The Fundamental Theorem in Complex Function Theory

This study is to determine the location of the difficulty of complex variable courses. Complex variable courses require a strong understanding, high concentration, good learning readiness, reasoning power and higher order logic in understanding direct objects. The direct objects referred to are Facts, Concepts, Skills, and Principles. In addition to direct objects in obtaining maximum results, indirect objects are needed. Indirect objects that are meant are tools used in the form of media, support and encouragement from others in obtaining good results. The results of the mid-semester grades for the 2014/2015 academic year were about 13% of students who got good grades and the rest were not good. In fact, this complex variable course is the main course of several courses. Another fact is that in the 2019/2020 semester VII, there are 35% in the poor category and 50% sufficient and the rest are good and very good. From 35% of students with poor learning outcomes took remedial, 1 person graduated with sufficient marks, while the rest did not pass. The specific objectives of the research are 1) to determine the mastery in complex variable courses, 2) the location of the difficulty and difficulty factors of complex variables, 3) to know the mastery and difficulty of indirect objects. This type of research approach is descriptive qualitative. Research wants to describe and describe how the actual event. The research was conducted in the Complex Variable Analysis course, Semester VII, implementation October-January and a total of 12 subjects. Techniques for obtaining data from observations, interviews and documentation. The analysis technique, presenting data, reducing and drawing conclusions. Conclusion,The difficulty of students lies in the facts and concepts that exist in the direct object. In the interview session, 71.42% said that the difficulty factor was the mastery of facts and concepts. In online implementation, students experience percentage difficulties when writing symbols and operating the media used. This is a new problem for mathematics students in achieving good learning outcomes with objective assessment standards.

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

A number of results concerning infinite series are presented in this chapter. The material spans convergence tests for series of constants, uniform convergence for series of functions, and power series. It is not the intention here to give a comprehensive account of this theory, but rather to review material that is relevant to later chapters. This chapter is intended to make the rest of the book more self contained and remind the reader of some basic results. It is assumed that the reader has a background in real analysis of a single variable and elementary complex analysis; consequently, no attempt is made to prove these results or provide examples. Proofs and examples can be found in textbooks on real analysis such as Apostol (Mathematical Analysis, Addison-Wesley, 2nd ed., 1974), Fulks (Advanced Calculus, 3rd ed., Wiley, 1978), and Little et al. (Real Analysis via Sequences and Series, Springer-Verlag, 2015) and complex analysis (Ahlfors, Complex Analysis, McGraw-Hill, 2nd ed., 1966; Brown and Churchill, Complex Variables and Applications, McGraw-Hill, 7th ed., 2004; Priestley, Introduction to Complex Analysis, Rev. ed., Oxford University Press, 1990). The exception is in the final section on double sequences and series. This material is often not part of courses on real or complex analysis. Here we provide proofs, examples and exercises.

The present paper deals with the definition and study of new kinds of hypergeometric matrix functions within complex analysis. We also get the radius of regularity and matrix recurrence relations are then developed for l(m,n)-hypergeometric matrix function of two complex variables. We give a different approach to prove the effect of the differential operator on this function. Finally, another hypergeometric matrix function, namely, p l(m,n)-hypergeometric matrix function of two complex variables are defined, its components when the positive integer p is greater than one, provide a matrix partial differential equation satisfied by these function and some of their properties are investigated.

The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results, such as the Bochner-Martinelli formula, the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations, and the Hartogs theorem, are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular, the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable, several complex variables, and several quaternionic variables with suitable modifications.

This paper discusses developments in complex analysis and CR geometry in the last forty years related to the Cauchy–Riemann equations, proper holomorphic mappings between balls, and positivity conditions in complex analysis. The paper includes anecdotes about some of the contributors to these developments.

The local residue has been presented since the early days of several complex variables and plays a fundamental role in complex analysis and geometry. Nevertheless, in many cases, there are very few practical tools in the computational point of view.

From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author's aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.

Holomorphic Functions on Nuclear Sequence Spaces

Complex-valued neural networks for nonlinear complex principal component analysis

Motivation. Analytic function theory on commutative complex extensions calls for an operator–theoretic calculus that simultaneously sees the algebra-induced coupling among components and supports boundary-to-interior mechanisms. Gap. While Dirac-type frameworks are classical in several complex variables and Clifford analysis, a coherent calculus aligning structural CR systems, a canonical first derivative, and a Cauchy-type boundary identity on the commutative model C(C)≅C4 has not been systematically developed. Purpose and Aims. This paper develops such a calculus for O-regular mappings on C(C) and establishes three pillars of the theory. Main Results. (i) A fully coupled Cauchy–Riemann system characterizing O-regularity; (ii) identification of a canonical first derivative g′(z)=∂x0g(z); and (iii) a Stokes-driven boundary annihilation law ∫∂Ωτg=0 for a canonical 7-form τ. On (pseudo)convex domains, ∂¯-methods yield solvability under natural compatibility and regularity assumptions. Stability (under algebra-preserving maps), Liouville-type, and removability results are also obtained, and function spaces suited to this algebra are outlined. Significance. The results show that a large portion of the classical holomorphic toolkit survives, in algebra-aware form, on C(C).

In this paper we deal with monogenic and k-monogenic Clifford-valued automorphic forms in two and several vector variables. We focus on the construction of several families of Eisenstein and Poincare series within the setting of this function theory. The functions that we construct provide non-trivial examples for generalizations of the classical Hilbert modular forms within the framework of real and complex Clifford analysis.

This Special Issue aims to present some of the newest results obtained from the study of complex-valued functions of one or several complex variables [...]

The purpose of this workshop was to discuss recent results in Several Complex Variables, Complex Geometry and Complex Dynamical Systems with a special focus on the exchange of ideas and methods among these areas. The main topics of the workshop included Holomorphic Dynamics and Nevanlinna’s Theory; L^2 -methods and Cohomologies; Plurisubharmonic Functions and Pluripotential Theory; Geometric Questions of Complex Analysis.

In this paper we intend to show that certain features encountered in the theory of holomorphic functions of several complex variables, also occur in the theory of biregular functions, which are regular functions of two variables of arbitrary dimension with values in a Clifford algebra. For these functions there exist integral representations with regular and non-regular kernels, just as in the complex case. We will give a representation formula of the second kind using differential forms. Another topic is the study of the domains of bi- regularity which are the analogues of the domains of holomorphy of complex analysis. We will make it clear that multi-valued functions cannot be avoided in this study. Some examples of domains of biregula- rity will be given, by constructing biregular functions in the domains which cannot be extended to any larger domain.