Picards famous theorem states that a meromorphic function of z, w fz, which is not a constant, assumes all values except at most 2. They generate a lot of applications in studying uniqueness of meromorphic functions, global solutions of differential equations, dynamics, and so on. It is well known that if f and g share four distinct values cm, then f is a m. Elliptic functionsimportant examples of meromorphic functions are elliptic functions. The independent significance of normality of subharmonic function family is emphasized. This chapter presents the analogues of the mittagleffler and weierstrass theorems for functions of several complex variables. Majorization problem for two subclasses of meromorphic. Riemann surfaces university of california, berkeley. Functions meromorphic in c are precisely those which can be written as a quotient of two entire functions. On entire and meromorphic functions that share one small. Akhter rasheed, saqib hussain, syed ghoos ali shah, maslina darus, saeed lodhi. Periodic meromorphic functions tsinghua university. Let f be a family of meromorphic functions on a plane domain d. Kotus, lu yinian skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites.
This section contains only the general definitions and some elementary results. Meromorphic functions, entire functions, nevanlinna theory, value distribution theory, deficient values, asymptotic values, riemann surfaces. Meromorphic functions over nonarchimedean fields pei. As stated above, what makes a meromorphic function unique is that it contains singularities that tend to infinity. In complex analysis, a meromorphic function on an open subset d of the complex plane is a function that is holomorphic on all d except a set of isolated points, which are poles for the function subcategories.
It follows then, that if the function is basically a set of neighborhoods with sprinkled asymptotes, the derivative can be found for one neighborhood at a time, assuming the values are finite with the exception of at most one singularity. On janowski type harmonic meromorphic functions with respect. Throughout this paper, a meromorphic function is assumed to be meromorphic in the whole complex plane. Thus, if d is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers. Click download or read online button to get meromorphic functions book now. Meromorphic functions with poles and zeros located in small angles 310 5. I have also thought of entire and everywhere analytic as interchangeable terminology. Meromorphic functions play a fundamental role in complex geometry, where they relate to divisors, bundles and sheaves. On properties of differences polynomials about meromorphic. A function which is holomorphic in a domain dexcept for poles is called meromorphic in d.
Pdf derivatives of meromorphic functions and sine function. Research article on harmonic meromorphic functions. The real difficulty arises when the biunivalency condition is imposed on the meromorphic functions and its inverse. In section 4 we examine linear subspaces of meromorphic functions generated on riemann surfaces. In this book, we will introduce nonarchimedean analogues of nevanlinna theory and its. I must have slept through something in my complex variables course, because all my life i have used the terms holomorphic, meromorphic, and analytic somewhat interchangeably. Meromorphic versus rational functions ucsd mathematics. Uniqueness, entire function, meromorphic function, shared value, di. Mathematica volumen 4, 197811979, 145149 an extension tieorem for meromorphic functions of several variables juhani riihentaus 1. The first result on meromorphic solutions of nonautonomous equations was the famous theorem of malmquist 41. Motivated from the above discussion on harmonic functions and class of meromorphic starlike functions with respect to symmetric point, we introduced the class of meromorphic harmonic univalent. There are two main theorems and defect relations that occupy a central place in n evanlinna theory. We prove four uniqueness theorems for meromorphic functions f and g sharing values 0, 1, 1which improve results of i. Let be a meromorphic function in the plane and let be a borel direction of order.
Shtffman 4, lemma 3 has obtained the following result concerning remov able singularities of analytic functions of several complex variables. Meromorphic functions sharing a small function wang, songmin and gao, zongsheng, abstract and applied analysis, 2007 common hypercyclic vectors for certain families of differential operators tsirivas, n. Then, a half line is a order borel direction if and only if it satisfies. A pole zero will create a vertical asymptote on a graph. Then, there exists a sequence of disks of the form where. The term comes from the ancient greek meros, meaning part every meromorphic function on d can be expressed as the ratio between two holomorphic functions with.
For a meromorphic function \fz\, we define the order and the lower order of \fz\ by. Meromorphic functions download ebook pdf, epub, tuebl, mobi. Majorization problem for two subclasses of meromorphic functions associated with a convolution operatorj. Then fis a rational function if and only if it has at worse a pole at in nity.
Jianhua zheng is a professor at the department of mathematical sciences, tsinghua university, china. On certain type of difference polynomials of meromorphic functions. Nevanlinna proved the following two wellknown theorems. Uniqueness of meromorphic functions concerning their. Namely, at each point of any teichmuller flow line, we bound the infinitesimal spectral gap for variations of the hodge norm along the flow line in terms of an easily estimated geometric quantity on the flat surface, which is greater than or equal to the flat systole. In complex analysis, a meromorphic function on an open subset d of the complex plane is a function that is holomorphic on all d except a set of isolated points, which are poles for the function. This book is an introduction to the theory of entire and meromorphic functions, with.
The text can be used as a manual for complex variable courses of. In the mathematical field of complex analysis, a meromorphic function on an open subset d of the complex plane is a function that is holomorphic on all of d except for a set of isolated points, which are poles of the function. On entire and meromorphic functions that share small. For a meromorphic function fwe put t0r,f r \ 1 tt,f t dt, n0r,a. To this end it develops fundamental methods of multivariable complex analysis that reach far beyond the applications we are going to give here. Value distribution of meromorphic functions jianhua. Research article on janowski type harmonic meromorphic functions with respect to symmetric point muhammad ghaffar khan,1 bakhtiar ahmad,2 maslina darus,3 wali khan mashwani,1 and shahid khan1 1institute of numerical sciences, kohat university of science and technology, kohat, pakistan 2govt. Complex meromorphic functions fp f and gpg sharing small function with finite weight. To find a meromorphic function with i a given principal part first cousin problem ii a given divisor second cousin problem. I have also thought of entire and everywhere analytic. Carleson measure and some classes of meromorphic functions.
A special class of meromorphic functions, namely meromorphic functions with linear poles, arise naturally in various contexts, in perturbative quantum. Filling disks of hayman type of meromorphic functions. If youre having trouble visualizing this, you can think of a pole in terms of asymptotes. The idea is to replace the complex plane, as domain of the multivalued function, by the graph of the function. Value distribution of meromorphic functions jianhua zheng. It is natural to wonder to what extent these give all meromorphic functions on c. We shall develop in this course nevanlinnas theory of meromorphic functions. Uniqueness of meromorphic functions sharing a small. Pdf twentyeight research questions on meromorphic functions and complex differential equations are listed and. On uniqueness polynomials and biurs for padic meromorphic. We say that f and g share the value a cm if f a and g a have the same zeros with the same multiplicities.
We also investigate the uniqueness problem of meromorphic functions with their difference operators and derivatives sharing. Meromorphic function an overview sciencedirect topics. Feb 01, 2011 let f and g be two nonconstant meromorphic functions, a z. Entire and meromorphic functions mathematisches seminar. One obvious way to give a meromorphic function on c is to write down a rational function, the quotient of two polynomials. By a meromorphic function we mean a meromorphic function in the complex plane c. Meromorphic functions with poles and zeros located close to a system of rays 279 3. In particular if d c we just say the function is meromorphic. Abstractlet f and g be two nonconstant meromorphic functions. On janowski type harmonic meromorphic functions with. We obtain some properties such as coe cient inequality, growth theorems, and extreme points. Introduction and the main results in this paper, a meromorphic function always means a function which is meromorphic in the whole complex plane c.
In this paper, we investigate the uniqueness problems of difference polynomials of meromorphic functions that share a value or a fixed point. We also investigate the uniqueness problem of meromorphic functions with their difference operators and derivatives sharing some values. Complements of graphs of meromorphic functions and complete vector fields. If two nonconstant meromorphic functions f and g share five values im, then f g. Let be a meromorphic function in the plane of order. M denote the class of functions that are analytic resp. Uniqueness of a meromorphic function and its derivative emis. This book is designed for scientists, engineers and post graduated students engaged in complex analysis and meromorphic functions. The basic notations and results of nevanlinna value distribution theory of meromorphic function are assumed to be known to the reader. The asymptotes of a meromorphic function are defined as the absolute value of z goes to infinity. If two distinct nonconstant meromorphic functions f and g. Pdf research questions on meromorphic functions and complex.
Mathematics is a beautiful subject, and entire functions is its most beautiful. Meromorphic functions over nonarchimedean fields peichu. Research article on harmonic meromorphic functions associated. Degree college mardan, 23200 mardan, pakistan 3department of mathematical sciences. We study the uniqueness of meromorphic functions that share one small function with more general differential polynomial.
The well known problem of feketeszego for this class is evaluated. Carleson measure and some classes of meromorphic functions, proc. Our discussion here can be repeated in the future for a meromorphic function with z0 as a pole. Periodic meromorphic functions 29 i imh, the imaginary part of h, takes values in z on f ii hf, the restriction of h to the maximal complex subspace f of the real span of f, is strictly positive definite. Given two nonconstant meromorphic functions f and g, it is said that they share a finite value a im ignoring multiplicities if f. In this paper, we investigate zeros of difference polynomials of the form fznhz, fsz, where fz is a meromorphic. Uniqueness of meromorphic functions and differential polynomials. Value distribution of meromorphic functions anatoly. In this picture, the function becomes projection to the waxis, which is wellde ned singlevalued. Pdf the order and type of meromorphic functions and entire. By making use of basic hypergeometric functions, a class of complex harmonic meromorphic functions with positive coe cients is introduced. Introduction let be the unit disk in the complex plane, and let daz be the euclidean area element on.
More speci cally, given a riemann surface of genus g, and divisor d, we generate subspaces ld of meromorphic functions and d of meromorphic 1forms. Meromorphic functions sharing one finite set sciencedirect. The problem of uniqueness of meromorphic functions in the field \mathbb c or in. Pdf uniqueness of meromorphic functions concerning their. We study the uniqueness of meromorphic functions and differential polynomials sharing one value with weight and prove two main theorems which generalize and improve some results earlier given by m. Let f be a meromorphic function in the complex plane, all of whose. A function g meromorphic in d is said to be subordinate to f co p if and only if there exists a function holomorphic in d. Uniqueness of meromorphic functions concerning their difference. After introducing principal parts and divisors we formulate the main problems that arise. Elliptic functions arose from attempts to evaluate certain integrals.
This theory has proved a tool of unparallelled precision for the study of the roots of equations fz a, f1z b, etc. Meromorphic functions with separated poles and zeros 261 2. Download fulltext pdf download fulltext pdf read fulltext. Germs of meromorphic functions in this section, we provide some terminology to describe the class of meromorphic functions under consideration.
Uniqueness and periodicity of meromorphic functions. In this paper we show that if two nonconstant meromorphic. This site is like a library, use search box in the widget to get ebook that you want. The unexpected and unusual behavior of the coefficients of meromorphic functions and their inverses.
For convenience, in this chapter we will provide full proofs of the results from 173 which are cited in chapter 5. Meromorphic solutions of algebraic differential equations. Uniqueness of meromorphic functions sharing two values. Should it leave out 3 distinct values, one could by means of a linear transformation of w locate these values at w 0, 1, picards own proof made use of the inverse. Uniqueness of meromorphic functions sharing a small function. Uniqueness of meromorphic functions when two linear. Let f, g be nonconstant meromorphic functions defined in the open complex plane c. Applications of meromorphic univalent functions associated.
For a meromorphic function fand a positive integer k, n kr,a. Let fz and gz be nonconstant meromorphic functions, a 2cf. Kung, chungchun yang, in encyclopedia of physical science and technology third edition, 2003. Meromorphic functions of several variables springerlink. Nevanlinna 20 showed that a meromorphic function on the. Nevanlinna 8 showed that a meromorphic function is uniquely determined by the inverse images, ignoring multiplicities, of 5. We prove a quantitative version of the nonuniform hyperbolicity of the teichmuller geodesic flow. Let f be a transcendental meromorphic function in c with the order. We then show that there is an algebraic relationship between.
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