scholarly journals Stable and Holomorphic Implementation of Complex Functions Using a Unit Circle-Based Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Binesh Thankappan
Keyword(s):  
Unit Circle ◽  
Infinite Series ◽  
Complex Number ◽  
Complex Function ◽  
Unstable System ◽  
Base Complex ◽  
Base Unit ◽  
Complex Functions ◽  
Constituent Elements ◽  
The Given

A stable and holomorphic implementation of complex functions in ℂ plane making use of a unit circle-based transform is presented in this paper. In this method, any complex number or function can be represented as an infinite series sum of progressive products of a base complex unit and its conjugate only, where both are defined inside the unit circle. With each term in the infinite progression lying inside the unit circle, the sum ultimately converges to the complex function under consideration. Since infinitely large number of terms are present in the progression, the first element of which may be deemed as the base unit of the given complex number, it is addressed as complex baselet so that the complex number or function is termed as the complex baselet transform. Using this approach, various fundamental operations applied on the original complex number in ℂ are mapped to equivalent operations on the complex baselet inside the unit circle, and results are presented. This implementation has unique properties due to the fact that the constituent elements are all lying inside the unit circle. Out of numerous applications, two cases are presented: one of a stable implementation of an otherwise unstable system and the second case of functions not satisfying Cauchy–Riemann equations thereby not holomorphic in ℂ plane, which are made complex differentiable using the proposed transform-based implementation. Various lemmas and theorems related to this approach are also included with proofs.

scholarly journals Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

Energies ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1488
Author(s):  
Damian Trofimowicz ◽  
Tomasz P. Stefański
Keyword(s):  
Unit Circle ◽  
Phase Analysis ◽  
Characteristic Equation ◽  
Digital Filter ◽  
Digital Filters ◽  
Computing Time ◽  
Complex Function ◽  
Stochastic Methods ◽  
Final Population ◽  
Candidate Regions

In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots.

scholarly journals Winding Numbers, Unwinding Numbers, and the Lambert W Function

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Lambert W Function ◽  
Complex Numbers ◽  
Winding Number ◽  
Line Integral ◽  
Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

2012 ◽  
Vol 446-449 ◽  
pp. 2080-2084 ◽  
Author(s):  
Xue Xia Zhang ◽  
Chan Li ◽  
Xiao Chao Cui ◽  
Wen Bin Zhao
Keyword(s):  
Crack Length ◽  
Stress Function ◽  
Function Theory ◽  
Complex Function ◽  
Mode Iii ◽  
Mechanical Problem ◽  
Complex Function Theory ◽  
Plane Shear ◽  
Periodic Cracks ◽  
The Given

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

scholarly journals Errors In Using Convergence Tests In Infinite Series

2020 ◽  
Vol 13 (2) ◽  
pp. 113-127
Author(s):  
Murat GENÇ ◽  
Mustafa AKINCI
Keyword(s):  
Undergraduate Students ◽  
Infinite Series ◽  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Test Results ◽  
Convergence Criteria ◽  
Convergence Test ◽  
Comparison Test ◽  
The Given

Abstract: The present study aimed to identify the errors made by pre-service elementary mathematics teachers while investigating the convergence of infinite series. A qualitative exploratory case study design was used with a total of 43 undergraduate students. Data were obtained from a test administered in a paper-and-pencil form consisting of seven open-ended questions. The data analysis was done using descriptive and content analysis techniques. Findings were presented as follows: inappropriate test selections; failure to check convergence criteria; incorrect use of a comparison test; limit comparison test error; re-test convergence test results; considering ∑ as a multiplicative function; misunderstanding of special series; considering that series has no character when the convergence test is inconclusive; confusing sequences with series; misunderstanding of the nth-term test; misinterpretation of convergence test results. Findings showed that students with insufficient procedural knowledge had difficulty in solving the given problem even if they understood it, whereas those with insufficient conceptual knowledge could not literally understand what they did even if they solved the problem. Therefore, the establishment of a moderate balance between procedural and conceptual knowledge in the learning of the convergence of series is essential in reducing the errors or learning difficulties for developing deep mathematical understanding

scholarly journals Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 206
Author(s):  
Ji-Eun Kim
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Complex Number ◽  
Complex Function ◽  
Taylor Series Expansion ◽  
Complex Numbers ◽  
Definition Of ◽  
Generalized Quaternion ◽  
Complex Valued ◽  
Complex Basis

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

scholarly journals Representación gráfica de las funciones complejas con el Mathematica

2018 ◽  
pp. 52-56
Keyword(s):  
Complex Variable ◽  
Dimensional Space ◽  
Complex Function ◽  
Three Dimensional ◽  
Graphical Display ◽  
Model Based ◽  
Complex Functions ◽  
Complex Valued ◽  
Three Dimensional Space

Representación gráfica de las funciones complejas con el Mathematica Graphical display of complex functions with Mathematica Ricardo Velezmoro y Robert Ipanaqué Universidad Nacional de Piura, Urb. Miraflores s/n, Castilla, Piura, Perú. DOI: https://doi.org/10.33017/RevECIPeru2015.0008/ Resumen La representación gráfica de las funciones de valor complejo, de una variable compleja, es un tema de mucho interés dado que la gráfica de una función de este tipo tendría que ser dibujada en un espacio tetra dimensional. Este artículo presenta una propuesta para representar tales gráficas mediante el uso de un modelo basado en una submersión, del espacio tetra dimensional en el espacio tridimensional; para luego, con ayuda del Mathematica llegar a obtener una representación de las mencionadas gráficas en una pantalla 2D. Adicionalmente, se implementarán algunos comandos en el Mathematica, los mismos que permitirán realizar las representaciones de variados e interesantes ejemplos. Descriptores: función compleja, visualización, submersión Abstract The graphical display of complex-valued functions of a complex variable is a subject of much interest since the graph of such a function would have to be drawn in a four-dimensional space. This article presents a proposal to display such graphs using a model based on a submersion, from four-dimensional space to three-dimensional space; then, with the help of Mathematica arrive at a representation of the graphs mentioned in a 2D screen. Additionally, some commands are implemented in Mathematica, the same that will make representations varied and interesting examples. Keywords: complex function, visualization, submersion

scholarly journals Attacks on the Faithfulness of the Burau Representation of the Braid Group $B_4$

2016 ◽  
Vol 8 (1) ◽  
pp. 5
Author(s):  
Mohammad Y. Chreif ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Unit Circle ◽  
Complex Number ◽  
Braid Group ◽  
Simple Algorithm ◽  
Necessary Condition ◽  
Burau Representation ◽  
Open Question

<p align="left">The faithfulness of the Burau representation of the 4-strand braid group, $B_4$, remains an open question.<br />In this work, there are two main results. First, we specialize the indeterminate $t$ to a complex number on the unit circle, and we find a necessary condition for a word of $B_4$ to belong to the kernel of the representation. Second, by using a simple algorithm,<br />we will be able to exclude a family of words in the generators from belonging to the kernel of the reduced Burau representation.</p>

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