Functions Of Complex Variables Research Articles

Dynamic inverse plane characteristics of circular holes in a strip-shaped medium under line source excitation

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of circular holes in the strip domain under line source excitation were studied. The total wave field in the strip domain is constructed using the concept of infinite mirroring, which eliminates the mutual impact of the wave field under the action of the upper and lower limits of the strip domain. To determine the overall wave field, solve the wave field's unknown coefficients in accordance with the boundary conditions. The right number of mirror images is found using the dynamic stress distribution of a circular hole in the band-shaped domain as a specific calculation example. The effect of altering the incident frequency and band-shaped domain width on the dynamic stress concentration coefficient surrounding the circular hole is also explored. As a consequence of the findings, the dynamic stress concentration coefficient around the circular hole in the zone domain at low frequencies is substantially bigger than it is at medium and high frequencies. Under identical conditions, the dynamic stress concentration coefficient will exhibit periodic extreme values and converge as the distance increases between the lower boundary and the circular hole’s center.

An Exploration of Cultivating Students' Creative Thinking in Teaching Functions of Complex Variables

As an important branch of higher mathematics, complex function occupies a core position in many fields such as mathematics, physics, and engineering, which not only involves rich theoretical knowledge but also shows great value in practical applications. Therefore, the teaching of complex functions is extremely important for students of mathematics and related majors. However, traditional teaching methods often emphasize theory over practice, lack effective ways to stimulate students' innovative thinking and limit the overall development of students' abilities. Based on this, this study will explore how to cultivate students' innovative thinking more effectively through the teaching of functions of a complex variable, aiming at finding and evaluating different teaching strategies and methods to better stimulate students' interest, improve their innovative and practical abilities, and promote the overall development of students' comprehensive abilities.

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

Еdgе fiеld of dеflеctorplаtes with expanding screens

Deflector plates consist of two parallel conductive plates that create a deflecting electric field. They can be used to control the flow of charged particles ― electrons or ions. The effect of the еdgе fiеld of deflector plates leads to a change in the velocity of charged particles in the longitudinal and transverse directions, consequently of which their real trajectories change, deviating from ideal ones, which violates the space-time resolution of corpuscular optical devices in which they are used. Apart from that, the electric field at the input to the plates of deflector can vary over time, which must also be taken into account when the deflector diverts the beam of charged particles. Thus, in many cases, the use of deflector plates with open ends is inappropriate, since uncontrolled scattering fields are formed. In this article, we can consider the field of deflector plates with expanded screens at the output, so that deflecting beams of charged particles can be used at the output of deflector plates. Using the methods of the theory of complex variable functions, analytical expressions for the edge field of deflector plates with grounded screens were obtained. Firstly, by grounding the screens and shielding the plates from the deflection field, we can localize the edge electric field and reduce the uncontrolled scattering fields, and secondly, such a field can be accurately calculated analytically.

Dynamic anti-planar characteristics of circular holes in laminated structures under the action of line source

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of the circular hole in the laminated structure under line source excitation were studied. To deal with the interface problem within the laminated structure, the idea of “fit” is adopted to construct continuity conditions that satisfy displacement and stress. The circular hole boundary stress-free condition is used, the wave field is constructed using the mirror method, and the unknown coefficients of the scattered wave field are solved through Fourier series expansion, and finally the total wave field in the laminated structure is obtained. Taking the dynamic stress distribution of a circular hole in a laminated structure as a specific calculation example, the effects of changes in line source position, layer thickness, and medium hard-soft ratio on the dynamic stress concentration coefficient around the circular hole are discussed. The results show that when the line source load is applied directly above the circular hole and the lower layer is narrow, the dynamic stress concentration coefficient will have a sudden change compared with other situations. When other conditions are the same, under low, medium, and high-frequency wave fields, the dynamic stress concentration caused by different soft-to-hard ratios shows completely different rules. These rules can be used to better detect defects in laminated structures.

Theoretical modeling and dynamic analysis of designable grillages for repairing material surface defect

Designing structural grillages is an innovative solution to repair material surface defects. The grillages have advantages as lightweight and directionally enhanced, but their stability and stress concentration under dynamic loads may pose significant risks. Three analytical models of grillages above a semicircular defect are constructed. The analytical solution of the frequency-domain wave field around the semicircular defect under SH wave incidence is solved. The complex variable function method is introduced to simplify the wave field expression. The accuracy is verified by comparing it with classic examples. Applying the Modified Time-Frequency Transform to obtain the time-domain wave field. Based on the Multipoint Transient Input Method (MTIM), the transient surface displacement response of the grillage above the defect is derived, mainly discussing the superposition behavior of input waves in the grillage. MTIM is based on the analytical solution of local time-domain wave field and takes the time histories of the surface characteristic points as the wave input. It has been validated as an effective method for studying the dynamic response of structures. The numerical results show that grillage form, incident angle, and wave velocity ratio are important factors affecting the structural responses of grillage above the defect. This research aims to reveal the propagation characteristics of waves in grillages in filling material defects, providing theoretical guidance for the design and application of grillages in this field.

Stress Intensity Factors for Thermoelectric Bonded Materials Weakened by an Inclined Crack

Thermoelectric Bonded Materials (TEBM) weakened by an Inclined Crack Problems (ICP) subjected to remote stress was presented in this study. The problems are addressed by employing the Modified Complex Variable Function (MCVF) method, which incorporates the Continuity Conditions (CC) for the Resultant Electric Force (REF) and Displacement Electric Function (DEF) to formulate the Hypersingular Integral Equations (HSIEs) associated with these problems. By applying the curved length coordinate method, the unknown functions of Crack Opening Displacement (COD), Electric Current Density (ECD), and Energy Flux Load (EFL) are mapped onto the square root singularity function. The resulting equations are then numerically solved using appropriate quadrature formulas, with the traction along the crack utilized as the right-hand term. The obtained COD, ECD and EFL functions is then used to compute the dimensionless Stress Intensity Factors (SIFs) in order to determine the stability behavior of TEBM weakened by an ICP. The numerical results provided demonstrate the dimensionless SIFs at the crack tips. These results exhibit excellent agreement with previous studies conducted on the subject. Additionally, it is observed that the dimensionless SIFs at the crack tips are influenced by factors such as the ratio of Elastic Constants (ECR), the geometry of the cracks, and the coefficients associated with the Electric Current Density (ECD).

Recent progress in the theory of functions of several complex variables and complex geometry

Recent progress in the theory of functions of several complex variables and complex geometry

Recent progress in the theory of functions of several complex variables and complex geometry

Дан обзор недавних новых результатов в области обратных теорем о $L^2$-существовании и $L^2$-продолжении, составляющих две основные части $L^2$-теории. Эти результаты используются для получения критерия положительности по Гриффитсу и условий положительности по Накано для (сингулярных) эрмитовых метрик голоморфных векторных расслоений, а также для доказательства сильной открытости и устойчивости пучков мультипликативных подмодулей, связанных с сингулярными неотрицательными по Накано эрмитовыми метриками на голоморфных векторных расслоениях.

Stress analysis of elliptical tunnels in an orthogonally anisotropic elastic full‐space under non‐uniform in‐situ stress

AbstractNon‐uniformly distributed in‐situ stress around the tunnel is a significant factor inducing various geohazards during tunnel excavation in the rock mass, and the anisotropy of the rock mass would further differentiate the in‐situ stress state around the tunnel. To ensure the safety of tunnel construction and operation, it is necessary to deeply analyze the stress distribution law and mechanical mechanism of the orthogonally anisotropic elastic media the around tunnel. This paper presents a complex variable function method for the elliptical tunnel in an orthogonally anisotropic elastic full‐space under non‐uniform in‐situ stress. To obtain the analytical solution, the affine transform and conformal mapping are applied to consider the orthogonal anisotropy of material and the ellipticity of the tunnel. A numerical series fitting method is introduced to describe the non‐uniformly distributed load, and then the stress solution around tunnel is further obtained. The comparisons with the existing literature, numerical solutions and model tests are provided to verify the correctness of the proposed method. Finally, the convergence study and the numerical analysis on the stress distribution are given graphically.

Investigation of the Application of Complex Function Theory in Underground Mine Design: A Case Study

This study, with the engineering background of the design of a stope involving a sublevel mining method in a certain underground metal mine, explored the application of stress-solving methods based on the complex variable function theory in actual engineering. Three mathematical calculation models based on the functions of a complex variable were established. Through triangle interpolation, mapping functions of a plane with a roadway section and a plane with the stope section were determined. An improved Schwarz alternating method was adopted to study the stability of the roadway and the influence of an adjacent roadway from the perspective of the stress field. In addition, in light of the distribution characteristics of a gangue in the stope, the design parameters of a pillar were optimized, with the pillar’s optimal dimensions determined. The results showed that when the magnitudes of two far-field principal stresses in the rock mass are relatively close, the distribution of the surrounding rock stress around the roadway is more uniform, and tensile stress is less likely to occur. The excavation of a neighboring roadway exacerbates to some extent the side stress of the other roadway, especially the compressive stress concentration on the side closer to the neighboring roadway. However, when the distance between the two roadways is significantly larger than the roadway size, this effect is not pronounced. In the engineering case studied in this research, the thickness of the pillar is approximately linearly positively correlated with the safety factor of the pillar approximately linearly negatively correlated with the recovery rate. Overall, this research explored the application of the complex variable function theory in an underground mine design, demonstrating its accuracy and practicality.

Teaching and Learning Computational Mathematics with Intensive Application of the Virtual Campus

Nowadays, the field of professional development in engineering requires appropriate and relevant training for a satisfactory insertion in the labor market. This requires that educational institutions provide not only the specific knowledge of the career being studied but also must complement it with other skills necessary for adequate performance in the work environment. Based on these premises, this paper describes the design of a classroom experience to be implemented in the subject of Advanced Calculus (AC), a third-level subject of the Mechanical Engineering course. The proposed activities are developed in the Computer and Multidisciplinary Laboratory of Basic Sciences available at the faculty and seek to encourage the use of the virtual campus, where the mediation between the content of the subject and the concrete applications of simple engineering models are the axis to develop the topic: Analytical Functions of Complex Variables (AFCV).

Reliability analysis of plate parts with multiple interacting hole-edge cracks using complex variable function

In this paper, a successive conformal mapping technique is used to propose an analytical method for solving the stress intensity factor (SIF) of asymmetrical hole-edge cracks on the basis of Muskhelishvili's complex variable function analysis method. Based on this analytical solution, the superposition method is used to get the SIF under the interaction of multiple cracks. Subsequently, according to the plastic collapse criterion, the residual strength under the interaction of two collinear asymmetric elliptical hole-edge cracks with different sizes on the side rail of a dumper's frame is analyzed through the method established. The failure probability of the plate part is obtained by the Kriging meta-model. This study provides a theoretical guide for the operation and maintenance plan of plate parts with multiple cracks.

Stress redistribution in a multilayer chamber for compressed air energy storage in abandoned coalmine: Elastic analytical insights and material choice

AbstractCompressed air energy storage (CAES) is attracting attention as one of large‐scale renewable energy storage systems. Its gas storage chamber is one of key components for its success. A successful utilization of an abandoned coalmine roadway depends on the stability of the gas storage chamber. The chamber is a multilayer structure and the redistribution of the stress and displacement in each layer is critical to the chamber stability. So far, this redistribution mechanism under any lateral pressure coefficient and internal air pressure has been unclear and thus a quick and easy evaluation on the CAES chamber stability becomes difficult. In this study, the redistributions of stress and displacement in each layer are analytically solved based on complex variable function theory. First, the stress and displacement in three CAES multilayer structures are obtained under any lateral pressure coefficient and internal air pressure. Then, a comprehensive parameter b1 is obtained to describe the redistribution of stress and displacement in rock mass, lining layer, and grout layer. Its linkage with lateral pressure coefficient, internal air pressure, and material properties is analytically expressed. Thirdly, an analytical expression is obtained for the influence range of internal air pressure on chamber stress. Finally, the role and selection of lining and grouting layers are explored at different lateral pressure coefficient and internal air pressure. It is found that the CAES chamber stability can be effectively and quickly evaluated by these analytical solutions and comprehensive parameters and enhanced by a material with low bulk modulus but high shear modulus.

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

Mechanisms of coupled mode I-II fracture initiation in rocks subjected to heat flow-induced fracture gas-steam pressure and heat flow stress

The construction of tunnel projects in the Sichuan-Tibet region has led to the occurrence of rock fractures caused by high temperature and high humidity, which are inevitable challenges. When there is fracture water in rock fractures and it encounters a heat flow, the water vaporizes into fracture steam, which undergoes thermal expansion to generate fracture gas-steam pressure. The heat flow direction is at an angle to the crack, causing heat flow stress in the rock. To understand the mechanism of crack initiation and propagation in rocks under the joint action of uniform heat flow-induced fracture gas-steam pressure and heat flow stress, the control equation for fracture gas-steam pressure is derived and expressions for stress intensity factor, crack temperature field, stress field, and crack initiation strength criterion are established based on the theory of complex variable functions. The key parameters are analyzed and discussed, finding that under low heat flux density, small incidence angles, and large crack inclination angles, KI values are higher, and the crack initiation is mainly in tension. Under high heat flow density, large incidence angles, and crack inclination angles near 45°, KII values are higher, and the crack initiation is mainly in shear. The coupling effect of KI and KII is more sensitive to heat flow density and inclination angle. The critical initiation angle and initiation strength increase with the increase of heat flow density and incidence angle, with heat flow density being the determining factor.

Fractional derivatives of fuzzy complex-variable functions and application to fuzzy fractional complex differential equations

In this paper, through theory of fuzzy complex analysis, we introduce some representations of r-cuts of trivial fuzzy zero, nontrivial fuzzy zero and fuzzy complex numbers in polar form on complex plane and establish some of their basic properties. Based on this, the addition and scalar multiplication operations of fuzzy complex numbers are established and the calculus of fuzzy complex-variable functions is studied. Moreover, we introduce a notion of fuzzy Ortigueira fractional derivative of fuzzy complex-variable functions and develop a theory of fuzzy fractional calculus on complex plane including some geometric features. As applications, the existence and uniqueness theorem of solutions for nonlinear fuzzy fractional complex-variable differential equations is established and some examples are provided.

Visco-Elastic Interface Effect on the Dynamic Stress of Symmetrical Tunnel Embedded in a Half-Plane Subjected to P Waves

A semi-analytical method is developed to study the visco-elastic interface effect on the dynamic stress of a non-circular symmetrical tunnel in a half-plane subjected to P waves, and the interaction between the visco-elastic interface around the tunnel and the traction free boundary at the half-plane is analyzed. The complex variable function method combined with the wave function expansion method is introduced to obtain the theoretical expressions of waves around the non-circular tunnel. To satisfy the traction-free boundary condition at the half-plane, the large circle assumption is applied. The expanded coefficients are determined by using appropriate boundary conditions with stiffness parameters and viscosity coefficients of the interface. Selected numerical results are presented, and the interface effect on the dynamic stress under different embedded depths is discussed. It is found that the interacting effect of the visco-elastic interface and the traction-free boundary is quite related with the wave frequency. Comparison with existing results is also given to validate the semi-analytical method.

On Polyanalytic Functions in Several Complex Variables

We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces mathcal {H}, being the weighted L_2 function spaces on a connected simply connected domains D subset mathbb {C}^n. The typical examples considered in the paper are the unit ball mathbb {B}^n and the whole space mathbb {C}^n. Our approach is based on the use of the two tuples of operators a=(a1,a2,…,an)andb=(b1,b2,…,bn),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathbf {\\mathfrak {a}}= (\\mathfrak {a}_1, \\ \\mathfrak {a}_2, \\ \\ldots , \\ \\mathfrak {a}_n) \\quad \ extrm{and} \\quad \\mathbf {\\mathfrak {b}}= (\\mathfrak {b}_1, \\ \\mathfrak {b}_2, \\ \\ldots , \\ \\mathfrak {b}_n), \\end{aligned}$$\\end{document}which act invariantly in some linear space and satisfy therein the commutation relations aj,bℓ=δj,ℓI,aj,aℓ=0,bj,bℓ=0,j,ℓ=1,2,...,n.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left[ \\mathfrak {a}_j, \\mathfrak {b}_{\\ell }\\right] = \\delta _{j,\\ell }I, \\quad \\left[ \\mathfrak {a}_j, \\mathfrak {a}_{\\ell }\\right] = 0, \\quad \\left[ \\mathfrak {b}_j, \\mathfrak {b}_{\\ell }\\right] = 0, \\quad j,\\ell = 1,2,...,n. \\end{aligned}$$\\end{document}We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space mathcal {H}, and impose several additional conditions. The exact formulation is given in the Extended Fock space construction. Further we specify the obtained description to different concrete realizations of the above operators and Hilbert spaces mathcal {H}, illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.