The present book embodies the researches carried out at the Aligarh Muslim University, Aligarh. The Bessel Functions originally introduced by Bessel in 1824, while discussing a problem in astronomy. The Bessel Functions received the most extensive treatment among all the Special Functions. W. Voigt (1850-1919) introduced the functions K(x,y) and L(x,y) which play an important role in several diverse fields of physics. Later Srivastava and Miller (1987) established a link between Bessel Functions and generalized Voigt functions. The Voigt functions were subsequently modified by others e.g. Exton, Srivastava and Chen, Klusch, Siddiqui, Srivastava, Pathan, etc.

In this book we consider the concept of generalized function in mathematics . We give a brief discussion of some known concepts and theorems like operations on a space of generalized function and the definition of Dirac delta function . Also we discuss why we need the spaces of new generalized functions , and we define some spaces of new generalized functions in which the operations of multiplication , Fourier transformation , and differentiation are defined . Ramadan

Bessel function is defied for a first time by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel. Bessel functions are also called cylinder function or cylindrical harmonic function because they are found in the solution to Laplace’s equation in cylindrical coordinated. Bessel equation arises in problems involving vibrations, or heat conduction in regions possessing circular symmetry; therefore Bessel function have many application in physics and engineering in connection with the propagation of waves, elasticity, fluid motion and especially in many problem of potential theory and diffusion involving cylindrical symmetry. This work consists three chapters. The first chapter remained about the power series, second order linear differential equation, singularity point, Sturm-Liouville problem and then gamma function which help to express factorial. In the second chapter it is discussed about the Bessel equation and its solution which is Bessel functions with the plot of Bessel function. The third chapter discuss about the modified Bessel equation and it’s solution, which is the special case of the Bessel equation.

Generalized functions are the extension of classical functions and allow us to generalize the concept of derivative to all continuous functions. The generalized functions have many applications in various fields especially in science and engineering where the non-continuous phenomena that naturally lead to differential equations whose solutions are not ordinary functions, such as the Dirac-delta function therefore generalized functions can help us to develop an operational calculus in order to investigate linear ordinary differential equations as well as partial differential equations with constant and non-constant coefficients through their fundamental solutions. In the present work we extend certain operations such as multiplications and convolutions to the generalized functions by using the neutrix limit approach. The essential use of the neutrix limit is to extract the finite part from a divergent quantity as one has usually done to subtract the divergent terms via rather complicated procedures in the renormalization theory. In fact we can consider the neutrices as the generalization of the Hadamard finite parts

Kamps (1995) suggested a new theoretical approach, which is called generalized order statistics (gos). This new model includes ordinary order statistics (oos), sequential order statistics, progressive type II censored order statistics and record values. The concept of gos enables a common approach to structural similarities and analogies. The distributional and inferential properties of oos and record values turn out to remain valid for gos (cf. Cramer and Kamps, 2001). Thus, the concept of gos provides a large class of models with many interesting and useful properties for both the description and the analysis of practical problems. Due to this reason, the question arises whether the general distribution theory of gos as well as their properties can be obtained by analogy with that for oos. The latter has been extensively investigated in the literature, e.g., see, David, 1981. The main purpose of this thesis is to investigate the asymptotic behavior of some important functions of gos, in view of theory of statistics and its applications. Some of these functions are non-linear, e.g., extremal product and extremal quotient and other are linear e.g., range and midrange.

The subject of the special functions of the single complex variable and several complex variables occupies a great attention in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study this subject, taking it into account as one of several theses, which presented and still being presented by Specialists in complex analysis. The main aim of this thesis de?ning and studying of some special functions which contain two complex variables such as Bessel matrix function, Tricomi matrix function, Horn matrix function and Struve matrix function and also providing special inequalities of Bessel functions and modi?ed Bessel functions and Tricomi functions of two scalar index of two complex variables and of their ratios.

The continuous distribution’s samples and their historical sequences in addition to the positive results they achieved within several practical applications in various productive and planning aspects are very important. It is notable that there is negligence in shaping the form of those continuous distributions. The multiple varieties of studied aspects make those samples and some of their mixtures which proved themselves during the last decade in many productive studies. The common assumption during the last period of time which says that the samples we study might be considered pure under some conditions which are not enough and impossible to achieve in addition of being very expensive and lately became very far from reality and that was affirmed by most of the studied samples. The researcher suggested the form which produces forms and various continuous distributions mixtures within several information values, which is the idea of the researcher in order to find the most general formula for various forms used in for the continuous distribution . The general gamma distribution model is considered as one of such distributions which established their role since it is shown

This book is desined as an introduction to theory and applications of integral transform to problems in linear differential equation,optical system analysis signal processing. The conventional canonical cosine transforms has been extended up to the distribution of compact support by using the kernel method, the same approach has been employed in case of sine transform. The most striking result in both the chapters are inversion theorems for generalized canonical cosine transforms and canonical sine transforms, Uniqueness theorems etc. Some operation transform formulae for generalized canonical cosine transforms and for generalized canonical sine transform and some basic properties of canonical cosine and sine transforms including modulation theorems and properties of Kernels have been provrd

It is often overlooked that there are naturally no less than infinitely many differential algebras of generalized functions. These infinitely many differential algebras of generalized functions are equally naturally subjected to a basic dichotomic sheaf theoretic singularity test regarding their significantly different abilities to deal with large classes of singularities. The property of a vector space or algebra of generalized functions of being a flabby sheaf proves to be essential in being able to deal with large classes of singularities. A review is presented of the way singularities are dealt with in five of the infinitely many types of differential algebras of generalized functions. These five types of algebras, in the order they were introduced in the literature are : the nowhere dense algebras, chains of algebras, the Colombeau algebras, space-time foam algebras, and local algebras. The first, third and fourth of them turned out to be the ones most frequently used in a variety of applications. Five fundamentally important issues related to singularities are pursued.

This book presents new research results on channel characterization and modeling. MIMO channel characterization and modeling techniques are investigated. The first part exposes a generalized and systematic representation of systems and channels in the spatio-temporal domains. It is a generalization of the concepts introduced in the research literature: it consists of a set of sixteen kernel functions and eight system functions, in time, frequency, space and wave-vector domains. A series of examples illustrates the concept. In the second part, a space-time signal measurement technique is introduced along with a wave-vector-frequency spectrum estimation method. These are employed to create a large virtual antenna array from a small number of antenna elements. A new calibration technique to estimate measured wave vector spectra is presented. In the last part, a generalized view of the Doppler phenomenon, based on the characterization technique is presented. Here, the Doppler spectra, caused by scatterer mobility under different scatterer velocity distributions, are investigated. Fixed-wireless scenarios, with stationary transmitters and receivers with mobile scatterers are analyzed.

The complex analysis is one of the classical branches of mathematics and has its roots in the XVIII century. Two important directions of complex analysis are the theory of conformal representations and the geometric theory of analytic functions. Applications and extensions of these theories have been developed in numerous fields, including differential equations, harmonic functions, meromorphic functions, integral operators, Banach spaces and others. This book, pertaining to the geometric function theory, provides several new classes of meromorphic functions, some new integral operators and preserving properties results, regarding the new introduced operators. These new classes are extensions of the well-known starlike, convex and close-to-convex classes. After a review of some basic concepts and standard results, related to the subordination and superordination, sandwich-type results are obtained, using the generalized Briot-Bouquet differential subordinations and superordinations. The book is useful for mathematicians, physicists, engineers, students, researches and other specialists who want to have a good training in the field of geometric function theory.

This work deals maily with following areas of XRD line profile analysis: In chapter II methods of precise and accurate estimation of crystallographic parameters using statistical techniques such as maximum likelyhood method & Min-Max method, have been developed and applied to extract different crystallographic parameters. In Chapter III methods were developed to ascertain which analytical function can describe an experimentally observed XRD peak best. In chapter IV short-commings of Voigt & pseudo- Voigt functions when subjected to Warren Averbach and variance method of size-strain analysis ,were discussed. In Chapter V a new method of structure refinement - an alternative to the rietveld method, was proposed and applied to powder data of tungsten oxide .This method requires no apriori assumption regarding the functional nature of observed peaks .

The subjects of special functions and orthogonal matrix polynomials of the single complex variable have a great situation in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study these subjects and is considered to be one of several theses, which presented and still presented by the school established by Sayyed in one of the branches of complex analysis. This work investigates some subjects in complex analysis such as generalizations of Hermite matrix polynomials, the Ultraspherical matrix polynomials. Some results, which are important in this field, have been obtained in this work. In this work, Some of a new concepts of some special functions and matrix polynomials are introduced and studied in: (i) Give a review on some properties of Hermite matrix polynomials. (ii) Introduce and study of Ultraspherical matrix polynomials and their properties. We conclude our study by listing some of unsolved problems which as listed as open problems in future work:Introduce and study of Hermite-Bessel, Hermite-Laguerre, Laguerre-Laguerre, Legendre-Hermite and Hermite-Legendre matrix functions

This thesis deals with proton, neutron as well as deuteron structure functions determined from deep inelastic scattering experiments approximated for low-x region. Structure functions are calculated from complete, particular and unique solutions of GLDAP evolution equations which are deduced from pertubative quantum chromodynamics. t and x-evolutions of structure functions at low-x region are predicted. Theoretical predictions are compared with experimental data and recent global parameterizations.

The aim of this thesis was the development of the study of lacunary interpolation by spline functions and their applications by changed the boundary condition for quantic and sixtic spline functions, and the algorithm was used to find the new absolute error between the original function and the spline function, and also the error bounded between the derivatives of original function and the derivatives of spline functions. Firstly, the object of this work is to show that the change of the boundary conditions and the class of spline functions have effect on minimizing error bounds theoretically and practically, and for application, the )NEB( algorithm was used. Secondly, in this study, (0, 4) lacunary interpolation was generalized by quantic spline function to obtain, the existence, uniqueness, and error bounds for the generalized (0, 4) lacunary interpolation by quantic spline. Finally, the lacunary interpolation problem consisted of finding the sixth degree spline of deficiency four, interpolating the data given on the function value with first and fourth order in the interval [0,1]. Also, an extra initial condition was prescribed on the second derivative of the functions.

## on generalized bessel functions and voigt functions в наличии / купить интернет-магазине

## On Generalized Bessel Functions And Voigt Functions

The present book embodies the researches carried out at the Aligarh Muslim University, Aligarh. The Bessel Functions originally introduced by Bessel in 1824, while discussing a problem in astronomy. The Bessel Functions received the most extensive treatment among all the Special Functions. W. Voigt (1850-1919) introduced the functions K(x,y) and L(x,y) which play an important role in several diverse fields of physics. Later Srivastava and Miller (1987) established a link between Bessel Functions and generalized Voigt functions. The Voigt functions were subsequently modified by others e.g. Exton, Srivastava and Chen, Klusch, Siddiqui, Srivastava, Pathan, etc.

## New Generalized Functions

In this book we consider the concept of generalized function in mathematics . We give a brief discussion of some known concepts and theorems like operations on a space of generalized function and the definition of Dirac delta function . Also we discuss why we need the spaces of new generalized functions , and we define some spaces of new generalized functions in which the operations of multiplication , Fourier transformation , and differentiation are defined . Ramadan

## Bessel Function And The Modified Bessel Function

Bessel function is defied for a first time by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel. Bessel functions are also called cylinder function or cylindrical harmonic function because they are found in the solution to Laplace’s equation in cylindrical coordinated. Bessel equation arises in problems involving vibrations, or heat conduction in regions possessing circular symmetry; therefore Bessel function have many application in physics and engineering in connection with the propagation of waves, elasticity, fluid motion and especially in many problem of potential theory and diffusion involving cylindrical symmetry. This work consists three chapters. The first chapter remained about the power series, second order linear differential equation, singularity point, Sturm-Liouville problem and then gamma function which help to express factorial. In the second chapter it is discussed about the Bessel equation and its solution which is Bessel functions with the plot of Bessel function. The third chapter discuss about the modified Bessel equation and it’s solution, which is the special case of the Bessel equation.

## Generalized Functions Using Neutrix Calculus

Generalized functions are the extension of classical functions and allow us to generalize the concept of derivative to all continuous functions. The generalized functions have many applications in various fields especially in science and engineering where the non-continuous phenomena that naturally lead to differential equations whose solutions are not ordinary functions, such as the Dirac-delta function therefore generalized functions can help us to develop an operational calculus in order to investigate linear ordinary differential equations as well as partial differential equations with constant and non-constant coefficients through their fundamental solutions. In the present work we extend certain operations such as multiplications and convolutions to the generalized functions by using the neutrix limit approach. The essential use of the neutrix limit is to extract the finite part from a divergent quantity as one has usually done to subtract the divergent terms via rather complicated procedures in the renormalization theory. In fact we can consider the neutrices as the generalization of the Hadamard finite parts

## On The Asymptotic Generalized Order Statistics And Related Functions

Kamps (1995) suggested a new theoretical approach, which is called generalized order statistics (gos). This new model includes ordinary order statistics (oos), sequential order statistics, progressive type II censored order statistics and record values. The concept of gos enables a common approach to structural similarities and analogies. The distributional and inferential properties of oos and record values turn out to remain valid for gos (cf. Cramer and Kamps, 2001). Thus, the concept of gos provides a large class of models with many interesting and useful properties for both the description and the analysis of practical problems. Due to this reason, the question arises whether the general distribution theory of gos as well as their properties can be obtained by analogy with that for oos. The latter has been extensively investigated in the literature, e.g., see, David, 1981. The main purpose of this thesis is to investigate the asymptotic behavior of some important functions of gos, in view of theory of statistics and its applications. Some of these functions are non-linear, e.g., extremal product and extremal quotient and other are linear e.g., range and midrange.

## Some Topics in Complex Analysis and its Apps. in Special Functions

The subject of the special functions of the single complex variable and several complex variables occupies a great attention in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study this subject, taking it into account as one of several theses, which presented and still being presented by Specialists in complex analysis. The main aim of this thesis de?ning and studying of some special functions which contain two complex variables such as Bessel matrix function, Tricomi matrix function, Horn matrix function and Struve matrix function and also providing special inequalities of Bessel functions and modi?ed Bessel functions and Tricomi functions of two scalar index of two complex variables and of their ratios.

## Extended Generalized Functions and Its Distributions Applications

The continuous distribution’s samples and their historical sequences in addition to the positive results they achieved within several practical applications in various productive and planning aspects are very important. It is notable that there is negligence in shaping the form of those continuous distributions. The multiple varieties of studied aspects make those samples and some of their mixtures which proved themselves during the last decade in many productive studies. The common assumption during the last period of time which says that the samples we study might be considered pure under some conditions which are not enough and impossible to achieve in addition of being very expensive and lately became very far from reality and that was affirmed by most of the studied samples. The researcher suggested the form which produces forms and various continuous distributions mixtures within several information values, which is the idea of the researcher in order to find the most general formula for various forms used in for the continuous distribution . The general gamma distribution model is considered as one of such distributions which established their role since it is shown

## Integral Transform of Generalized Functions and Their Applications

This book is desined as an introduction to theory and applications of integral transform to problems in linear differential equation,optical system analysis signal processing. The conventional canonical cosine transforms has been extended up to the distribution of compact support by using the kernel method, the same approach has been employed in case of sine transform. The most striking result in both the chapters are inversion theorems for generalized canonical cosine transforms and canonical sine transforms, Uniqueness theorems etc. Some operation transform formulae for generalized canonical cosine transforms and for generalized canonical sine transform and some basic properties of canonical cosine and sine transforms including modulation theorems and properties of Kernels have been provrd

## Singularities and Differential Algebras of Generalized Functions

It is often overlooked that there are naturally no less than infinitely many differential algebras of generalized functions. These infinitely many differential algebras of generalized functions are equally naturally subjected to a basic dichotomic sheaf theoretic singularity test regarding their significantly different abilities to deal with large classes of singularities. The property of a vector space or algebra of generalized functions of being a flabby sheaf proves to be essential in being able to deal with large classes of singularities. A review is presented of the way singularities are dealt with in five of the infinitely many types of differential algebras of generalized functions. These five types of algebras, in the order they were introduced in the literature are : the nowhere dense algebras, chains of algebras, the Colombeau algebras, space-time foam algebras, and local algebras. The first, third and fourth of them turned out to be the ones most frequently used in a variety of applications. Five fundamentally important issues related to singularities are pursued.

## Generalized Spatio-Temporal Channel and System Characterization

This book presents new research results on channel characterization and modeling. MIMO channel characterization and modeling techniques are investigated. The first part exposes a generalized and systematic representation of systems and channels in the spatio-temporal domains. It is a generalization of the concepts introduced in the research literature: it consists of a set of sixteen kernel functions and eight system functions, in time, frequency, space and wave-vector domains. A series of examples illustrates the concept. In the second part, a space-time signal measurement technique is introduced along with a wave-vector-frequency spectrum estimation method. These are employed to create a large virtual antenna array from a small number of antenna elements. A new calibration technique to estimate measured wave vector spectra is presented. In the last part, a generalized view of the Doppler phenomenon, based on the characterization technique is presented. Here, the Doppler spectra, caused by scatterer mobility under different scatterer velocity distributions, are investigated. Fixed-wireless scenarios, with stationary transmitters and receivers with mobile scatterers are analyzed.

## Integral operators on spaces of complex functions

The complex analysis is one of the classical branches of mathematics and has its roots in the XVIII century. Two important directions of complex analysis are the theory of conformal representations and the geometric theory of analytic functions. Applications and extensions of these theories have been developed in numerous fields, including differential equations, harmonic functions, meromorphic functions, integral operators, Banach spaces and others. This book, pertaining to the geometric function theory, provides several new classes of meromorphic functions, some new integral operators and preserving properties results, regarding the new introduced operators. These new classes are extensions of the well-known starlike, convex and close-to-convex classes. After a review of some basic concepts and standard results, related to the subordination and superordination, sandwich-type results are obtained, using the generalized Briot-Bouquet differential subordinations and superordinations. The book is useful for mathematicians, physicists, engineers, students, researches and other specialists who want to have a good training in the field of geometric function theory.

## Statistical methods and profile fitting functions in powder XRD

This work deals maily with following areas of XRD line profile analysis: In chapter II methods of precise and accurate estimation of crystallographic parameters using statistical techniques such as maximum likelyhood method & Min-Max method, have been developed and applied to extract different crystallographic parameters. In Chapter III methods were developed to ascertain which analytical function can describe an experimentally observed XRD peak best. In chapter IV short-commings of Voigt & pseudo- Voigt functions when subjected to Warren Averbach and variance method of size-strain analysis ,were discussed. In Chapter V a new method of structure refinement - an alternative to the rietveld method, was proposed and applied to powder data of tungsten oxide .This method requires no apriori assumption regarding the functional nature of observed peaks .

## A Study of Some Special Functions and Polynomials

The subjects of special functions and orthogonal matrix polynomials of the single complex variable have a great situation in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study these subjects and is considered to be one of several theses, which presented and still presented by the school established by Sayyed in one of the branches of complex analysis. This work investigates some subjects in complex analysis such as generalizations of Hermite matrix polynomials, the Ultraspherical matrix polynomials. Some results, which are important in this field, have been obtained in this work. In this work, Some of a new concepts of some special functions and matrix polynomials are introduced and studied in: (i) Give a review on some properties of Hermite matrix polynomials. (ii) Introduce and study of Ultraspherical matrix polynomials and their properties. We conclude our study by listing some of unsolved problems which as listed as open problems in future work:Introduce and study of Hermite-Bessel, Hermite-Laguerre, Laguerre-Laguerre, Legendre-Hermite and Hermite-Legendre matrix functions

## Studies on Hadron Structure Functions and GLDAP Evolution Equations

This thesis deals with proton, neutron as well as deuteron structure functions determined from deep inelastic scattering experiments approximated for low-x region. Structure functions are calculated from complete, particular and unique solutions of GLDAP evolution equations which are deduced from pertubative quantum chromodynamics. t and x-evolutions of structure functions at low-x region are predicted. Theoretical predictions are compared with experimental data and recent global parameterizations.

## Development of Some Lacunary Interpolation by S. F. and Application

The aim of this thesis was the development of the study of lacunary interpolation by spline functions and their applications by changed the boundary condition for quantic and sixtic spline functions, and the algorithm was used to find the new absolute error between the original function and the spline function, and also the error bounded between the derivatives of original function and the derivatives of spline functions. Firstly, the object of this work is to show that the change of the boundary conditions and the class of spline functions have effect on minimizing error bounds theoretically and practically, and for application, the )NEB( algorithm was used. Secondly, in this study, (0, 4) lacunary interpolation was generalized by quantic spline function to obtain, the existence, uniqueness, and error bounds for the generalized (0, 4) lacunary interpolation by quantic spline. Finally, the lacunary interpolation problem consisted of finding the sixth degree spline of deficiency four, interpolating the data given on the function value with first and fourth order in the interval [0,1]. Also, an extra initial condition was prescribed on the second derivative of the functions.