Power Series and Elementary Functions

2015 ◽  
pp. 147-196
Keyword(s):  
Power Series ◽  
Elementary Functions

Approximations to the time evolution of an Izhikevich neuron

2014 ◽  
Vol 25 (10) ◽  
pp. 1450052 ◽  
Author(s):  
August Romeo ◽  
Hans Supèr
Keyword(s):  
Simple Model ◽  
Power Series ◽  
Small Parameter ◽  
Time Evolution ◽  
Scale Effects ◽  
Spiking Neuron ◽  
Elementary Functions ◽  
Constant Input ◽  
Parameter Expansion

Possible ways of obtaining information about the solutions of Izhikevich's "simple model" for a spiking neuron are considered. The method of power series in time is reviewed. From a different viewpoint, in the case of constant input and weak recovery scale effects, advantage is taken of a small-parameter expansion. The obtained approximations can be expressed in terms of elementary functions.

ON THREE INTEGRALS ARISING IN THE THEORY OF ANISOTROPIC CRITICALITY

1996 ◽  
Vol 10 (09) ◽  
pp. 377-383
Author(s):  
YONKO T. MILLEV
Keyword(s):  
Power Series ◽  
Coupling Constant ◽  
Elementary Functions ◽  
Reduced Dimensionality ◽  
Series Representations ◽  
Anisotropic Dispersion ◽  
By Products ◽  
Double Power Series

Three integrals encountered in earlier studies of anisotropic criticality (anisotropic dispersion laws characterised by the coupling constant f) are solved in terms of elementary functions. The nature of the singularity for the case f→1 which corresponds to an effectively reduced dimensionality has been revealed. Double power-series representations for the integrals are also found and some mathematical implications and by-products are discussed. The advance relates to a prospective full-scope analysis of dipolar criticality.

scholarly journals Power Series with Inverse Binomial Coefficients and Harmonic Numbers

2017 ◽  
Vol 70 (1) ◽  
pp. 199-206
Author(s):  
Khristo N. Boyadzhiev
Keyword(s):  
Power Series ◽  
Generating Function ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Elementary Functions

Abstract In this note we compute the generating function for the numbers terms of elementary functions and dilogarithms.

scholarly journals METHOD OF OBTAINING APPROXIMATE FORMULAS

2018 ◽  
Vol 2 ◽  
pp. 72-78
Author(s):  
Konstantin Ludanov
Keyword(s):  
Power Series ◽  
Parameter Method ◽  
Accurate Approximation ◽  
Maclaurin Series ◽  
Elementary Functions ◽  
Approximation Of Functions ◽  
Analytic Approximations ◽  
Approximate Function ◽  
Similar Range ◽  
Two Parameter

The two-parameter method of approximating the sum of a power series in terms of its first three terms of the expansion, which allows one to obtain analytic approximations of various functions, decomposes into a Maclaurin series. As an approximation function of this approximation, it is proposed to use elementary functions constructed in the Nth degree, but with a "compressed" or "stretched" variable x due to the introduction of the numerical factor M (x ≡ ε ∙ m, M ≠ 0) into it. The use of this method makes it possible to significantly increase the range of very accurate approximation of the obtained approximate function with respect to a similar range of the output fragment of a series of three terms. Expressions for both the approximation parameters (M and N) are obtained in a general form and are determined by the coefficients of the second and third terms of the Maclaurin series. Also expressions of both approximation parameters are found for the case if the basis function and the approximant function decompose into the Maclaurin series in even powers of the argument. A number of examples of approximation of functions on the basis of the analysis of power series into which they decompose are given.

Divergence of geometrical optics series at the boundary of its applicability: An analytical example in elementary functions (2D Gaussian beam in free space)

Open Physics ◽  
2006 ◽  
Vol 4 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Yury Kravtsov ◽  
Stefan Buske
Keyword(s):  
Power Series ◽  
Gaussian Beam ◽  
Free Space ◽  
Geometrical Optics ◽  
Elementary Functions ◽  
Beam Diffraction

AbstractAn analytical example in elementary functions is presented (2D Gaussian beam diffraction in free space), which demonstrates the divergence of the geometrical optics (GO) series when the conditions for its applicability are violated. This example shows that accounting for higher terms in GO power series leads to divergence and therefore becomes completely useless beyond the boundaries of GO applicability.

scholarly journals Analysis of Transients in a Three-Level DC–DC Flying Capacitor Converter. Time Domain Approach

2019 ◽  
Vol 4 (1) ◽  
pp. 33-45
Author(s):  
Boris Reznikov ◽  
Alexander Ruderman ◽  
Valentina Galanina
Keyword(s):  
Power Series ◽  
Pulse Width Modulation ◽  
State Space Model ◽  
Voltage Source ◽  
Voltage Reference ◽  
Elementary Functions ◽  
Discrete State ◽  
Source Power ◽  
Arithmetic Average ◽  
The Matrix

AbstractThe paper considers a discrete state-space model for transients in a three-level flying capacitor DC–DC converter. A transition matrix is obtained for a pulse width modulation (PWM) period. The matrix elements are expanded into a power series using a selected small parameter. The matrix eigenvalues that determine the natural balancing dynamics transients are presented in the form of power series as well. Four separate transients are constructed based on four possible PWM period initial states (topologies). Inductor current and capacitor voltage transients are found for the voltage source power-up as the arithmetic average of the four separate transients. The discrete solutions are replaced by continuous ones. The resulting transients that are elementary functions of the circuit parameters, PWM period, and voltage reference demonstrate good agreement with the simulation results.

scholarly journals DEFINITION OF THE ELEMENTARY FUNCTIONS ON THE PRINCIPLE OF THE MATHEMATICAL ANALYSIS

2020 ◽  
Vol 1 (191) ◽  
pp. 12-16
Author(s):  
Yurii Volkov ◽  
◽  
Natalia Vojnaloviсh ◽  
Keyword(s):  
Sixteenth Century ◽  
Power Series ◽  
Mathematical Analysis ◽  
Elementary Functions ◽  
Integral Calculus ◽  
Educational Literature ◽  
Power Operations ◽  
Elementary Methods ◽  
Definition Of ◽  
Basic Level

The elementary functions occupy the special place in school maths and at the study of mathematical analysis in universities .Usually the study of elementary functions begins with the definition of basic elementary functions. Elementary methods are used, without regard to difficulties and imperfections of similar methods. Often rely on intuition, although, it will be desirable to give definition and studies of properties of functions at more logical level. This is old problem, but it is also actual today. The mathematicians were interested in the problem of introduction the definition of the power operations and the logarithms since the sixteenth century, but a main contribution to the decision of this problem belongs to Euler, his book "Introductio in analisin infinitorum - Lausanannae, 1748" became basic level in development of mathematical analysis and by inalienable part in educational literature. Later to the questions of methodology of introduction of definitions basic the elementary functions an attention has been given a number of well-known mathematicians such as F.Kiein, N.Bourbaki, R.Kurant et al. Basic idea: of using the methods of mathematical analysis for the construction of more complete theory. But this suggestion wasn’t very successful in methodology of study of the elementary functions not only at school but also at higher educational establishments. There are many different ways of studying basic elementary functions (logarithmic, exponential, sine, cosine) with the using of differential, integral calculus and the theory of power series are shown in this article.

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