of poles of sectionally meromorphic functions

The numerical methods of Abd-Ellal, Delves and Reid for locating poles of meromorphic functions inside smooth closed contours C in the complex plane are generalized to apply to sectionally meromorphic functions satisfying a homogeneous or nonhomogeneous Riemann-Hilbert boundary value problem on the simple closed contours of their discontinuity (which may intersect C). In the first case, the zeros of the meromorphic functions can be determined as well. Two simple illustrations of the present generalization are also presented.     

How poles of orthogonal rational functions affect their Christoffel functions

We show that even a relatively small number of poles of a sequence of orthogonal rational functions approaching the interval of orthogonality, can prevent their Christoffel functions from having the expected asymptotics. We also establish a sufficient condition on the rate for such asymptotics, provided the rate of approach of the poles is sufficiently slow. This provides a supplement to recent results of the authors where poles were assumed to stay away from the interval of orthogonality.     

On the poles of topological zeta functions

We study the topological zeta function associated to a polynomial with complex coefficients. This is a rational function in one variable, and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote has a pole in . We show that is a subset of ; for and , the last two authors proved before that these are exactly the poles less than . As the main result we prove that each rational number in the interval is contained in .