Arbitrary Complex Numbers Research Articles

On methods of summability based on integral functions

Suppose throughout thatand thatis an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

On Borel‐type methods of summability

Suppose throughout that l, a n ( n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ]. Let

On the Intrinsic Derivative of Generalized Order

vative (-.,)It T. of the tensor T::: as a gemetrical genera6t t0 lization of EL. Post' s deri vati ve { f (D)}t such that if f (D) DM to0 where M is an arbitrary complex number and particularly a negative integer -m it gives the M-th intrinsic derivative as a generalization of the fractional derivative and the m-th order tensorial integral of tensors respectively. For this purpose, at first, we shall show that the excovariant and crossed extensors of integral infinite ranges are derived from tensors by Post' s differentiation and then, from such generalized derived extensors, we construct the required derivatives of covariant tensors in first a nonholonomic space whose original space is affinely connected and treat of their properties.

Applications of the theory of cluster sets to a class of meromorphic functions

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined bywhere m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned byfrom which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.

On the Physical Realizability of Linear Non-Reciprocal Networks

The necessary and sufficient conditions are given that a matrix with arbitrary complex number elements be the impedance, admittance or scattering matrix of a physical linear reciprocal or non-reciprocal network. A canonical form for non-reciprocal network synthesis is presented which applies to any linear n-terminal pair (n-port) system at any fixed frequency. If the network is passive the only circuit element required in addition to lossless inductors, capacitors, transformers and positive resistors is the gyrator. If the network is active, negative resistors and gyrators must be used in addition to conventional passive elements. Some discussion of matrixes with frequency variable elements is also given.

Note on 𝑆-numbers

and Khintchine [2] has shown that the measure of all real S-numbers with 7=1 is zero. On the other hand, Mahler [3] proved that almost all (real or complex) numbers are S-numbers. His proof shows that this statement is true with 7=4, and he conjectured that for almost all real numbers one can takey = 1 +e, and that for almost all complex numbers one can take y=1/2 +e, for arbitrary e >0. Let yr be the infimum of all numbers y' such that almost all real numbers are S-numbers with y 1/; i.e., he proved the conjecture for m = 2. (It is well known for m = 1.) By combining a theorem due to Fel'dman [6] with Mahler's original argument, it is shown here that y, 5 2, yc < 3/2. Lemma 5 of Fel'dman's paper is as follows: Let f(z) = ao+ + +amzm, where ao, * * *, am are rational integers with Iai( <a, let t1y .. * *,(m be its zeros, which are supposed distinct, and let r be an arbitrary complex number. If 5 =mini (1 v-(4), then

Contributions to the analytic theory of 𝐽-fractions

in which the coefficients ap, bp are complex numbers, and z is a complex parameter, have been called J-fractions because of their connection with the infinite matrices known as J-matrices. The theory of J-fractions with real coefficients includes the Stieltjes continued fraction theory and certain of its extensions. In a recent paper, Hellinger and Wall [3 ](1) treated the case where ap is real and bp is an arbitrary complex number with nonnegative imaginary part. In these cases, the J-fraction obviously has the property that all the quadratic forms