scholarly journals Complex Function Differentiability

2009 ◽  
Vol 17 (2) ◽  
pp. 67-72 ◽  
Author(s):  
Chanapat Pacharapokin ◽  
Hiroshi Yamazaki ◽  
Yasunari Shidama ◽  
Yatsuka Nakamura
Keyword(s):  
Differential Calculus ◽  
Single Point ◽  
Complex Function ◽  
Complex Numbers ◽  
Complex Functions ◽  
Algebraic Properties ◽  
Differential Complex ◽  
Complex Valued

Complex Function Differentiability For a complex valued function defined on its domain in complex numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential complex functions are shown.

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 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.

MODELS OF HOPFIELD-TYPE QUATERNION NEURAL NETWORKS AND THEIR ENERGY FUNCTIONS

2005 ◽  
Vol 15 (01n02) ◽  
pp. 129-135 ◽  
Author(s):  
MITSUO YOSHIDA ◽  
YASUAKI KUROE ◽  
TAKEHIRO MORI
Keyword(s):  
Neural Networks ◽  
Information Processing ◽  
Energy Function ◽  
Point Of View ◽  
Complex Numbers ◽  
Energy Functions ◽  
Fully Connected ◽  
Complex Valued

Recently models of neural networks that can directly deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. Furthermore models of neural networks that can deal with quaternion numbers, which is the extension of complex numbers, have also been proposed. However they are all multilayer quaternion neural networks. This paper proposes models of fully connected recurrent quaternion neural networks, Hopfield-type quaternion neural networks. Since quaternion numbers are non-commutative on multiplication, some different models can be considered. We investigate dynamics of these proposed models from the point of view of the existence of an energy function and derive their conditions for existence.

Gleason Parts of Real Function Algebras

1981 ◽  
Vol 33 (1) ◽  
pp. 181-200 ◽  
Author(s):  
S. H. Kulkarni ◽  
B. V. Limaye
Keyword(s):  
Real Function ◽  
Banach Algebras ◽  
Complex Function ◽  
Klein Surfaces ◽  
The Real ◽  
Uniform Algebras ◽  
Function Algebras ◽  
Complex Valued ◽  
Systematic Exposition

Although the theory of complex Banach algebras is by now classical, the first systematic exposition of the theory of real Banach algebras was given by Ingelstam [5] as late as 1965. More recently, further attention to real Banach algebras was paid in 1970 [1], where, among other things, the (real) standard algebras on finite open Klein surfaces were introduced. Generalizing these considerations, real uniform algebras were studied in [7] and [6].In the present paper, an attempt is made to develop the theory of real function algebras (see Section 1 for the definition) along the lines of the complex function algebras. Although the real function algebras are not structurally different from the real uniform algebras introduced in [7], they are easier to deal with since their elements are actually (complex-valued) functions.

scholarly journals Handling of Complex Numbers in the CHProgramming Language

1993 ◽  
Vol 2 (3) ◽  
pp. 77-106 ◽  
Author(s):  
Harry H. Cheng
Keyword(s):  
Variable Number ◽  
Language Design ◽  
Complex Numbers ◽  
Computer Language ◽  
Real Numbers ◽  
Mathematical Functions ◽  
Complex Arithmetic ◽  
Complex Functions ◽  
Complex Features ◽  
Branch Cuts

The handling of complex numbers in the CHprogramming language will be described in this paper. Complex is a built-in data type in CH. The I/O, arithmetic and relational operations, and built-in mathematical functions are defined for both regular complex numbers and complex metanumbers of ComplexZero, Complexlnf, and ComplexNaN. Due to polymorphism, the syntax of complex arithmetic and relational operations and built-in mathematical functions are the same as those for real numbers. Besides polymorphism, the built-in mathematical functions are implemented with a variable number of arguments that greatly simplify computations of different branches of multiple-valued complex functions. The valid lvalues related to complex numbers are defined. Rationales for the design of complex features in CHare discussed from language design, implementation, and application points of views. Sample CHprograms show that a computer language that does not distinguish the sign of zeros in complex numbers can also handle the branch cuts of multiple-valued complex functions effectively so long as it is appropriately designed and implemented.

Kernels with only a finite number of characteristic values

1971 ◽  
Vol 70 (2) ◽  
pp. 257-262
Author(s):  
Dale W. Swann
Keyword(s):  
Finite Number ◽  
Higher Order ◽  
Complex Numbers ◽  
Finite Characteristic ◽  
Characteristic Values ◽  
Image Position ◽  
Complex Valued

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we meanand let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfyingFor such kernels, the iterated kernels,are well-defined (1), as are the higher order tracesCarleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinantthe latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

scholarly journals HOW TO REPRESENT THE GENETIC CODE?

2004 ◽  
Vol 2 (2) ◽  
pp. 13
Author(s):  
N.S. Santos-Magalhães ◽  
E.A. Bouton ◽  
H.M. De Oliveira
Keyword(s):  
Genetic Code ◽  
Dna Sequences ◽  
Single Point ◽  
Molecular Genetic ◽  
Dna Mapping ◽  
Greenwich Meridian ◽  
Modern Genomic ◽  
Complex Valued ◽  
Human Kind

The advent of molecular genetic comprises a true revolution of far-reaching consequences for human-kind, which evolved into a specialized branch of the modern-day Biochemistry. The analysis of specicgenomic information are gaining wide-ranging interest because of their signicance to the early diag-nosis of disease, and the discovery of modern drugs. In order to take advantage of a wide assortmentof signal processing (SP) algorithms, the primary step of modern genomic SP involves convertingsymbolic-DNA sequences into complex-valued signals. How to represent the genetic code? Despitebeing extensively known, the DNA mapping into proteins is one of the relevant discoveries of genetics.The genetic code (GC) is revisited in this work, addressing other descriptions for it, which can beworthy for genomic SP. Three original representations are discussed. The inner-to-outer map buildson the unbalanced role of nucleotides of a codon. A two-dimensional-Gray genetic representationis oered as a structured map that can help interpreting DNA spectrograms or scalograms. Theseare among the powerful visual tools for genome analysis, which depends on the choice of the geneticmapping. Finally, the world-chart for the GC is investigated. Evoking the cyclic structure of thegenetic mapping, it can be folded joining the left-right borders, and the top-bottom frontiers. As aresult, the GC can be drawn on the surface of a sphere resembling a world-map. Eight parallels oflatitude are required (four in each hemisphere) as well as four meridians of longitude associated tofour corresponding anti-meridians. The tropic circles have 11.25o, 33.75o, 56.25o, and 78.5o (Northand South). Starting from an arbitrary Greenwich meridian, the meridians of longitude can be plottedat 22.5o, 67.5o, 112.5o, and 157.5o (East and West). Each triplet is assigned to a single point on thesurface that we named Nirenberg-Kohamas Earth. Despite being valuable, usual representations forthe GC can be replaced by the handy descriptions oered in this work. These alternative maps arealso particularly useful for educational purposes, giving a much rich interpretation and visualizationthan a simple look-up table.

scholarly journals Visualising Complex Polynomials: A Parabola Is but a Drop in the Ocean of Quadratics

2018 ◽  
Vol 10 (6) ◽  
pp. 91
Author(s):  
Harry Wiggins ◽  
Ansie Harding ◽  
Johann Engelbrecht
Keyword(s):  
Complex Function ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Complex Polynomials ◽  
Singular Case ◽  
Hyperbolic Paraboloid ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Coefficients ◽  
Roots Of A Polynomial

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called "imaginary" roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic.  We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

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