Independent Complex Variables Research Articles

Differentiable functions in a three-dimensional associative noncommutative algebra

We consider a three-dimensional associative noncommutative algebra Ã2 over the field C, which contains the algebra of bicomplex numbers B(C) as a subalgebra. In this paper we consider functions of the form Φ(ζ)=f1(ξ1, ξ2,ξ3)I1+ f2(ξ1, ξ2,ξ3)I2+ f3(ξ1, ξ2,ξ3)ρ of the variable ζ= ξ1I1+ ξ2I2+ ξ3ρ, where ξ1, ξ2, ξ3 are independent complex variables and f1, f2, f3 are holomorphic functions of three complex variables. We construct in an explicit form all functions defined by equalities dΦ =dζ·Φ´(ζ) or dΦ = Φ´(ζ) ·dζ. The obtained descriptions we apply to representation of the mentioned class of functions by series. Also we established integral representations of these functions.

Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices

Suppose that $\mathbf{X}_{n}=(x_{jk})$ is $N\times n$ whose elements are independent complex variables with mean zero, variance 1. The separable sample covariance matrix is defined as $\mathbf{B}_{n}=\frac{1}{N}\mathbf{T}_{2n}^{1/2}\mathbf{X}_{n}\mathbf{T}_{1n}\mathbf{X}_{n}^{*}\mathbf{T}_{2n}^{1/2}$ where $\mathbf{T}_{1n}$ is a Hermitian matrix and $\mathbf{T}_{2n}^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $\mathbf{T}_{2n}$. Its linear spectral statistics (LSS) are shown to have Gaussian limits when $n/N$ approaches a positive constant under some conditions.

An identity for the Dedekind eta-function involving two independent complex variables

The authors prove a new identity for the Dedekind eta-function that involves third powers of the eta-function, with each of the two cubes being a function of a different complex variable.

Double Dirichlet series over function fields

We construct a finite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a finite field , q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s and w. We prove that these series satisfy a finite, non-abelian group of functional equations in the two complex variables (s, w) and are rational functions in q-s and q-w with a specified denominator. The group is D6, the dihedral group of order 12.

Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions: III

The main object of this paper is the mean square I h (s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for I h (1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for I h (1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for I h (1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

Explicit Formulas and Asymptotic Expansions for Certain Mean Square of Hurwitz Zeta-Functions.

The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for Ih(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

The unsteady lift on a swept blade tip

Highly swept blades are now commonly used in modern aeroengines, and in this paper we solve a model problem of relevance to the understanding and prediction of noise generation by the interaction between incident vortical disturbances and such a blade. In order to include the potentially significant effects of the blade-tip region, we consider a (semi-infinite span) quarter-plane aligned at a non-zero sweep angle to supersonic mean flow, with a single harmonic gust incident on the quarterplane from upstream. The solution is completed using a novel application of the Wiener–Hopf technique, in which the usual factorisation is carried out with respect to two independent complex variables separately, and closed-form expressions for the practically important lift per unit span are derived for both the subsonic and the supersonic leading-edge regimes. The dependence of the unsteady response on the sweep angle and the gust wavenumbers is examined, and in particular we demonstrate that the magnitude of the effects of the tip region is significantly reduced by increasing the blade sweep or by considering gusts of higher frequency. It also becomes clear that the magnitude of the unsteady response can be either decreased or increased by sweeping the blade, in a way which proves highly dependent on the particular values of the flow parameters. In addition, we consider the two critical values of the gust trace speed along the leading edge which correspond to the sonic velocities in the two spanwise directions, and for which the chordwise oscillation of the unsteady lift distribution on an infinite-span blade vanishes. In these two cases, the lift per unit span (integrated over the infinite chord) is clearly singular, but we demonstrate that the effect of including the blade tip is to smooth out just one of these singularities, and replace it instead by a relatively large, but finite, value.

Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions

The main object of this paper is the mean square Ih(s) of higher derivatives of Hurwitz zeta functions ζ(s, α). We shall prove asymptotic formulas for Ih(1/2 + it) as t → +∞ with the coefficients in closed expressions (Theorem 1). We also prove a certain explicit formula for Ih(1/2 + it) (Theorem 2), in which the coefficients are, in a sense, not explicit. However, one merit of this formula is that it contains sufficient information for obtaining the complete asymptotic expansion for Ih(1/2 + it) when h is small. Another merit is that Theorem 1 can be strengthened with the aid of Theorem 2 (see Theorem 3). The fundamental method for the proofs is Atkinson's dissection argument applied to the product ζ(u, α)ζ(v, α) with the independent complex variables u and v.

S-order nondiagonal quasiprobabilities.

This paper presents a correspondence between the annihilation a^ and creation a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ operators and two independent complex variables \ensuremath{\alpha} and \ensuremath{\beta}, which makes possible the definition of positive nondiagonal quasiprobabilities for any Cahill-Glauber s order. A generalized version of the Cahill and Glauber s-ordered displacement operator displacing the annihilation and creation operators by \ensuremath{\alpha} and \ensuremath{\beta} is defined. Corresponding to this n/Inondiagonal displacement, a nondiagonal ordering operator is introduced so that a map of s-ordered operators into c numbers a^\ensuremath{\rightarrow}\ensuremath{\alpha}, a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$\ensuremath{\rightarrow}\ensuremath{\beta} can be defined. The Drummond-Gardiner projector and the Cahill-Glauber diagonal ordering operator are obtained as particular cases. In order to use the nondiagonal correspondence in the determination of quantum expectation values of observables, several families of quasiprobabilities are defined, generalizing Drummond-Gardiner (normal order and nondiagonal) positive-P-function and Cahill-Glauber (s-order and diagonal) quasiprobabilities. It is demonstrated that these nondiagonal quasiprobabilities exist as well-behaved functions where at least one is non-negative, for any state and any order. The time evolution of these quasiprobabilities is discussed both in the Schr\odinger and in the Heisenberg pictures. In the Heisenberg picture, a method for obtaining c-number stochastic differential equations (SDE's) directly from operator equations using the s-order nondiagonal correspondence is described. The main difference between this method and the Langevin approach is that in the latter a diagonal correspondence is used, leading eventually to wrong results. The use of these SDE's to solve quantum optical problems is discussed, and an application to the nonlinearly damped degenerate parametric oscillator is made. In order to obtain the SDE, various (inequivalent) truncation schemes are necessary.

The solution of boundary value problems of the theory of shells with an unknown boundary

The elliptical boundary value problem with an unknown free boundary is solved using analytic continuation into the domain of independent complex variables. The unknown region of solution is estimated using the method of characteristics based on the Schwarz mapping principle and the correspondence between the solutions of the Cauchy problem in the elliptical and hyperbolic region. A numerical solution of the problem in question is also obtained using Galerkin's method and Green's functions.

Analytic solutions of the heat equation and some formulas for laguerre and hermite polynomials

We consider analytic solutions of the heat equation u xx + u yy=u t defined in a cylinder and show that any such solution can be expanded in a series of polynomial solutions to the heat equation. If we define the independent complex variables z and [zbar] by =z=x + iy, [zbar] x−y where x and y are independent complex variables, it is shown that any real-valued analytic solution of the heat equation is uniquely determined by its values on [zbar]=0 or i=0. Using this result, and expressing the above mentioned polynomial solutions to the heat equation in terms of Laguerre polynomials, we obtain some generating functions for Laguerre polynomials, as well as connection formulas between products of Hermite polynomials and Laguerre polynomials of argument r 2=x 2 + y 2. These connection formulas generalize a well known result of Feldheim.

On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations

Asymptotic stability of delay differential equations of the retarded type is studied in terms of zero criteria for polynomials in two independent complex variables. The major new result of this note is that asymptotic stability independent of delay can be expressed in terms of a (finitely) implementable algebraic criterion involving a two-variable polynomial.

Probability distribution of the sum of N complex random variables

A method is described for the investigation of the statistics of the sum of N independent complex variables. This method allows one to obtain the exact mean value of the squared modulus of the sum and its moments, for any value of the number N of variables. In particular, the case has been examined where the variables are lognormally distributed in amplitude and normally distributed in phase. This case is of interest in the field of atmospheric optics for the investigation of the effect of the diameter of the receiving aperture.

Normal threshold approximation to analycity and Mandelstam representation

Normal threshold approximation to analyticity of two-body scattering amplitude formulated in the spaceC3 of independent complex variables\(\hat s,\hat t,\hat u\) and restricted by the assumption of polynomial boundedness in all three variables leads to a three-dimensional integral representation which is shown to reproduce Mandelstam representation on the complex plane\(\hat s + \hat t + \hat u = \sum\limits_{i = 1}^4 {m_i^2 } \).

Bounds for Solutions of a System of Linear Partial Differential Equations on Domains with Bergman-Silov Boundary

The method of integral operators has been used by Bergman and others (4; 6; 7; 10; 12) to obtain many properties of solutions of linear partial differential equations. In the case of equations in two variables with entire coefficients various integral operators have been introduced which transform holomorphic functions of one complex variable into solutions of the equation. This approach has been extended to differential equations in more variables and systems of differential equations. Recently Bergman (6; 4) obtained an integral operator transforming certain combinations of holomorphic functions of two complex variables into functions of four real variables which are the real parts of solutions of the system1where z1, z1*, z2, z2* are independent complex variables and the functions Fj (J = 1, 2) are entire functions of the indicated variables.

The solution of boundary value problems by multiple Laplace transformations

By use of the multiple Laplace transform a partial differential equation and its associated boundary conditions characterizing a boundary value problem in n independent real variables can be transferred directly into an algebraic equation in n independent complex variables. This algebraic equation can be solved for the multiple transform of the solution of the boundary value problem. Multiple inversion of this transform then gives the desired solution. The general theory underlying such solution of boundary value problems in two and three independent variables is advanced in detail. Use of this theory is illustrated by solution of two specific problems.

On the analytic extension of functions defined by double power series

defining a function f(x, y) of the two independent complex variables x, y withiii the regions xI 0, > ?), to determine the value of this funietion at any point in its domain of existence. t It is our special purpose to demonstrate the following theorem: THEOREM. If in the series (1) the function a (qu, v) considered as a function of the two independent, comnplex variables U, v ( ut = a + i'3, v = 'y + i8 ) is single valued and analytic for those valui es qf u, v for which a O 0, 'y _ 0 and if there exists a constant c such that for the same values of ut, v