Analytical Study of Helical Latching Ferrite Phaser Structure

The basic parabolic cylinder (bpc) distributions of the firstBasic parabolic cylinder (bpc) distributionof the first kind and second kindsBasic parabolic cylinder (bpc) distributionof the second kind were derived by OgasawaraOgasawara (Commun Stat Theor Methods [17]) to obtain the absolute momentsMomentsabsolute of the normal variable underStripe truncation stripe (zebraicTruncationzebraic or tigerishTruncationtigerish) truncationTruncationstripe. The bpc distribution of the third kindBasic parabolic cylinder (bpc) distributionof the third kind or simply bpc is introduced as a general bpc, whose special cases are the bpc’s of the first and second kinds. TheWeighted or incomplete parabolic cylinder function weighted or incomplete parabolic cylinder functionParabolic cylinder functionincomplete is obtained using theWeighted or incomplete Kummer confluent hypergeometric function weighted or incomplete Kummer confluent hypergeometric functionKummer confluent hypergeometric function given by OgasawaraOgasawara [J Multivar Anal [18]) to have the absolute momentsMomentsabsolute. The multivariate bpc distributionMultivariate basic parabolic cylinder (bpc) distribution is also derived to obtain the absolute momentsMomentsabsolute of the normal vectorNormalvector underTruncationsectional sectional truncationSectional truncation, which is the multivariate version ofTruncationstripe stripe truncationStripe truncation.

The borderlines of the area in which the circular waveguide, entirely filled with azimuthally magnetized ferrite, works as a phase shifter for the normal TE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">01</sub> mode, are determined. For the purpose the special features of the cut-off regime of the configuration and of its limiting one, observed at higher frequencies in which the propagation stops, provided the magnetic bias is negative, are used. For a fixed numerical equivalent of the normalized in an appropriate way guide radius, the off-diagonal ferrite permeability tensor element and the phase constant of the wave are counted by the roots of the structure's characteristic equation, derived through the Kummer confluent hypergeometric function. This is done repeatedly for a varying negative (positive) value of the imaginary part of its complex first parameter. The process continues, until the computed value of the element coincides with that of the same, corresponding to the cut-off state for the radius chosen (to the limiting one, if the latter is greater than certain number) with a prescribed accuracy. The relevant calculated numerical equivalent of the phase constant is accepted as the one, searched for. The differential phase shift equals the constant mentioned in the first case, whereas in the second one, it is found from it by a simple arithmetic. Changing the normalized radius, the boundaries of the domain studied are traced. The outcomes are presented graphically in a normalized form.

Some remarks on the location of non-asymptotic zeros of Whittaker and Kummer hypergeometric functions

We explore the application of fundamental solutions, constructed as irreducible linear combinations with variable coefficients of two Kummer confluent hypergeometric functions, to solve the one-dimensional stationary Schrödinger equation. This approach offers a new technique for tackling this equation, particularly well-suited for singular potentials exhibiting non-monotonic behavior, a challenge for traditional methods. We focus on a specific family of four-parameter potential wells defined on the positive semi-axis using the Lambert-W function. For these wells, the potential energy increases toward the origin but tends to zero at infinity. We present solutions in the form of linear combinations of Kummer functions and their derivatives, along with power functions and exponents. We then explore two specific well types within this family, analyzing the energy spectrum of the quantum system for each case.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

The nonreciprocal propagation of rotationally symmetric TE-modes along a circular guide, uniformly filled with latching ferrite magnetised azimuthally to remanence, is studied theoretically. The boundary-value problem is solved in terms of Kummer confluent hypergeometric functions of complex parameter and variable. An asymptotic method for solution of the characteristic equation is developed, providing a basis for numerical computation of the eigenvalue spectrum and phase characteristics of the structure with a fair degree of engineering accuracy. The normalised nonreciprocal phase characteristics and the differential phase shift of the waveguide under rotationally symmetric TE-mode excitation are presented for both latched states of ferrite remanent magnetisation. The potentialities of the structure as digital nonreciprocal phase shifter and/or bistable ferrite switch using intrinsic advantages of ferrite remanent magnetisation are discussed.

Examples of the application of confluent hypergeometric functions in miscellaneous areas of theoretical physics are presented. Emphasis is placed on the use of Kummer function in the field of microwaves: the cases of normal and slow rotationally symmetric TE modes propagation in the azimuthally magnetized circular ferrite waveguide are considered. Some of the properties of function mentioned in the complex (real) domain, of importance in the solution of boundary-value problem stated for normal (slow) waves, are established. A theorem for the identity of purely imaginary and real zeros of the complex resp. real Kummer function for certain parameters, is proved numerically. The terms for wave transmission are obtained as four bilaterally open intervals of variation of quantities, specifying the fields. It turns out that the normal (slow) modes may exist in one (two) region(s). The theoretically predicted phase curves for the first waves of the two TE sets examined show that the structure explored is suitable for ferrite control components design.

The propagation of normal TE 0n modes in the circular waveguide, comprising a coaxially positioned ferrite cylinder with azimuthal bias and a dielectric toroid, of equal relative permittivities, is studied in terms of complex Kummer confluent hypergeometric functions and real Bessel and Neumann ones. The existence of certain positive real numbers (named L 3± numbers), related to the positive purely imaginary roots of the characteristic equation of configuration, is ascertained. It is found out that its phase curves for both signs of ferrite magnetization are finite, with limits, determined by the cutoff frequencies and by special envelope lines, depending on the quantities L 3± .

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

Risk aggregation in multivariate dependent Pareto distributions

The real negative zeros equation and equation — a finite positive integer) in the first parameter k, resp. â of the real Kummer confluent hypergeometric function equation with â, ĉ and x — all real, â 0 and x > 0, â = ĉ/2 + k (k = â – ĉ/2 — real, negative, −∞ < k < −ĉ/2 ), resp., â — arbitrary real negative, are investigated numerically for selected values of n, assuming x and ĉ as parameters. The results are presented in the form of Tables and graphs. Analyzing them, certain properties of the zeros are revealed and summarized in three theorems which are substantiated numerically and analytically. Theorem 1 states that when x → +∞, then equation and equation tend to the real positive numbers equation, relevantly equation, and in case x → 0, both quantities of interest go to −∞. Theorem 2 determines the number n of zeros studied. It ascertains that it is finite and depends on their values. According to it n = 1; 2; …, p, where equation, correspondingly equation, (equation is the greatest integer less than or equal to equation). Theorem 3 specifies the value of the first zero at x = ĉ to equation, accordingly to equation. An application of the quantities considered for ĉ = 3 is given in the theory of azimuthally magnetized circular ferrite waveguide under slow equation modes excitation.

A class of identities relating Whittaker and Bessel functions

Mass, momentum and energy conservation laws, including the radiation stress, are used to derive an equation of the eigenvalues of rip current spacing. A coastal region with linear bottom slope is divided into two parts: Offshore region and surfzone separated by the breaker line. Wave set-up, wave energy and mean current are assumed to be composed of basic state, which is a function of the distance from the coast to offshore only, and of superposed two-dimensional perturbations. In the case of normal incidence of waves, basic steady current system vanishes and perturbations are found to be of cellular shape. According to the boundary conditions at the coast, stream function of perturbed motion in the surfzone can be represented by the confluent hypergeometric function, while in the offshore zone it is approximated by the modified Bessel function. Interpolation of the stream functions in the surf and offshore regions enables us to obtain a characteristic relation which gives the eigenvalues of nondimensional alongshore spacing of rip current system as a function of a parameter determined by the bottom friction coefficient, width of the surfzone and breaker height.

The theoretical analysis of hybrid surface wave propagation at chiral-plasma planar interface is presented in this manuscript. The dispersion relation is obtained for the proposed structure by applying boundary conditions at the chiral-plasma interface. The characteristic of hybrid surface wave can be tuned by changing the plasma frequency, plasma collisional frequency, and chirality parameter. It is shown that phase constant , attenuation or propagation loss and propagation length are dependent on plasma and chiral parameters. The effect of plasma frequency, plasma collisional frequency, and chirality parameter on the normalized phase constant , attenuation constant, and propagation length are presented computationally. Moreover, to verify the surface wave's condition normalized filed distribution is also presented. The present work may have potential applications in chiral and plasma-sensing based waveguide and enantiomeric sensing devices.

This paper deals with extended M-series, which is extension of the generalized M-series [12]. Mittag-Leffler function, $\omega$ - hypergeometric function, generalized $\omega-$ Gauss hypergeometric function, $\omega-$ confluent hypergeometric function, Bessel-Maitland function can be deduced as special cases of our finding. Moreover, we obtain some theorem for extended M-series by using fractional calculus operators and many results associated with Riemann-Liouville, Weyl and ErdelyiKober operators. We begin our study from the following definitions.