Meromorphic functions

An exposition is given, partly historical and partly mathematical, of the Riemann zeta function � ( s ) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count non-trivial zeros of � ( s ). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.

Given a meromorphic function $f$, we present an accurate numerical method that computes all the zeros and poles of $f$ that lie inside a Jordan curve $\gamma$, together with their respective multiplicities and orders. An upper bound for the total number of poles of $f$ that lie inside $\gamma$ is assumed to be known. Our algorithm is based on numerical integration along $\gamma$ and formal orthogonal polynomials. It uses the logarithmic derivative $f'/f$. Initial approximations for the zeros and poles are not needed. Numerical examples illustrate the effectiveness of our approach.

Mittag-Leffler identity is obtained from basic definition of meromorphic function. This identity is used to obtain Matsubara frequency sums. Application in the theory of superconductivity is presented as an integral part of finite temperature field theory. The summation is done with the infinite series expansion of coth(πy). We do it from meromorphic function.

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

In this paper, we derive some subordination and superordination results associated with the family of Jung-Kim-Srivastava integral operators defined on the space of meromorphic functions. Several sandwich-type results are also obtained. 2010 Mathematics Subject Classifications. Primary 30C45; Secondary 30C80. .

We introduce a unified subclass of the function class 􀀶 of bi-univalent functions defined in the open unit disc.
Furthermore, the estimates and for |a2| and |a3 | are obtained . Moreover, a relevant connections with known results are
pointed out.

Mittag-Leffler identity is obtained from basic definition of meromorphic function. This identity is used to obtain Matsubara frequency sums. Application in the theory of superconductivity is presented as an integral part of finite temperature field theory. The summation is done with the infinite series expansion of coth(πy). We do it from meromorphic function.

Our goal in this paper is to introduce some new sequences of some mero-morphic function spaces, which will be called b q and q K-sequences. Our study is motivated by the theories of normal, Q # K and meromorphic Besov functions. For a non-normal function f the sequences of points {a n } and {b n } for which lim n→∞ (1 − |a n | 2) f # (a n) = +∞ and lim n→∞ ∆ f # (z) q (1 − |z| 2) q−2 (1 − |ϕ a n (z)| 2) s dA(z) = +∞ or lim n→∞ ∆ f # (z) 2 K(z, a n)dA(z) = +∞ are considered and compared with each other. Finally, non-normal mero-morphic functions are described in terms of the distribution of the values of these meromorphic functions.