Meromorphic Functions Of Several Variables Research Articles

Normal Families of Meromorphic Functions in Several Variables

Normal Families of Meromorphic Functions in Several Variables

Some Results Regarding Partial Differential Polynomials and the Uniqueness of Meromorphic Functions in Several Variables

Some Results Regarding Partial Differential Polynomials and the Uniqueness of Meromorphic Functions in Several Variables

Difference uniqueness theorems on meromorphic functions in several variables

In this paper, we mainly investigate the uniqueness problem on meromorphic functions in $\mathbb{C}^m$ sharing small functions with their difference operators or shifts, and we obtain some interesting results that act as some extensions of previous results from one complex variable to several complex variables.

THE TUMURA–CLUNIE THEOREM IN SEVERAL COMPLEX VARIABLES

AbstractIt is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

Hayman conjecture for p-adic meromorphic functions in several variables

In this paper we prove a generalized version of the Hayman Conjecture for p-adic meromorphic functions in several variables.

A Difference Picard Theorem for Meromorphic Functions of Several Variables

It is shown that if n ∈ ℕ, c ∈ ℂn, and three distinct values of a meromorphic function f: ℂn sr 1 of hyper-order gV(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂn sr ℂn, τ (z) = z + c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green’s Picard-Type Theorem in the special case where gV(f) < 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the Lemma on the Logarithmic Derivative and of the Second Main Theorem of Nevanlinna theory for meromorphic functions ℂn → ℙ P1 are given, and their applications to partial difference equations are discussed.

On the convergence of certain sequences of rational approximants to meromorphic functions in several variables

In a previous paper, the author introduced a new class of multivariate rational interpolants, which are called Optimal Padé-type Approximants (OPTA). There, for this class of rational interpolants, which extends classical univariate Padé Approximants, a direct extension of the “de Montessus de Ballore's Theorem” for meromorphic functions in several variables is established. In the univariate case, this theorem ensures uniform convergence of a row of Pade Approximants when the denominator degree equals the number of poles (counting multiplicities) in a certain disc. When one overshoots the number of poles when fixing the denominator degree, convergence in measure or capacity has been proved and, under certain additional restrictions, the uniform convergence of a subsequence of the row. The author tackles the latter case and studies its generalization to functions in several variables by using OPTA.

On Rational Interpolation to Meromorphic Functions in Several Variables

In this paper, a new approach to construct rational interpolants to functions of several variables is considered. These new families of interpolants, which in fact are particular cases of the so-called Padé-type approximants (that is, rational interpolants with prescribed denominators), extend the classical Padé approximants (for the univariate case) and provide rather general extensions of the well-known Montessus de Ballore theorem for several variables. The accuracy of these approximants and the sharpness of our convergence results are analyzed by means of several examples.

Malmquist type theorem and factorization of meromorphic solutions of partial differential equations

In this paper, a general definition of factorization of meromorphic functions of several variables is given. We have proved a general Malmquist type theorem of partial differential equations and prove the pseudo-prime ness of meromorphic solutions of system of Riccati partial differential equations with polynomial coefficients. Also it is proved that any finite order meromorphic solution of linear partial differential equations with rational coefficients must be pseudo-prime. The main tool used in the paper is Nevanlinna type of higher-dimensional value distribution theory.

Spectral analysis of periodic systems with degenerate weight (expansions in Bloch solutions)

6. LITERATURE CITED L. I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Am. Math. Soc., Providence (1974). L. I. Ronkin, "On a general approach for the investigation of the distribution of the values of entire and meromorphic functions on parallel complex lines," in: All-Union Conference on the Theory of Functions of a Complex Variable. Abstracts of Communications [in Russian], Akad. Nauk Ukr. SSR, Kharkov (1971), pp. 189-190. S. Yu. Favorov, "On functions of class B and their applications in the theory of meromorphic functions of several variables," Teor. Funktsii Funktsional. Anal. i Prilozhen. (Kharkov), No. 20, 150-160 (1974). S. Yu. Favorov, "On the growth of plurisubharmonic functions," Sib. Mat. Zh., 24, No. i, 168-174 (1983). P. Lelong, Fonctionnelles Analytiques et Fonctions Entieres (n Variables), Les Presses de l'Universite de Montreal, Montreal (1968). N. Sibony and P.-M. Wong, "Some results of global analytic sets," in Seminaire P. Lelong H. Skoda (Analyse) Annees 1978/79, Lecture Notes in Math., No. 822, 221-237 (1980).

Approximation divisors of a holomorphic mapping and the deficiencies of meromorphic functions of several variables

Approximation divisors of a holomorphic mapping and the deficiencies of meromorphic functions of several variables

Boundary behaviour of meromorphic functions of several variables

Boundary behaviour of meromorphic functions of several variables

An extension theorem for meromorphic functions of several variables

An extension theorem for meromorphic functions of several variables

On families of meromorphic maps into the complex projective space

In [10], P. Montel defined the notion of a quasinormal family of meromorphic functions and obtained several results relating to this. Afterwards, in [13], H. Rutishauser generalized some of them to the case of meromorphic functions of several variables. By definition, a quasi-normal family of meromorphic functions on a domain D in Cn is a family such that any sequence in has a subsequence which converges compactly outside a thin analytic subset of D.