Integral Representations in Complex Analysis: From Classical Results to Recent Developments

2020 ◽  
pp. 449-471
Author(s):  
Michael Range
Keyword(s):  
Complex Analysis ◽  
Integral Representations ◽  
Recent Developments

scholarly journals Multidimensional Approaches to Examining Digital Literacies in the Contemporary Global Society

2019 ◽  
Vol 7 (2) ◽  
pp. 36-46 ◽  
Author(s):  
Kewman M. Lee ◽  
Sohee Park ◽  
Bong Gee Jang ◽  
Byeong-Young Cho
Keyword(s):  
Digital Literacy ◽  
Complex Analysis ◽  
Literacy Practices ◽  
Integrative Approach ◽  
Digital Literacies ◽  
Multiple Perspectives ◽  
Interpretive Inquiry ◽  
Recent Developments ◽  
Global World ◽  
Cognitive Perspectives

Literacy scholars have offered compelling theories about and methods for understanding the digital literacy practices of youth. However, little work has explored the possibility of an approach that would demonstrate how different perspectives on literacies might intersect and interconnect in order to better describe the multifaceted nature of youth digital literacies. In this conceptual article, we adopt the idea of theoretical triangulation in interpretive inquiry and explore how multiple perspectives can jointly contribute to constructing a nuanced description of young people’s literacies in today’s digitally mediated global world. For this purpose, we first suggest a triangulation framework that integrates sociocultural, affective, and cognitive perspectives on digital literacies, focusing on recent developments in these perspectives. We then use an example of discourse data from a globally connected online affinity space and demonstrate how our multidimensional framework can lead to a complex analysis and interpretation of the data. In particular, we describe the substance of one specific case of youth digital literacies from each of the three perspectives on literacy, which in turn converge to provide a complex account of such literacy practices. In conclusion, we discuss the promise and limitations of our integrative approach to studying the digital literacy practices of youth.

A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line

2008 ◽  
Vol 464 (2095) ◽  
pp. 1823-1849 ◽  
Author(s):  
N Flyer ◽  
A.S Fokas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Plane ◽  
Complex Analysis ◽  
Variable Coefficients ◽  
Integral Representations ◽  
New Method ◽  
Half Line ◽  
Time Step ◽  
Partial Differential

A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

scholarly journals Linearization: Geometric, Complex, and Conditional

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
Asghar Qadir
Keyword(s):  
Group Theory ◽  
Differential Equations ◽  
Original Work ◽  
Complex Analysis ◽  
Lie Symmetry ◽  
Lie Symmetry Analysis ◽  
Systematic Method ◽  
Lie Group Theory ◽  
Systems Of Differential Equations ◽  
Recent Developments

Lie symmetry analysis provides a systematic method of obtaining exact solutions of nonlinear (systems of) differential equations, whether partial or ordinary. Of special interest is the procedure that Lie developed to transform scalar nonlinear second-order ordinary differential equations to linear form. Not much work was done in this direction to start with, but recently there have been various developments. Here, first the original work of Lie (and the early developments on it), and then more recent developments based on geometry and complex analysis, apart from Lie’s own method of algebra (namely, Lie group theory), are reviewed. It is relevant to mention that much of the work isnotlinearization but uses the base of linearization.

scholarly journals Lindelöf Representations and (Non-)Holonomic Sequences

10.37236/275 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Finite Difference ◽  
Generating Functions ◽  
Complex Analysis ◽  
Integral Representations ◽  
Asymptotic Estimates ◽  
Linear Recurrences ◽  
Polynomial Coefficients ◽  
Difference Sequences ◽  
Analytic Expressions ◽  
Linear Differential

Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindelöf approach.

scholarly journals On Lommel Matrix Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2335
Author(s):  
Ayman Shehata
Keyword(s):  
Entire Function ◽  
Recurrence Relations ◽  
Complex Analysis ◽  
Order Type ◽  
Explicit Representation ◽  
Integral Representations ◽  
Matrix Polynomials ◽  
New Class ◽  
Special Cases ◽  
Matrix Recurrence Relations

The main aim of this paper is to introduce a new class of Lommel matrix polynomials with the help of hypergeometric matrix function within complex analysis. We derive several properties such as an entire function, order, type, matrix recurrence relations, differential equation and integral representations for Lommel matrix polynomials and discuss its various special cases. Finally, we establish an entire function, order, type, explicit representation and several properties of modified Lommel matrix polynomials. There are also several unique examples of our comprehensive results constructed.

scholarly journals On functions with the boundary Morera property in domains with piecewise-smooth boundary

2021 ◽  
Vol 31 (1) ◽  
pp. 50-58
Author(s):  
A.M. Kytmanov ◽  
S.G. Myslivets
Keyword(s):  
Complex Analysis ◽  
Smooth Boundary ◽  
Holomorphic Extension ◽  
Continuous Functions ◽  
Integral Representations ◽  
Piecewise Smooth ◽  
Multidimensional Case ◽  
Piecewise Smooth Boundary ◽  
Multidimensional Complex Analysis ◽  
Extension Of Functions

The problem of holomorphic extension of functions defined on the boundary of a domain into this domain is actual in multidimensional complex analysis. It has a long history, starting with the proceedings of Poincaré and Hartogs. This paper considers continuous functions defined on the boundary of a bounded domain $ D $ in $ \mathbb C ^ n $, $ n> 1 $, with piecewise-smooth boundary, and having the generalized boundary Morera property along the family of complex lines that intersect the boundary of a domain. Morera property is that the integral of a given function is equal to zero over the intersection of the boundary of the domain with the complex line. It is shown that such functions extend holomorphically to the domain $ D $. For functions of one complex variable, the Morera property obviously does not imply a holomorphic extension. Therefore, this problem should be considered only in the multidimensional case $ (n> 1) $. The main method for studying such functions is the method of multidimensional integral representations, in particular, the Bochner-Martinelli integral representation.

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