1. The document presents methods for obtaining derivatives and partial derivatives of a generalized hypergeometric function of two variables, denoted H(x,y).
2. The H(x,y) function is defined in terms of gamma functions and involves multiple parameters and variables. Formulas are given for the derivatives and partial derivatives of H(x,y) with respect to these parameters.
3. The integrals defining H(x,y) are shown to converge under certain conditions on the parameters. Asymptotic formulas are also provided for the behavior of H(x,y) as the variables x and y approach zero or infinity.
Effect of orientation angle of elliptical hole in thermoplastic composite pla...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
On Some Integrals of Products of H -FunctionsIJMER
The object of the present paper is to evaluate an integral involving products of three H -function of different arguments which not only provides us the Laplace transform, Hankel transform ([8],p.3), Meijer’s Bessel transform ([8],p.121) and various other integral transforms of the product of two H -functions but also generalizes the result given earlier by many writers notably by Bailey ([3],
p.38), Meijer ([8],p.422) and Slater ([20], p.54(3.7.2).
This document introduces two new fractional integration operators associated with the H-function of two variables. The H-function of two variables is defined and some of its key properties are established. These operators generalize previous results given by other authors. The behavior of the H-function for small and large values of the variables is also described. When certain parameters are set to specific values, the H-function reduces to known cases.
On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Quantum mechanics 1st edition mc intyre solutions manualSelina333
Quantum Mechanics 1st Edition McIntyre Solutions Manual
Download at: https://goo.gl/SdC7Ef
quantum mechanics david mcintyre solutions pdf
quantum mechanics mcintyre pdf
quantum mechanics a paradigms approach solutions pdf
quantum mechanics mcintyre solutions pdf
quantum mechanics a paradigms approach solution manual
quantum mechanics mcintyre solutions manual pdf
hidden life of prayer
Effect of orientation angle of elliptical hole in thermoplastic composite pla...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
On Some Integrals of Products of H -FunctionsIJMER
The object of the present paper is to evaluate an integral involving products of three H -function of different arguments which not only provides us the Laplace transform, Hankel transform ([8],p.3), Meijer’s Bessel transform ([8],p.121) and various other integral transforms of the product of two H -functions but also generalizes the result given earlier by many writers notably by Bailey ([3],
p.38), Meijer ([8],p.422) and Slater ([20], p.54(3.7.2).
This document introduces two new fractional integration operators associated with the H-function of two variables. The H-function of two variables is defined and some of its key properties are established. These operators generalize previous results given by other authors. The behavior of the H-function for small and large values of the variables is also described. When certain parameters are set to specific values, the H-function reduces to known cases.
On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Quantum mechanics 1st edition mc intyre solutions manualSelina333
Quantum Mechanics 1st Edition McIntyre Solutions Manual
Download at: https://goo.gl/SdC7Ef
quantum mechanics david mcintyre solutions pdf
quantum mechanics mcintyre pdf
quantum mechanics a paradigms approach solutions pdf
quantum mechanics mcintyre solutions pdf
quantum mechanics a paradigms approach solution manual
quantum mechanics mcintyre solutions manual pdf
hidden life of prayer
+2 maths pta question bank come book english mediumMdk Raja
This document contains study material for matrices and determinants, and vector algebra for class 12.
For matrices and determinants, it provides examples and exercises on finding the adjoint and inverse of matrices, solving homogeneous and non-homogeneous systems of linear equations using matrix inversion and determinant methods, consistency of systems of equations, and word problems involving systems of linear equations.
For vector algebra, it gives exercises to prove various vector identities involving addition, subtraction and multiplication of vectors, and word problems involving vectors. The exercises cover topics like verifying properties of vectors under rotation, solving problems related to lines and planes.
The material is intended to help students prepare for 10 mark questions that may be asked in their full test on matrices
Certain D - Operator for Srivastava H B - Hypergeometric Functions of Three V...inventionjournals
The aim of this paper is to derive certain relations involving Srivastava’s hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, which express the aforementioned H B - hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly. Also, we have derived certain new integral representations for the H B - hypergeometric functions of three variables defined earlier by Lauricella [ 8 ] and Srivastava [14 ] .
Decomposition formulas for H B - hypergeometric functions of three variablesinventionjournals
: In this paper we investigate several decomposition formulas associated with hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 5 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell's hypergeometric functions.
Solving Poisson’s Equation Using Preconditioned Nine-Point Group SOR Iterativ...inventionjournals
The document describes a preconditioned nine-point group successive over-relaxation (SOR) iterative method for solving Poisson's equation. Poisson's equation is discretized using a finite difference scheme, resulting in a large sparse system of linear equations. The nine-point group SOR method is formulated for solving this system. A new type of preconditioner is then applied to the nine-point group SOR method to improve its rate of convergence. Numerical results demonstrate that the preconditioned nine-point group SOR method has improved convergence properties compared to the unpreconditioned version.
The document describes a Perfect Bayesian Equilibrium model. It involves two players (P1 and P2) who choose between two actions (a1 and a2). The players have different payoffs depending on the joint action. The model specifies the players' types (u1 and u2), actions, payoffs, and beliefs to represent a Perfect Bayesian Equilibrium where each player chooses the optimal action given their beliefs about the other player's action. It also describes how to calculate the expected utility for each player based on the other player's strategy and the equilibrium concept requires strategies to be optimal given beliefs.
International journal of applied sciences and innovation vol 2015 - no 1 - ...sophiabelthome
This document presents a derivative-free cubic convergent extrapolated Newton's method for finding zeros of nonlinear equations. The method is a two-step iterative process that approximates derivatives using backward difference approximations, requiring three functional evaluations per iteration. It is proven to have third-order convergence like the extrapolated Newton's method. Numerical examples are given to illustrate the efficiency of the new method compared to Newton's method and extrapolated Newton's method.
This document contains sections and problems related to vectors and geometry in space. It covers vectors in the plane, space coordinates and vectors in space, the dot and cross products of vectors, lines and planes in space, surfaces in space, and cylindrical and spherical coordinates. There are review exercises and problem solving questions provided at the end.
This document is a solutions manual for a textbook on communication systems. It provides step-by-step solutions to problems from each chapter of the textbook. The problems cover topics such as signal representations using Fourier series and integrals, power calculations for periodic signals, and bandpass signal representations. The solutions demonstrate techniques for analyzing and working with signals commonly encountered in electrical communication systems.
Boundary value problem and its application in i function of multivariableAlexander Decker
This document presents a mathematical model of a boundary value problem and its application to the I-function of multivariable. It defines the boundary value problem for temperature distribution in a rectangular plate under given boundary conditions. It then evaluates the solution of the boundary value problem using the I-function of multivariable, involving products of general polynomials. Several integral formulas are used to derive the solution as a sum involving arbitrary functions and constants.
This document provides 30 equations and inequalities and asks the reader to solve them on the set of real numbers. It uses variables like x, square roots, exponents, and basic arithmetic operations. The problems range from simple one-variable equations to more complex expressions with multiple variables. The goal is to calculate the value(s) of the variable(s) that satisfy each equation or inequality.
Communication systems solution manual 5th editionTayeen Ahmed
This document contains solutions to problems from Communications Systems, 5th edition. It includes solutions for time domain representations of pulses, their frequency domain counterparts obtained using Fourier transforms, properties of even and odd functions, and bounds on the bandwidth of time-limited signals. The solutions demonstrate properties like linearity, time-shifting, differentiation in the time and frequency domains, and the relationships between a function and its Fourier transform.
This document contains solutions to various equations and inequalities involving radicals on the set of real numbers. It is divided into 6 sections, with multiple problems provided in each section ranging from simple single-term radical equations to more complex multi-term radical equations and inequalities. The document provides the step-by-step workings for solving each problem.
This document contains mathematical formulas and definitions from various topics including:
- Set operations and identities involving sets A, B, and C.
- Properties of rational numbers including ratios in lowest terms.
- Algebra topics such as FOIL, factoring formulas, and special products involving binomials.
- Properties of fractions, equations, inequalities, and their relationships.
- Formulas for linear equations, the quadratic formula, distance, midpoint, and equations of circles.
- Properties that classify functions as even or odd based on their behavior as the input variable x approaches negative values.
The document presents a probability loophole in the CHSH inequality. It shows that if measurement settings for Alice (A) and Bob (B) are restricted to intervals on the real number line, then the probability that a local hidden variable (LHV) model reproduces the quantum prediction of E(a,b) = -1 for the CHSH setting (a,b) = (1,1) is greater than zero. This is due to the existence of LHV models where the integrals evaluating E(x,y) for different settings are equal to each other, violating the CHSH inequality.
Main aim of this article is the discussion of Univalent complex functions, Cap like complex functions, and star like complex functions, close-to-cap like complex functions.
This document contains mathematical expressions and logical steps involving inequalities. It begins by defining an inequality constraint on a variable x from -2 to 18. It then proceeds through several steps of algebraic manipulations and logical deductions to derive new constraints on x based on analyzing different cases for the values of x. These cases are broken into thresholds such as x > 4, 2 + 2 < x < 4, and -1 < x < 0, with the constraints on x determined within each range.
A Boundary Value Problem and Expansion Formula of I - Function and General Cl...IJERA Editor
In the present paper, we make a model of a boundary value problem and then obtain its solution involving
products of I -function and a general class of polynomials.
The document is a mathematics examination with 7 multiple choice questions:
1) It asks which set of ordered pairs is a function.
2) It asks which quadrant the point (-5,6) is located in.
3) It asks for the coordinates of a point located 4 units left of the y-axis and 3 units above the x-axis.
4) It asks for the domain of a set of ordered pairs.
5) It asks to evaluate a function f(x)=2x^2 - 3 at x=-2.
6) It asks for the range of a set of ordered pairs.
7) It asks to evaluate a function f(x
This document describes the use of visual art in counseling and therapy. It discusses how art allows individuals to symbolically express conflicts in a unique way. The document outlines the phases of "layer by layer work" where clients use different materials to create art and add dialogue to uncover deeper meanings and layers. Towards the end of sessions, clients present their work to the group to honor each individual's process without questioning or explanations. Visual art taps into the unconscious and inspires growth and connection for many populations, especially when combined with other creative arts.
The OŠ Jure Kaštelana was originally founded in 1967 as the Primary School 25th of May and was renamed in 1991 to Primary school Jure Kastelan to honor Jure Kastelan on his birthday. The school currently has 532 students and 63 employees including 44 teachers and participates in eco projects and international programs like Comenius and Erasmus+.
+2 maths pta question bank come book english mediumMdk Raja
This document contains study material for matrices and determinants, and vector algebra for class 12.
For matrices and determinants, it provides examples and exercises on finding the adjoint and inverse of matrices, solving homogeneous and non-homogeneous systems of linear equations using matrix inversion and determinant methods, consistency of systems of equations, and word problems involving systems of linear equations.
For vector algebra, it gives exercises to prove various vector identities involving addition, subtraction and multiplication of vectors, and word problems involving vectors. The exercises cover topics like verifying properties of vectors under rotation, solving problems related to lines and planes.
The material is intended to help students prepare for 10 mark questions that may be asked in their full test on matrices
Certain D - Operator for Srivastava H B - Hypergeometric Functions of Three V...inventionjournals
The aim of this paper is to derive certain relations involving Srivastava’s hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, which express the aforementioned H B - hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell hypergeometric functions. Other closely-related results are also considered briefly. Also, we have derived certain new integral representations for the H B - hypergeometric functions of three variables defined earlier by Lauricella [ 8 ] and Srivastava [14 ] .
Decomposition formulas for H B - hypergeometric functions of three variablesinventionjournals
: In this paper we investigate several decomposition formulas associated with hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 5 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell's hypergeometric functions.
Solving Poisson’s Equation Using Preconditioned Nine-Point Group SOR Iterativ...inventionjournals
The document describes a preconditioned nine-point group successive over-relaxation (SOR) iterative method for solving Poisson's equation. Poisson's equation is discretized using a finite difference scheme, resulting in a large sparse system of linear equations. The nine-point group SOR method is formulated for solving this system. A new type of preconditioner is then applied to the nine-point group SOR method to improve its rate of convergence. Numerical results demonstrate that the preconditioned nine-point group SOR method has improved convergence properties compared to the unpreconditioned version.
The document describes a Perfect Bayesian Equilibrium model. It involves two players (P1 and P2) who choose between two actions (a1 and a2). The players have different payoffs depending on the joint action. The model specifies the players' types (u1 and u2), actions, payoffs, and beliefs to represent a Perfect Bayesian Equilibrium where each player chooses the optimal action given their beliefs about the other player's action. It also describes how to calculate the expected utility for each player based on the other player's strategy and the equilibrium concept requires strategies to be optimal given beliefs.
International journal of applied sciences and innovation vol 2015 - no 1 - ...sophiabelthome
This document presents a derivative-free cubic convergent extrapolated Newton's method for finding zeros of nonlinear equations. The method is a two-step iterative process that approximates derivatives using backward difference approximations, requiring three functional evaluations per iteration. It is proven to have third-order convergence like the extrapolated Newton's method. Numerical examples are given to illustrate the efficiency of the new method compared to Newton's method and extrapolated Newton's method.
This document contains sections and problems related to vectors and geometry in space. It covers vectors in the plane, space coordinates and vectors in space, the dot and cross products of vectors, lines and planes in space, surfaces in space, and cylindrical and spherical coordinates. There are review exercises and problem solving questions provided at the end.
This document is a solutions manual for a textbook on communication systems. It provides step-by-step solutions to problems from each chapter of the textbook. The problems cover topics such as signal representations using Fourier series and integrals, power calculations for periodic signals, and bandpass signal representations. The solutions demonstrate techniques for analyzing and working with signals commonly encountered in electrical communication systems.
Boundary value problem and its application in i function of multivariableAlexander Decker
This document presents a mathematical model of a boundary value problem and its application to the I-function of multivariable. It defines the boundary value problem for temperature distribution in a rectangular plate under given boundary conditions. It then evaluates the solution of the boundary value problem using the I-function of multivariable, involving products of general polynomials. Several integral formulas are used to derive the solution as a sum involving arbitrary functions and constants.
This document provides 30 equations and inequalities and asks the reader to solve them on the set of real numbers. It uses variables like x, square roots, exponents, and basic arithmetic operations. The problems range from simple one-variable equations to more complex expressions with multiple variables. The goal is to calculate the value(s) of the variable(s) that satisfy each equation or inequality.
Communication systems solution manual 5th editionTayeen Ahmed
This document contains solutions to problems from Communications Systems, 5th edition. It includes solutions for time domain representations of pulses, their frequency domain counterparts obtained using Fourier transforms, properties of even and odd functions, and bounds on the bandwidth of time-limited signals. The solutions demonstrate properties like linearity, time-shifting, differentiation in the time and frequency domains, and the relationships between a function and its Fourier transform.
This document contains solutions to various equations and inequalities involving radicals on the set of real numbers. It is divided into 6 sections, with multiple problems provided in each section ranging from simple single-term radical equations to more complex multi-term radical equations and inequalities. The document provides the step-by-step workings for solving each problem.
This document contains mathematical formulas and definitions from various topics including:
- Set operations and identities involving sets A, B, and C.
- Properties of rational numbers including ratios in lowest terms.
- Algebra topics such as FOIL, factoring formulas, and special products involving binomials.
- Properties of fractions, equations, inequalities, and their relationships.
- Formulas for linear equations, the quadratic formula, distance, midpoint, and equations of circles.
- Properties that classify functions as even or odd based on their behavior as the input variable x approaches negative values.
The document presents a probability loophole in the CHSH inequality. It shows that if measurement settings for Alice (A) and Bob (B) are restricted to intervals on the real number line, then the probability that a local hidden variable (LHV) model reproduces the quantum prediction of E(a,b) = -1 for the CHSH setting (a,b) = (1,1) is greater than zero. This is due to the existence of LHV models where the integrals evaluating E(x,y) for different settings are equal to each other, violating the CHSH inequality.
Main aim of this article is the discussion of Univalent complex functions, Cap like complex functions, and star like complex functions, close-to-cap like complex functions.
This document contains mathematical expressions and logical steps involving inequalities. It begins by defining an inequality constraint on a variable x from -2 to 18. It then proceeds through several steps of algebraic manipulations and logical deductions to derive new constraints on x based on analyzing different cases for the values of x. These cases are broken into thresholds such as x > 4, 2 + 2 < x < 4, and -1 < x < 0, with the constraints on x determined within each range.
A Boundary Value Problem and Expansion Formula of I - Function and General Cl...IJERA Editor
In the present paper, we make a model of a boundary value problem and then obtain its solution involving
products of I -function and a general class of polynomials.
The document is a mathematics examination with 7 multiple choice questions:
1) It asks which set of ordered pairs is a function.
2) It asks which quadrant the point (-5,6) is located in.
3) It asks for the coordinates of a point located 4 units left of the y-axis and 3 units above the x-axis.
4) It asks for the domain of a set of ordered pairs.
5) It asks to evaluate a function f(x)=2x^2 - 3 at x=-2.
6) It asks for the range of a set of ordered pairs.
7) It asks to evaluate a function f(x
This document describes the use of visual art in counseling and therapy. It discusses how art allows individuals to symbolically express conflicts in a unique way. The document outlines the phases of "layer by layer work" where clients use different materials to create art and add dialogue to uncover deeper meanings and layers. Towards the end of sessions, clients present their work to the group to honor each individual's process without questioning or explanations. Visual art taps into the unconscious and inspires growth and connection for many populations, especially when combined with other creative arts.
The OŠ Jure Kaštelana was originally founded in 1967 as the Primary School 25th of May and was renamed in 1991 to Primary school Jure Kastelan to honor Jure Kastelan on his birthday. The school currently has 532 students and 63 employees including 44 teachers and participates in eco projects and international programs like Comenius and Erasmus+.
Zagreb is the capital and largest city of Croatia, with a population of around 790,000. It holds the leading economic and political position in the country. The first recorded mention of Zagreb's name was in 1094, and in 1242 it received autonomy under the Golden Bull.
The document summarizes a lesson about the human body and movement. Students worked in groups to name and describe parts of the human face and body. They then practiced controlling their movements and improvising tasks. The lesson used lectures, questions, discussions and group work. Students were split into colored groups to complete tasks about the body. They also learned and performed a dance. Finally, students created movements for different body parts and presented them for informal self-assessment.
Creative Arts in Counseling: Chapter 5 Presentation by Jera DykesJera Dykes
This document provides an overview of how visual arts are used in counseling. It discusses the pioneers of this approach and some key benefits, such as helping clients express unconscious conflicts symbolically. Various techniques are described for different populations across the lifespan, from drawing tests with children to using photography with adults. Examples of activities include body outlines, serial drawings, clay work, and using published pictures. The document emphasizes that the creative process is more important than the artwork itself. Visual arts can be combined with other modalities like writing, music, and psychodrama. Overall, it positions visual arts therapy as a helpful approach for many clients and populations.
This document discusses dance therapy and its use as a method of self-knowledge. It describes how dance therapy focuses on connecting the self, others, and one's environment through space. The body stores memories and experiences from life, and dance therapy can help reveal insights from movement. Benefits include increased self-awareness, self-esteem, communication skills, and understanding of one's strengths and weaknesses. Dance is discussed as a tool for personal development and alternative form of communication.
This document discusses how dance can be integrated into the ESL classroom to benefit kinesthetic learners. It provides several examples of dance activities that target different language skills, such as having students physically represent "being" words or using parts of speech like verbs and adverbs to create dances. Interviewing dance educator Carolee Bongiorno, it outlines her philosophy of making students feel comfortable expressing themselves through dance to boost engagement and language learning. The document advocates that with commitment and respect for students, teachers can successfully incorporate dance into language lessons regardless of dance experience.
Dr. Miriam Galindo uses play and art therapy in her private clinical psychology practice to counsel children. As children often express themselves more clearly through art than words, art therapy allows children to represent and discuss life issues, feelings, and traumas. It also helps distract children from struggles and acts as a stress reliever while developing coping skills, such as allowing physically abused children to communicate painful experiences that are otherwise difficult to discuss.
Dance therapy can benefit people with physical and emotional disabilities. It improves motor skills, balance, muscle coordination and helps develop coping skills through movement and dance. Dance therapy has been used since the 1940s to express and deal with feelings in a psychological treatment. It can address both physical and emotional needs by strengthening the body and helping people move through difficult emotions. Both physical and emotional dance therapy can treat conditions like autism and learning disabilities.
The document discusses the key concepts of guidance and counseling. It defines guidance as the process of assisting individuals to determine their abilities and make wise choices. Counseling is defined as a process where a counselor helps a counselee interpret facts to make decisions. The document outlines various principles of guidance, including organizing services based on student needs and interests, providing continuous services, and ensuring services address the whole student.
Art therapy is a therapeutic process that uses art-making to increase self-awareness, cope with symptoms, and enhance well-being. While art therapy requires specific training, other mental health clinicians can incorporate art into treatment by following guidelines. Clinicians should provide a safe environment, understand materials, obtain informed consent, and know when to consult an art therapist. Art-making can be therapeutic for clients and clinicians through creative self-expression.
Expressive art therapy uses various forms of creative expression like art, music, and play as alternatives to traditional talk therapy for children. It aims to improve social skills and behavior through nonverbal communication. The document recommends more research on established expressive art therapy programs with larger sample sizes and case studies, especially for treating conditions like autism.
Art therapy can help children with a variety of issues such as death of a loved one, abuse, learning disabilities, emotional issues, and more. It combines traditional psychotherapy techniques with creative art processes to improve self-awareness, self-esteem, and positive behaviors. To become an art therapist requires a bachelor's degree in psychology or art, then a master's degree with credits in both fields and an art therapy certificate. Financial aid like grants, scholarships, loans, and work-study can assist with education costs. Art therapists earn a typical salary range of $30,000-$50,000 but some earn up to $149,000.
Art therapy uses the creative process of art making to improve physical, mental, and emotional well-being. It is based on the belief that creative self-expression through art helps people resolve conflicts, develop skills, and increase self-esteem. The document discusses the history and founders of art therapy including Freud, Jung, Naumberg, and Kramer. It then describes the speaker's work using art therapy at an outpatient substance abuse treatment center, giving examples of techniques used like body maps, anger lamps, and altered books to help patients process emotions, reduce stress, and work through trauma.
Dance Movement Therapy with clients with Eating DisordersMonarch Cove
In this webinar, Dr. Marybeth Weinstock discusses the use of dance movement therapy in eating disorder treatment. Marybeth is the DMT for Monarch Cove Eating Disorder Treatment Center in Pacific Grove, California
This document discusses different types of counselling:
1) Directive counselling is counsellor-centered where the counsellor directs the client to resolve issues by informing and advising them.
2) Non-directive counselling is client-centered where the counsellor creates an environment for the client to work out their own problems through free expression of feelings and developing insight.
3) Eclectic counselling combines elements of directive and non-directive counselling by taking a flexible middle approach depending on the client and situation. The counsellor plays an active but not overbearing role.
The document discusses natural coordinates, also called area coordinates, for a three-node triangular element in finite element analysis. It defines the natural coordinates (L1, L2, L3) of any point P within the triangle in terms of the Cartesian coordinates of the three nodes and the fractions of the total triangle area (A1, A2, A3) associated with each node. The natural coordinates represent the point P shifted to each node and satisfy the properties L1 + L2 + L3 = 1 and L1, L2, L3 ≥ 0.
IRJET- On Certain Subclasses of Univalent Functions: An ApplicationIRJET Journal
This document presents research on certain subclasses of univalent functions. Specifically, it studies the class WR(λ,β,α,μ,θ) which consists of analytic and univalent functions with negative coefficients in the open disk, defined by the Hadamard product with the Rafid operator. The paper obtains coefficient bounds and extreme points for this class. It also examines the weighted mean, arithmetic mean, and derives some additional results for the class.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document contains the solutions to a final exam for a communication networks course. It includes multiple problems involving calculating metrics like average transmissions, generating functions, and mean calculations for queueing models. It also contains a multicast routing problem that involves building a minimum cost tree and calculating its cost.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Generalised Class of Unbiased Seperate Regression Type Estimator under Stra...IOSRJM
In this paper a generalized class of regression type estimators using the auxiliary information on population mean and population variance is proposed under stratified random sampling. In order to improve the performance of the proposed class of estimator, the Jack-knifed versions are also proposed. A comparative study of the proposed estimator is made with that of separate ratio estimator, separate product estimator, separate linear regression estimator and the usual stratified sample mean. It is shown that the estimators through proposed allocation always give more efficient estimators in the sense of having smaller mean square error than those obtained through Neyman Allocation
Finite Triple Integral Representation For The Polynomial Set Tn(x1 ,x2 ,x3 ,x4 )iosrjce
Recently,we introduced “An unification of certain generalized Geometric polynomial Set T
n
(x1
,x2
,x3
,x4
) ,with the help of generating function which contains Appell function of four variables in
the notation of Burchnall and Choundy [4] associated with Lauricella function. This generated
hypergeometric polynomial Set covers as many as thirty four orthogonal and non -orthogonal
polynomials.In the present paper an attempt has been made to express a Triple finite integral
representation of the polynomial set T
n
(x1
,x2
,x3
,x4
).
Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)paperpublications3
Abstract: Ramanujan’s lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously many authors proved the first six of Ramanujan’s tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan’s tenth order mock theta function identities which are expressed by mock theta functions and also a definite integral. The properties of modular forms are used for the proofs of theta function identities and L. J. Mordell’s transformation formula for the definite integral.Keywords: Mock Theta Functions from Ramanujan’s Lost Notebook.
Title: Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)
Author: MOHAMMADI BEGUM JEELANI SHAIKH
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
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International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
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1. ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 10 || October – 2014 ||
International Journal of Computational Engineering Research (IJCER)
www.ijceronline.com Open Access Journal Page 24
On The Derivatives and Partial Derivatives of A Certain
Generalized Hyper geometric Function
1
Yashwant Singh, 2
Nandavasanthnadiger Kulkarni
1
Department of Mathematics, Government College, Kaladera, Jaipur, Rajasthan, India
2
Department of Mathematics, Dayananda Sagar college of arts, science and commerce, Kumaraswamy layout,
Bangalore, Karnataka, India
I. INTRODUCTION
The H -function of two variables defined and represented by Singh and Mandia [12] in the following manner:
1 2 2 3 2 1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1 1 2 2 2 2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, ; , , ; , , , , ; , ,, : , : ,
, : , ; ,
, ; , , , , ; , , , , ;
,
j j j j j j j j j j j j j
p n n p n n p
j j j j j j j j j j j j j
q m m q m m q
a A c K c e E R e Eo n m n m n
x x
p q p q p qy y b B d d L f F f F S
H x y H H
=
1 2
1 2 32
1
, ( ) ( )
4 L L
x y d d
(1.1)
Where
1
1 1
1
1
1
1 1
1
,
1
n
j j j
j
p q
j j j j j j
j n j
a A
a A b B
(1.2)
2 2
2 2
2 2
1 1
2
1 1
1
1
j
j
n m
K
j j j j
j j
p q
L
j j j j
j n j m
c d
c d
(1.3)
3 3
3 3
3 3
1 1
3
1 1
1
1
j
j
n m
R
j j j j
j j
p q
S
j j j j
j n j m
e E f F
e E f F
(1.4)
ABSTRACT
In this paper, methods involving derivatives and the Mellin transformation are employed in obtaining finite
summations for the H -function of two variables and certain special partial derivatives for the H -function
of two variables with respect to parameters.
KEY WORDS: Derivatives, Partial derivatives, H -function of two variables, Mellin transformation.
(2000 Mathematics Subject Classification: 33C99)
2. On The Derivatives and Partial Derivatives…
www.ijceronline.com Open Access Journal Page 25
Where x and y are not equal to zero (real or complex), and an empty product is interpreted as unity
, , ,i i i j
p q n m are non-negative integers such that 0 , ( 1, 2, 3; 2, 3)i i j j
n p o m q i j . All the
1 1 2 2
( 1, 2, ..., ), ( 1, 2, ..., ), ( 1, 2, ..., ), ( 1, 2, ..., ),j j j j
a j p b j q c j p d j q
3 3
( 1, 2, ..., ), ( 1, 2, ..., )j j
e j p f j q are complex
parameters. 2 2
0 ( 1, 2, ..., ), 0 ( 1, 2, ..., )j j
j p j q (not all zero simultaneously), similarly
3 3
0 ( 1, 2, ..., ), 0 ( 1, 2, ..., )j j
E j p F j q (not all zero simultaneously). The exponents
3 2 2 3 3 3
( 1, 2, ..., ), ( 1, ..., ), ( 1, 2, ..., ), ( 1, ..., )j j j j
K j n L j m q R j n S j m q can take on non-
negative values.
The contour 1
L is in -plane and runs from i to i . The poles of 2
( 1, 2, ..., )j j
d j m lie to
the right and the poles of 2 1
1 ( 1, 2, ..., ), 1 ( 1, 2, ..., )
jK
j j j j j
c j n a A j n to
the left of the contour. For 2
( 1, 2, ..., )j
K j n not an integer, the poles of gamma functions of the numerator
in (1.3) are converted to the branch points.
The contour 2
L is in -plane and runs from i to i . The poles of 3
( 1, 2, ..., )j j
f F j m lie to
the right and the poles of 3 1
1 ( 1, 2, ..., ), 1 ( 1, 2, ..., )
jR
j j j j j
e E j n a A j n to
the left of the contour. For 3
( 1, 2, ..., )j
R j n not an integer, the poles of gamma functions of the numerator in
(1.4) are converted to the branch points.
The functions defined in (1.1) is an analytic function of x and y , if
1 2 1 2
1 1 1 1
0
p p q q
j j j j
j j j j
U
(1.5)
3 31 1
1 1 1 1
0
p qp q
j j j j
j j j j
V A E B F
(1.6)
The integral in (1.1) converges under the following set of conditions:
1 1 2 2 2 2 1
1 2 21 1 1 1 1 1 1
0
n p m q n p q
j j j j j j j j j
j j n j j m j j n j
L K
(1.7)
3 31 1 2 2 1
1 2 21 1 1 1 1 1 1
0
n pn p m q q
j j j j j j j j j
j j n j j m j j n j
A A F F S E R E B
(1.8)
1 1
| arg | , | arg |
2 2
x y (1.9)
The behavior of the H -function of two variables for small values of | |z follows as:
3. On The Derivatives and Partial Derivatives…
www.ijceronline.com Open Access Journal Page 26
[ , ] 0 (| | | | ), m ax | |, | | 0H x y x y x y
(1.10)
Where
21
m in R e
j
j m
j
d
21
m in R e
j
j m
j
f
F
(1.11)
For large value of | |z ,
' '
[ , ] 0 | | , | | , m in | |, | | 0H x y x y x y
(1.12)
Where
21
1
' m ax R e
j
j
j n
j
c
K
,
31
1
' m ax R e
j
j
j n
j
e
R
E
(1.13)
Provided that 0U and 0V .
If we
take 2 2 2 3 3 3
1( 1, 2, ..., ), 1( 1, ..., ), 1( 1, 2, ..., ), 1( 1, ..., )j j j j
K j n L j m q R j n S j m q i
n (2.1), the H -function of two variables reduces to H -function of two variables due to [9].
If we set 1 1 1
0n p q , the H -function of two variables breaks up into a product of two H -function of
one variable namely
2 2 3 3 1 , 1 , 1 , 1 ,2 2 3 3 3
2 2 3 3
1 , 1 , 1 , 1 ,2 2 2 3 3 3
: , ; , , : , ; , ,0 ,0: , : ,
0 ,0: , : ,
: , , , ; : , , , ;
j j j j j j j j j j
n n p n n p
j j j j j j j j j j
m m q m m q
c K c e E R e Em n m n
x
p q p q y d d L f F f F S
H
=
2 2 3 3 1 , 1 ,1 , 1 , 3 3 32 2 2
2 2 3 3
1 , 1 , 1 , 1 ,2 2 2 3 3 3
, ; , ,, ; , ,, ,
, ,
, , , ; , , , ;
j j j j jj j j j j
n n pn n p
j j j j j j j j j j
m m q m m q
e E R e Ec K cm n m n
p q p q
d d L f F f F S
H x H y
(1.14)
If 0 , we then obtain
1 2 2 3 3 1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1 1 2 2 3 3
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, ; , , ; , , , , ; , ,0 , : , : ,
2
, : , : ,
, ; , , , , ; , , , , ;
j j j j j j j j j j j j j
p n n p n n p
j j j j j j j j j j j j j
q m m q m m q
a A c K c e E R e En m n m n
x
p q p q p q
y b B d d L f F f F S
H
=
1 2 2 3 3 1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1 1 2 2 3 3
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, ; , , ; , , , , ; , ,0 , : , : ,
, : , : ,
, ; , , , , ; , , , , ;
j j j j j j j j j j j j j
p n n p n n p
j j j j j j j j j j j j j
q m m q m m q
a A c K c e E R e En m n m n
x
p q p q p q y b B d d L f F f F S
H
(1.15)
1 2 2 3 3 1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1 1 2 2 3 3
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, ; , , ; , , , , ; , ,10 , : , : ,
, : , : , 1 , ; , , , , ; , , , , ;
j j j j j j j j j j j j j
p n n p n n p
j j j j j j j j j j j j j
q m m q m m q
a A c K c e E R e En m n m n
x
p q p q p q
b B d d L f F f F S
y
H
=
1 2 2 3 3 1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1 1 2 2 3 3
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1 , ; , 1 , , 1 , ; , 1 , , 1 , ;0 , : , : ,
, : , : ,
1 , ; , 1 , ; , 1 , , 1 , ; , 1 ,
j j j j j j j j j j j j j
q m m q m m q
j j j j j j j j j j j j j
p n n p n n p
b B d d L f F f F Sn m n m n
x
p q p q p q y a A c K c e E R e E
H
(1.16)
4. On The Derivatives and Partial Derivatives…
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II. MAIN RESULTS
If t be an arbitrary parameter and ', '' be positive real numbers, then it can be verified that
1 2 2 3 2
1 1 2 2 2 2
'
1 , 1 ,1 21
''
2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, 1: , : ,
'' ' '' 3
1, 1: , ; ,1 2
(1 ; ', ''), , ; , , ; , ,
, ; ,(1 ; ', ''), , , , ; , , , , ;
,
j j j j j j j j
p n n
j j j j j j j j j j j j j
q m m q m m q
o n m n m n
e
p q p q p q
e n a A c K c
z t
z t b B e d d L f F f F S
D H z t z t t H
1 , 1 , 1 ,2 2 3 3 3
, , ; , ,j j j j j
p n n p
e E R e E
(2.1)
And
'2 2 3 3 1 , 1 , 1 , 1 ,2 2 3 3 31
''2 2 3 3
2
1 , 1 , 1 , 1 ,2 2 2 3 3 3
: , ; , , : , ; , ,0 ,0: , : ,
1
0 ,0: , : ,
: , , , ; : , , , ;
j j j j j j j j j j
n n p n n p
j j j j j j j j j j
m m q m m q
c K c e E R e Em n m n
z te
p q p q
z t d d L f F f F S
t H
=
2 2 3 3 1 , 1 ,1 , 1 , 3 3 32 2 2
2 2 3 3
1 , 1 , 1 , 1 ,2 2 2 3 3 3
( 1) ( 1)
, ; , ,, ; , ,, ,
' ''2 2
, ,1 2, , , ; , , , ;
j j j j jj j j j j
n n pn n p
j j j j j j j j j j
m m q m m q
e e
e E R e Ec K cm n m n
p q p q
d d L f F f F S
t H z t t H z t
(2.2)
Differentiating (2.2) two times w.r.t. t and simplifying, it follows by induction that
2 2 3 3
2 2 3 3
1 2
1 2
' 1
1
''
2
1 , 1 , 1 , 1 ,2 2 2 3 3 3
0 ,0: , 1: , 1
0 ,0: 1, 1: 1, 1
, 0 1 2
1
: 1 , ';1 , ;
2
1 1
: , , , ; , 1 , ';1 : , , , ; , 1 , '';1
2 2
!
! !
j j
j j j j j j j j j j
m m q m m q
n
m n m n
p q p q
n n
n n n
e
n c K
z t
e ez t
d d L f F f F S
n
H
n n
2
1 , 1 , 1 , 1 ,2 2 3 3 3
1
, , : 1 , '';1 , ; , ,
2
j j j j j j j j
n n p n n p
e
c n e E R e E
(2.3)
(2.3)readily admits an extension and we have
1 2 2 3 2
1 1 2 2 2 2
1 2
1 2
' 1
1 , 11
''
2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, : , 1: , 1
, : 1, 1; 1, 1
, 0 1 2
, ; , 1
1 1
, ; , , , , ; , 1 , ';1 , , , , ; , 1 , '';1
2 2
!
! !
j j j
p
j j j j j j j j j j j j j
q m m q m m q
n
o n m n m n
p q p q p q
n n
n n n
a A n
z t
e ez t
b B d d L f F f F S
n
H
n n
2
1 , 1 , 1 , 1 ,2 2 2 3 3 3
1 1
, ';1 , , ; , , , 1 , '';1 , , ; , ,
2 2
j j j j j j j j j j
n n p n n p
e e
c K c n e E R e E
(2.4)
Considering various other forms that (2.1)admits, similar other results can be obtained.
In the next place, in view of (2.1) we note that the 2-dimensional Mellin-transformation ([6],11.2)
''
M of the
H -function of two variables is given by
''( ) ( , )M H Q
Provided
2 2
1 1
1
m in R e m ax R e
j j
j
j m j n
j j
d c
K
'2 2 3 3 1 , 1 , 1 , 1 ,2 2 3 3 31
''2 2 3 3
2
1 , 1 , 1 , 1 ,2 2 2 3 3 3
(1 ; ', ''): , ; , , : , ; , ,0 ,1: , : ,
1,1: , : ,
(1 ; ', ''): , , , ; : , , , ;
j j j j j j j j j j
n n p n n p
j j j j j j j j j j
m m q m m q
e n c K c e E R e Em n m n
z t
p q p q
z t e d d L f F f F S
H
5. On The Derivatives and Partial Derivatives…
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3 3
1 1
1
m in R e m ax R e
j j
j
j m j n
j j
f e
R
F E
We also note that, since (1.7 (30) of Erdelyi [7]) for a positive integer N ,
1
1
( 1) ! ( ) '( )
( ) ( ) ; ( ) ,
( ) ! ( ) ( )
kN
k
N a a
a N a a
k N k a k a
Partial differentiation of the gamma product 1 ' '' 1 ' ''
2 2
e e
w.r.t. the
arbitrary parameter e at e N can be expressed as a finite sum
1 ' '' 1 ' ''
2 2
e N
e e
e
1
1
1 ' ''
1 ( 1) ! 2
1 ' ''
2 2 ( ) !
1 ' ''
2
kN
k
N
N N
Nk N k
k
,
Where ', '' are positive real numbers.
Thus for 0, 0,n N we have
1 2 2 3 2
1
1 1 2 2 2 2
2
1 , 1
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, 2: , : ,
2 , : , ; ,
; ', '' , ; ', '' , , ; ,, ,
2 2
1 1
, ; , , , , ; , 1 , ';1 , , , , ; , 1 , '';1
2 2
''
j j j j j
p
j j j j j j j j j j j j j
q m m q m m q
o n m n m n
z
p q p q p q z
e e
a A c
e e
b B d d L f F f F S
M H
2
1 , 1 , 1 , 1 ,2 2 2 3 3 3
1
; , , , 1 , '';1 , , ; , ,
2
j j j j j j j j
n n p n n p
e
K c n e E R e E
e N
1
1
1 ' '' 1 ' ''
! ( 1) ! 2 2
2 ( ) !
1 ' ''
2
kN
k
N N
N N
Nk N k
k
( , )Q
1 2 2 3 2
1
1 1 2 2 2 2
2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
1
, 3: , : ,
3 , 1: , ; ,
1
; ', '' , ; ', '' , ; ', ''
2 2 2
, ; , ; ', '' , , , , ; , , , , ;
2
! ( 1)
''
2 ( ) !
j j j j j j j j j j j j j
q m m q m m q
kN
o n m n m n
z
p q p q p q z
k
N N N
N
b B k d d L f F f F S
N
M H
k N k
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, , ; ,, , ; , , , , ; , ,j j j j j j j j j j j j j
p n n p n n p
a A c K c e E R e E
(2.5)
But for
( )
( )
, 1, 2
i
i
z u i
, (2.5) can be written as
6. On The Derivatives and Partial Derivatives…
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1 2 2 3 2
1
1 1 2 2 2 2
2
1 , 1 , 1 , 1 ,1 2 2 2 3
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, 2: , : ,
2 , : , ; ,
; ', '' , ; ', '' , , ; ,, , ; , , , , ; , ,
2 2
, ; , , , , ; , , , , ;
j j j j j j j j j j j j
p n n p n
j j j j j j j j j j j j j
q m m q m m q
o n m n m n
z
p q p q p q z
e e
a A c K c e E R e
b B d d L f F f F S
H
e
1 ,3 3
j
n p
E
e N
2
1
! ( 1)
2 ( ) !
N
NN
k
N u
k N k
1 2 2 3 2
1
1 1 2 2 2 2
2
1 , 1 , 1 ,1 2 2 2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, 2: , : ,
2
2 , : , ; ,
; ', '' , ; ', '' , , ; ,, , ; , , , , ;
2 2
, ; , , , , ; , , , , ;
j j j j j j j j j j
p n n p
j j j j j j j j j j j j j
q m m q m m q
N
k o n m n m n
zN k
p q p q p qu z
N N
a A c K c e E
b B d d L f F f F S
D u H
1 , 1 ,3 3 3
, ,j j j
n n p
R e E
(2.6)
If we express the derivative into a sum, carry out the differentiations, interchange the order of summation and
simplify, we obtain
1 2 2 3 2
1
1 1 2 2 2 2
2
1 , 1 , 1 , 1 ,1 2 2 2 3
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, 2: , : ,
2 , : , ; ,
; ', '' , ; ', '' , , ; ,, , ; , , , , ; , ,
2 2
, ; , , , , ; , , , , ;
j j j j j j j j j j j j
p n n p n
j j j j j j j j j j j j j
q m m q m m q
o n m n m n
z
p q p q p q z
e e
a A c K c e E R e
b B d d L f F f F S
H
e
1 ,3 3
j
n p
E
e N
1 2 2 3 2
1 1 2 2 2 2
1
, 2: , : ,
2 , : , ; ,
0
! ( 1)
2 !( ) !
N
o n m n m n
p q p q p q
p
N
H
p N p
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 31
2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
; ', '' , ; ', '' , , ; ,, , ; , , , , ; , ,
2 2
, ; , , , , ; , , , , ;
j j j j j j j j j j j j j
p n n p n n p
j j j j j j j j j j j j j
q m m q m m q
N N
p a A c K c e E R e E
z
z b B d d L f F f F S
(2.7)
Similar other results can be obtained by considering products or quotients of such gamma functions whose
partial derivatives w.r.t. the arbitrary parameter involved can be expressed as a finite sum.
For example, for the quotient
(1 ' '' )
(1 ' '' )
e N
e
,
We have
1 2 2 3 2
1
1 1 2 2 2 2
2
1 , 1 , 1 , 11 2 2 2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
, 1: , : ,
1, 1: , ; ,
; ', '' , ; ', '' , , ; ,, , ; , , , , ;
2
, ; ,( ; ', ''), , , , ; , , , , ;
j j j j j j j j j j j
p n n p
j j j j j j j j j j j j j
q m m q m m q
o n m n m n
z
p q p q p q z
e
e N a A c K c e E R
b B e d d L f F f F S
H
e
, 1 ,3 3 3
, ,j j
n n p
e E
7. On The Derivatives and Partial Derivatives…
www.ijceronline.com Open Access Journal Page 30
1 2 2 3 2
1 1 2 2 2 2
1
, 1: , : ,
1, 1: , ; ,
0
( 1)
!
( ) !
kN
o n m n m n
p q p q p q
k
N H
p N k
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 31
2
1 , 1 , 1 , 1 , 1 ,1 2 2 2 3 3 3
; ', '' , , ; , , ; , , , , ; , ,
, ; ,( ; ', ''), , , , ; , , , , ;
j j j j j j j j j j j j j
p n n p n n p
j j j j j j j j j j j j j
q m m q m m q
e N a A c K c e E R e E
z
z b B e k d d L f F f F S
(2.8)
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