scholarly journals Explicit Construction of the Inverse of an Analytic Real Function: Some Applications

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2154
Author(s):  
Joaquín Moreno ◽  
Miguel A. López ◽  
Raquel Martínez
Keyword(s):  
Power Series ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
Real Function ◽  
General Procedure ◽  
Explicit Construction ◽  
Catalan Numbers ◽  
Analytical Function ◽  
Series Development

In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given y=f(x), we provide the power series that defines its inverse x=hf(y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss–Catalan numbers.

Some the Application of the Taylor Series for the Analysis of Processes in Non-Linear Resistive Circuits

2014 ◽  
Vol 701-702 ◽  
pp. 1173-1176
Author(s):  
Vitaly Viktorovich Pivnev ◽  
Sergey Nikolaevich Basan
Keyword(s):  
Power Series ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
System Of Nonlinear Equations ◽  
Algebraic Equations ◽  
Electrical Circuits ◽  
It Implementation ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
The Way

The way of calculating the currents and voltages in nonlinear resistive electrical circuits , based on the use of power series (Taylor, Maclaurin) is considered . The advantage of this method lies in the fact that while it implementation it is not necessary to a system of nonlinear equations. To determine the numerical values ​​of the coefficients of the power series corresponding system of linear algebraic equations are solved. Nonlinear operations are limited to the calculation of the numerical values ​​of currents, voltages and their derivatives with respect to the pole equations of nonlinear elements.

Attitude reconstruction from strap-down rate gyros using power series

2021 ◽  
pp. 1-19
Author(s):  
Habib Ghanbarpourasl
Keyword(s):  
Power Series ◽  
Taylor Series ◽  
Direction Cosine ◽  
Analytical Description ◽  
Double Precision ◽  
Angular Velocity Vector ◽  
Higher Order Terms ◽  
Direction Cosine Matrix ◽  
The Stability ◽  
Better Than

Abstract This paper introduces a power series based method for attitude reconstruction from triad orthogonal strap-down gyros. The method is implemented and validated using quaternions and direction cosine matrix in single and double precision implementation forms. It is supposed that data from gyros are sampled with high frequency and a fitted polynomial is used for an analytical description of the angular velocity vector. The method is compared with the well-known Taylor series approach, and the stability of the coefficients’ norm in higher-order terms for both methods is analysed. It is shown that the norm of quaternions’ derivatives in the Taylor series is bigger than the equivalent terms coefficients in the power series. In the proposed method, more terms can be used in the power series before the saturation of the coefficients and the error of the proposed method is less than that for other methods. The numerical results show that the application of the proposed method with quaternions performs better than other methods. The method is robust with respect to the noise of the sensors and has a low computational load compared with other methods.

Every Power Series is a Taylor Series

1981 ◽  
Vol 88 (1) ◽  
pp. 51 ◽  
Author(s):  
Mark D. Meyerson
Keyword(s):  
Power Series ◽  
Taylor Series

Eigenvalues of the Barenblatt-Pattle similarity solution in nonlinear diffusion

1982 ◽  
Vol 383 (1784) ◽  
pp. 89-100 ◽  
Keyword(s):  
Power Series ◽  
Initial Data ◽  
Similarity Solution ◽  
Nonlinear Diffusion ◽  
Large Time ◽  
Infinite Sequence ◽  
Nonlinear Diffusion Equation ◽  
Time Behaviour ◽  
Series Development ◽  
Interactive Product

We consider the large-time behaviour of the nonlinear diffusion equation ∂ u /∂ t = r 1- μ ∂/∂ r ( r μ -1 u β ∂ u /∂ r ), u ≽ 0, β ≻ 0 for certain types of compact initial data. We show that the solution approaches the Barenblatt-Pattle similarity solution through an infinite sequence of negative real powers of t , which can be found in explicit form. These, together with their interactive product terms, determine the power-series development of u(r,t) as t → ∞.

scholarly journals An elementary proof for \sin x = x − \frac{x}{6} + o(x^3)

2016 ◽  
Vol 25 (2) ◽  
pp. 175-176
Author(s):  
RADU GOLOGAN ◽  
◽  
Keyword(s):  
Power Series ◽  
Elementary Proof ◽  
Series Development

Using only elementary trigonometrical calculations we prove the power series development for the sin and cos functions up to the terms of power three and four respectively.

scholarly journals Squaring Technique using Vedic Mathematics

2021 ◽  
Author(s):  
Jasmine Bajaj ◽  
Babita Jajodia
Keyword(s):  
Power Series ◽  
Taylor Series ◽  
Number System ◽  
Approximation Technique ◽  
Approximation Techniques ◽  
Vedic Mathematics ◽  
Decimal Number ◽  
Taylor Series Approximation ◽  
Series Approximation ◽  
Interesting Approach

Vedic Mathematics provides an interesting approach to modern computing applications by offering an edge of time and space complexities over conventional techniques. Vedic Mathematics consists of sixteen sutras and thirteen sub-sutras, to calculate problems revolving around arithmetic, algebra, geometry, calculus and conics. These sutras are specific to the decimal number system, but this can be easily applied to binary computations. This paper presented an optimised squaring technique using Karatsuba-Ofman Algorithm, and without the use of Duplex property for reduced algorithmic complexity. This work also attempts Taylor Series approximation of basic trigonometric and inverse trigonometric series. The advantage of this proposed power series approximation technique is that it provides a lower absolute mean error difference in comparison to previously existing approximation techniques.

Taylor Series and Power Series

2009 ◽  
pp. 227-246
Author(s):  
Klaus Weltner ◽  
Peter Schuster ◽  
Wolfgang J. Weber ◽  
Jean Grosjean
Keyword(s):  
Power Series ◽  
Taylor Series

scholarly journals Solving Systems of Volterra Integral and Integrodifferential Equations with Proportional Delays by Differential Transformation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Şuayip Yüzbaşı ◽  
Nurbol Ismailov
Keyword(s):  
Exact Solution ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
Transformation Method ◽  
Integrodifferential Equations ◽  
Differential Transformation Method ◽  
Differential Transformation ◽  
Proportional Delays ◽  
In Series ◽  
Linear And Nonlinear

In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. The method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems.

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