Related Experiment Video
Updated: Aug 22, 2025
10:46
A Method of Trigonometric Modelling of Seasonal Variation Demonstrated with Multiple Sclerosis Relapse Data
Published on: December 9, 2015
10.7K
Division of Power Series: Recursive and Non-Recursive Formulas
1Universidade Federal Fluminense, Departamento de Ciências Exatas, Av. dos Trabalhadores 420, Vila Sta. Cecília , 27255-125 Volta Redonda, RJ, Brazil.
Anais Da Academia Brasileira De Ciencias
|November 9, 2022
Summary
This paper introduces a novel formula for dividing power series, offering both recursive and non-recursive versions. The non-recursive approach optimizes computation for high-order series truncation, enhancing efficiency in mathematical analysis.
Area of Science:
- Mathematics
- Algebraic Geometry
Background:
- Power series manipulation is fundamental in various mathematical fields.
- Efficient computation of high-order series truncations presents a significant challenge.
Purpose of the Study:
- To introduce a new formula for the division of power series.
- To develop both recursive and non-recursive versions of the formula.
- To optimize computational efficiency for high-order series truncation.
Main Methods:
- Development of a recursive formula for power series division.
- Development of a non-recursive formula for power series division.
- Definition of fundamental sets of summation indices and calculation of a coefficient 𝛾 to account for index repetitions.
Main Results:
- A novel recursive formula for power series division has been established.
- A novel non-recursive formula for power series division has been established.
- A method for calculating the coefficient 𝛾 has been provided, enabling the practical application of the non-recursive formula.
Conclusions:
- The proposed formulas offer new tools for power series manipulation.
- The non-recursive formula significantly reduces computational cost for high-order series truncation.
- The methodology for calculating 𝛾 ensures accurate application of the non-recursive formula.
Related Concept Videos
01:24Exponential Fourier series
280
In audio signal processing, the exponential Fourier series plays a crucial role in sound synthesis, allowing complex sounds to be broken down into simpler sinusoidal components. This decomposition process is fundamental in analyzing and reconstructing musical notes and other audio signals. The exponential Fourier series expresses periodic signals as the sum of complex exponentials at both positive and negative harmonic frequencies, providing a powerful tool for signal analysis.
Euler's identity...
Euler's identity...
280
01:17Trigonometric Fourier series
347
Fourier series is a foundational mathematical technique that decomposes periodic functions into an infinite series of sinusoidal harmonics. This method enables the representation of complex periodic signals as sums of simple sine and cosine functions, facilitating their analysis and interpretation in various fields, including signal processing, acoustics, and electrical engineering.
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
The trigonometric Fourier series specifically expresses a periodic function with a defined period T using sine...
347
01:17Series and Parallel Inductors
611
In electrical circuits, integrating inductors into the toolkit of passive elements requires navigating the intricacies of series and parallel combinations involving these components. Practical circuits often feature configurations of multiple inductors, and understanding how to determine their equivalent inductance is vital.
For a series connection of N inductors, each carrying the same current, applying Kirchhoff's voltage law unveils a crucial relationship. Substituting the expression for...
For a series connection of N inductors, each carrying the same current, applying Kirchhoff's voltage law unveils a crucial relationship. Substituting the expression for...
611
01:18Parseval's Theorem
608
Parseval's theorem is a fundamental concept in signal processing and harmonic analysis. It asserts that for a periodic function, the average power of the signal over one period equals the sum of the squared magnitudes of all its complex Fourier coefficients. This theorem, named after Marc-Antoine Parseval, provides a powerful tool for analyzing the energy distribution in signals.
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which...
Interestingly, Parseval's theorem also holds for the trigonometric form of the Fourier series, which...
608
01:21Convergence of Fourier Series
193
The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
The Gibbs phenomenon refers to the persistent oscillations and overshoots that occur near discontinuities...
193
01:26The Power Flow Problem and Solution
315
Power flow problem analysis is fundamental for determining real and reactive power flows in network components, such as transmission lines, transformers, and loads. The power system's single-line diagram provides data on the bus, transmission line, and transformer. Each bus k in the system is characterized by four key variables: voltage magnitude Vk, phase angle δk, real power Pk, and reactive power Qk. Two of these four variables are inputs, while the...
315