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

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.

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

This monography presents new results in the theory of differential subordination and superordination, new classes of harmonic univalent and multivalent functions defined by an integral operator, properties of the analytic functions defined by integral Salagean operator, a class of analytic functions defined by Ruscheweyh operator and generalized almost starlike mappings associated with extension of operators for biholomorphic mappings.

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.

This book covers all about tensor factorization in a generalized way. Generalization is accomplished by use of various divergence functions as well as different hidden structures. The divergence functions are generalized by the beta divergences that are connected to the Tweedie Models. The hidden structure is generalized by use of invented abstract factorization notation. Various learning algorithms including coupled tensors are, then, derived accordingly.

The book is a step forward in the direction of the theory of analytic functions of a complex variable which is outstanding accomplishments of classical mathematics. Chapter three covers the definition of generalized logarithmic mean function and generalized logarithmic mean function of nth order of an entire Dirichlet series, and using these definitions some fruitful results are obtained. In chapter four, some growth relations of generalized logarithmic mean function and generalized logarithmic mean function of the nth derivative of an entire Dirichlet series are discussed. The relation between the orders of two entire functions is also obtained. The corollaries are also given in some theorems of this chapter. Further, this book includes some results for a class of entire functions of Dirichlet series.

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 is a collection of our recent published papers in area of "Selected Topics On Some Classes Of Analytic Functions". The book consists of seven chapters, organized as follows. In chapter 1, we introduce some basic definitions of univalent, p-valent functions, functions with bounded boundary rotation and functions with bounded argument. In chapter 2, we investigate some interesting properties for two subclasses of starlike and p-valent close-to-convex functions. In chapter 3, we give basic properties for three subclasses of close-to convex functions with complex order. In chapter 4, we compute the sharp radii of spirallike of the functions belonging to three classes of analytic functions. In chapter 5, we are concerned with basic properties of the classes consisting of bounded multivalent spirallike, bounded p-valent Robertson functions and multivalent Robertson functions defined by using a differential operator. In chapter 6, we derive several interesting and properties for two classes consisting of multivalent functions with bounded boundary rotation and complex order. In chapter 7, we get sharp coefficient bounds for functions in some classes of meromorphic functions.

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 .

We study superharmonic functions, reduced function, harmonic measures and Jensen measures in Brelot spaces. We introduce the concept of quasi multiply superharmonic functions on a product of Brelot spaces and study their properties. A main result obtained is characterizing the quasi superharmonic functions in terms of harmonic, finely harmonic and Jensen measures. Then we prove that a quasi multiply superharmonic function on a product of Brelot spaces equals its lower semicontinuous regularization out side of a 2-negligible set. Further we give a sufficient condition on a Brelot space under which it becomes an extension space for superharmonic functions. As a result we characterize the extreme Jensen measures in such spaces. Finally we study extreme Jensen measures relative to several classes of multiply superharmonic functions.

We develop new generalized integral operators of analytic functions (single or multi-valent) with negative coefficients and we study some of their properties. We give particular cases of our new integral operators and link them to existing integral operators. Therefore, we easily show that several integral operators from literature are covered by our generalized integral operators. Their properties are also conserved.

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.

## 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

## 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.

## 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

## New classes of functions defined by operators

This monography presents new results in the theory of differential subordination and superordination, new classes of harmonic univalent and multivalent functions defined by an integral operator, properties of the analytic functions defined by integral Salagean operator, a class of analytic functions defined by Ruscheweyh operator and generalized almost starlike mappings associated with extension of operators for biholomorphic mappings.

## 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.

## Generalized Tensor Factorization

This book covers all about tensor factorization in a generalized way. Generalization is accomplished by use of various divergence functions as well as different hidden structures. The divergence functions are generalized by the beta divergences that are connected to the Tweedie Models. The hidden structure is generalized by use of invented abstract factorization notation. Various learning algorithms including coupled tensors are, then, derived accordingly.

## Growth Properties of Logarithmic Means of Entire Dirichlet Series

The book is a step forward in the direction of the theory of analytic functions of a complex variable which is outstanding accomplishments of classical mathematics. Chapter three covers the definition of generalized logarithmic mean function and generalized logarithmic mean function of nth order of an entire Dirichlet series, and using these definitions some fruitful results are obtained. In chapter four, some growth relations of generalized logarithmic mean function and generalized logarithmic mean function of the nth derivative of an entire Dirichlet series are discussed. The relation between the orders of two entire functions is also obtained. The corollaries are also given in some theorems of this chapter. Further, this book includes some results for a class of entire functions of Dirichlet series.

## 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.

## Selected Topics On Some Classes Of Analytic Functions

This book is a collection of our recent published papers in area of "Selected Topics On Some Classes Of Analytic Functions". The book consists of seven chapters, organized as follows. In chapter 1, we introduce some basic definitions of univalent, p-valent functions, functions with bounded boundary rotation and functions with bounded argument. In chapter 2, we investigate some interesting properties for two subclasses of starlike and p-valent close-to-convex functions. In chapter 3, we give basic properties for three subclasses of close-to convex functions with complex order. In chapter 4, we compute the sharp radii of spirallike of the functions belonging to three classes of analytic functions. In chapter 5, we are concerned with basic properties of the classes consisting of bounded multivalent spirallike, bounded p-valent Robertson functions and multivalent Robertson functions defined by using a differential operator. In chapter 6, we derive several interesting and properties for two classes consisting of multivalent functions with bounded boundary rotation and complex order. In chapter 7, we get sharp coefficient bounds for functions in some classes of meromorphic functions.

## 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 .

## Superharmonic Functions in Brelot Spaces

We study superharmonic functions, reduced function, harmonic measures and Jensen measures in Brelot spaces. We introduce the concept of quasi multiply superharmonic functions on a product of Brelot spaces and study their properties. A main result obtained is characterizing the quasi superharmonic functions in terms of harmonic, finely harmonic and Jensen measures. Then we prove that a quasi multiply superharmonic function on a product of Brelot spaces equals its lower semicontinuous regularization out side of a 2-negligible set. Further we give a sufficient condition on a Brelot space under which it becomes an extension space for superharmonic functions. As a result we characterize the extreme Jensen measures in such spaces. Finally we study extreme Jensen measures relative to several classes of multiply superharmonic functions.

## New Integral Operators - Properties

We develop new generalized integral operators of analytic functions (single or multi-valent) with negative coefficients and we study some of their properties. We give particular cases of our new integral operators and link them to existing integral operators. Therefore, we easily show that several integral operators from literature are covered by our generalized integral operators. Their properties are also conserved.

## 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.