Power series with almost periodic coefficients

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

Explicit analytical expressions to represent the radiative properties of spherical particles in the small size-parameter range were obtained. These expressions were deduced following Penndorf’s approach of expanding the Mie coefficients in power series on the size parameter. However, in opposition to Penndorf’s original work—in which some errors were found and corrected—the Mie coefficients were expanded to the eighth power of the size parameter, which results in a five-term approximation to the extinction and scattering efficiencies. Also, expressions for the evaluation of the asymmetry factor and both polarized and unpolarized phase functions were deduced and presented. The results so obtained have proved to be very accurate, even for size parameters beyond the limit of validity of the approach utilized.

The tail decay of the hitting probabilities on level zero in M/G/1-type Markov renewal processes (MRPs) is studied. The Markov renewal process is transformed into a Markov chain so that the problem of tail decay is reformulated in terms of the decay of the coefficients of a suitable power series. The latter problem is reduced to analyze the analyticity domain of the power series. Two numerical methods are introduced to compute the convergence radius of the latter power series. Some numerical experiments are presented that show the effectiveness of the proposed algorithms.

4D arrays provide cost-effective beam steering capabilities considering radio-frequency switches (controlled by periodic sequences) instead of variable phase shifters. Their synthesis is based on the Fourier coefficients (which depend on configurable time parameters) of the corresponding periodic modulating sequences. Whereas Fourier series use trigonometric functions to synthesize any periodic continuous waveform, Walsh series relies on bipolar orthogonal sequences and, if such a series is truncated, the expanded function is approximated by a stairstep signal. In this paper we present a novel method to synthesize 4D arrays by means of a finite set of Walsh functions. The technique allows for implementing the analog time-modulated feeding network of an array employing single-pole double-throw switches and achieves excellent rejection levels of the undesired harmonics.

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension 3 using the framework of hypercomplex function theory. This first article on hand deals with generalized Fourier and Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in ℝ3 and their series coefficients can be explicitly (one-to-one) linked with each other. Finally, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented.

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.

Let h* be a complex (or real) oriented cohomology theory. It has recently been observed that the of h* plays a very important role in such a theory. Above all, D. Quillen [6] showed that the formal group FU of the complex cobordism theory MU* ( ) is isomorphic to the Lazard universal formal group, and its coefficients generate the ground ring MU* (pt). Unfortunately, in the case of quaternionic oriented cohomology theory (see § 2), there exists no such formal group, since the tensor product of quaternionic line bundles does not yield a quaternionic bundle. This situation makes it difficult, for example, to produce enough generators of MSp*(pt). However there are some substitutes for the formal group. In particular, using the total Pontrjagin class of a certain quaternionic vector bundle 2C (see § 3), N. Ja. Gozman [5] defined a subring A of MU* (pt) which is contained in the image of the forgetful homomorphism (p: MSp* (pt}-*MU*(pt). In this paper, we will generalize his approach to arbitrary quaternionic oriented cohomology theory 7i*, and define a subring Ah of h* (pt) which is generated by the coefficients of certain power series. Then it will be shown that

On applications of hypergeometric functions

(1958). On the Coefficients of Reciprocal Power Series. The American Mathematical Monthly: Vol. 65, No. 2, pp. 90-94.

The volume velocity function of the glottis can be computed from a pitch-synchronous spectral analysis of the speech pressure wave. The method consists of computing the Fourier coefficients of a pitch period, locating the formant frequencies, removing the formant poles from the spectrum, and regenerating the glottal waveform from the residual by Fourier synthesis. To obtain accurate results it is necessary to record the original speech without phase distortion while maintaining at least 55–60 db signal-to-noise ratio. The data reported here were derived from recordings originating with a WE-640-AA condenser microphone in a free-space room feeding a digital tape recorder. Two male speakers uttered six vowel sounds between h and d at different pitches and intensities. The glottal waveforms were examined to determine both shape and phase with respect to the speech wave as a function of the sound, the pitch and the intensity. Sequences of periods were studied for variations from period to period.

Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

Hill’s equation is a real linear second-order ordinary differential equation with a periodic coefficient f ( t ) f(t) : (0.1) y ′ ′ ( t ) + [ λ + ε f ( t ) ] y ( t ) = 0. \begin{equation} y^{\prime \prime } (t) +\left [ \lambda +\varepsilon f\left (t\right ) \right ] y(t) =0. \tag *{(0.1)} \end{equation} It has unbounded solutions for certain intervals of the real parameter λ \lambda , called instability intervals. Here these intervals, and the growth rate of the unbounded solutions, are determined for ε \varepsilon small, and also for λ \lambda large. This is done by constructing a fundamental pair of solutions which are power series in ε / λ 1 / 2 \varepsilon /\lambda ^{1/2} , with coefficients that are bounded functions of λ \lambda .

Chapter 2 - Series

A novel representation of a quasi-periodic modified von Mangoldt function L(n) on prime numbers and its decomposition into Fourier series has been investigated. We focus on some particular quantities characterizing the modified von Mangoldt function. The results indicate that prime number progression can be decomposed into periodic sequences. The main approach is to decompose it into sin or cosine function. Basically, it is applied to extract hidden periodicities in seemingly quasi periodic prime function. Numerical evidences were provided to confirm the periodic distribution of primes.

Application of theorem of minimum potential energy to a complex structure: Part II: three-dimensional analysis