Application Of The Theory Of Linear Singular Integral Equations And Contour Integration To Riemann Hilbert Problems For Determination Of New Decoupled Expressions
This document discusses applying the theory of linear singular integral equations to represent Chandrasekhar's H-functions, which are important in the study of radiative transfer, in a new form. The author derives a linear non-homogeneous singular integral equation for H-functions from the non-linear integral equation satisfied by H-functions. Plemelj's formulae are then applied to obtain a new linear integral equation for H-functions in the complex plane. This new representation of H-functions is not dependent on H-functions within the integral, unlike previous representations. The author shows the equivalence of applying the theory of linear singular integral equations to this problem and applying the Wiener-Hopf technique to the linear integral equation, as
This document discusses counterexamples to classical calculus theorems like the Inverse and Implicit Function Theorems in the context of Scale Calculus. Scale Calculus generalizes multivariable calculus to infinite-dimensional spaces and is foundational to Polyfold Theory. The authors construct examples of scale-differentiable and scale-Fredholm maps whose differentials are discontinuous when the base point changes, disproving naive extensions of the classical theorems. However, they show the scale-Fredholm notion in Polyfold Theory ensures continuity of differentials in specific coordinates, justifying its technical complexity and allowing for a perturbation theory without the classical theorems.
The document presents a criterion for determining when a pair of weighted composition operators acting on a Hilbert space of analytic functions satisfies the Hypercyclicity Criterion. Specifically:
1. The Hypercyclicity Criterion is introduced and shown to be a sufficient condition for hypercyclicity of operator pairs.
2. A theorem is presented proving that if two weighted composition operators satisfy certain conditions regarding their weights and composition map, then their adjoint operators satisfy the Hypercyclicity Criterion and are thus hypercyclic.
3. A corollary shows that the adjoint of a multiplication operator by a non-constant multiplier intersecting the unit circle also satisfies the Hypercyclicity Criterion.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
Digital Text Book :POTENTIAL THEORY AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS Baasilroy
This document summarizes potential theory and elliptic partial differential equations. It discusses Laplace's equation, which is a second order partial differential equation that is elliptic in nature. The document covers solutions to Laplace's equation, including the general solution for two and three variables. It also discusses Green's identities and theorems related to harmonic functions, including the maximum principle, minimum principle, and uniqueness theorem.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
This document summarizes key concepts related to solving the Schrodinger equation for the hydrogen atom. It begins by introducing the hydrogen atom as an important system to study and then presents the time-independent Schrodinger equation in spherical coordinates. It identifies the complete set of commuting observables for the hydrogen atom as the Hamiltonian, angular momentum, and z-component of angular momentum operators. It then separates the radial and angular dependencies of the wavefunction and shows that this leads to separate differential equations that can be solved.
Geometric properties for parabolic and elliptic pdeSpringer
This document discusses recent advances in fractional Laplacian operators and related problems in partial differential equations and geometric measure theory. Specifically, it addresses three key topics:
1. Symmetry problems for solutions of the fractional Allen-Cahn equation and whether solutions only depend on one variable like in the classical case. The answer is known to be positive for some dimensions and fractional exponents but remains open in general.
2. The Γ-convergence of functionals involving the fractional Laplacian as the small parameter ε approaches zero. This characterizes the asymptotic behavior and relates to fractional notions of perimeter.
3. Regularity of interfaces as the fractional exponent s approaches 1/2 from above, which corresponds to a critical threshold
This document discusses counterexamples to classical calculus theorems like the Inverse and Implicit Function Theorems in the context of Scale Calculus. Scale Calculus generalizes multivariable calculus to infinite-dimensional spaces and is foundational to Polyfold Theory. The authors construct examples of scale-differentiable and scale-Fredholm maps whose differentials are discontinuous when the base point changes, disproving naive extensions of the classical theorems. However, they show the scale-Fredholm notion in Polyfold Theory ensures continuity of differentials in specific coordinates, justifying its technical complexity and allowing for a perturbation theory without the classical theorems.
The document presents a criterion for determining when a pair of weighted composition operators acting on a Hilbert space of analytic functions satisfies the Hypercyclicity Criterion. Specifically:
1. The Hypercyclicity Criterion is introduced and shown to be a sufficient condition for hypercyclicity of operator pairs.
2. A theorem is presented proving that if two weighted composition operators satisfy certain conditions regarding their weights and composition map, then their adjoint operators satisfy the Hypercyclicity Criterion and are thus hypercyclic.
3. A corollary shows that the adjoint of a multiplication operator by a non-constant multiplier intersecting the unit circle also satisfies the Hypercyclicity Criterion.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
Digital Text Book :POTENTIAL THEORY AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS Baasilroy
This document summarizes potential theory and elliptic partial differential equations. It discusses Laplace's equation, which is a second order partial differential equation that is elliptic in nature. The document covers solutions to Laplace's equation, including the general solution for two and three variables. It also discusses Green's identities and theorems related to harmonic functions, including the maximum principle, minimum principle, and uniqueness theorem.
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
{\displaystyle z=f^{-1}(ze^{z})=W(ze^{z})} z=f^{-1}(ze^{z})=W(ze^{z})
By substituting {\displaystyle z'=ze^{z}} z'=ze^{z} in the above equation, we get the defining equation for the W function (and for the W relation in general):
{\displaystyle z'=W(z')e^{W(z')}} z'=W(z')e^{W(z')}
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions.[1] It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
This document summarizes key concepts related to solving the Schrodinger equation for the hydrogen atom. It begins by introducing the hydrogen atom as an important system to study and then presents the time-independent Schrodinger equation in spherical coordinates. It identifies the complete set of commuting observables for the hydrogen atom as the Hamiltonian, angular momentum, and z-component of angular momentum operators. It then separates the radial and angular dependencies of the wavefunction and shows that this leads to separate differential equations that can be solved.
Geometric properties for parabolic and elliptic pdeSpringer
This document discusses recent advances in fractional Laplacian operators and related problems in partial differential equations and geometric measure theory. Specifically, it addresses three key topics:
1. Symmetry problems for solutions of the fractional Allen-Cahn equation and whether solutions only depend on one variable like in the classical case. The answer is known to be positive for some dimensions and fractional exponents but remains open in general.
2. The Γ-convergence of functionals involving the fractional Laplacian as the small parameter ε approaches zero. This characterizes the asymptotic behavior and relates to fractional notions of perimeter.
3. Regularity of interfaces as the fractional exponent s approaches 1/2 from above, which corresponds to a critical threshold
The document discusses the mathematical background of the finite element method. It begins with an overview of concepts from functional analysis including vector spaces, scalar products, linear functionals, and Riesz representation theorem. Green's functions are introduced as the Riesz elements representing linear functionals. Green's identities are then used to derive classical influence functions for displacements and forces. Physically, Green's functions represent monopoles and dipoles with infinite energy, placing them outside the theory of weak boundary value problems. In nonlinear problems, Green's functions can be viewed as Lagrange multipliers at the linearization point.
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
This document summarizes and reviews results on contraction and extension maps in Hilbert spaces. It discusses important theorems related to representing linear operators using normal operators like self-adjoint and unitary operators. It also covers Neumark's theorem which shows that every generalized spectral family can be represented by the projection of an ordinary spectral family in an extended space. The document applies these results to problems in elasticity theory.
This document discusses equations of motion for higher-order Legendre transforms in quantum field theory. It begins by introducing Legendre transforms and their use in expressing potentials in terms of Green's functions. It then:
1) Derives Schwinger's equation and constraint equations for the generating functional of Green's functions. These determine the Green's functions uniquely.
2) Shows that iterating the equations expresses the generating functional as the sum of all vacuum loops, demonstrating the connected Green's functions are generated.
3) Notes that generalizing this proof to higher-order Legendre transforms would demonstrate their m-irreducibility properties, in line with previous work.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Interpreting Multiple Regressionvia an Ellipse Inscribed in a Square Extensi...Toshiyuki Shimono
The document discusses interpreting multiple regression results geometrically using ellipses. It presents 3 theorems:
1) The multiple correlation coefficient is the ratio of line segments from the origin to a point P inside the ellipse and its projection P' onto the ellipse.
2) Regression coefficients are values of linear scalar fields defined on the axes at point P.
3) Partial correlation coefficients are values of affine functions defined on the longest line segments through P parallel to the axes.
This paper reviews the fundamental concepts and
the basic theory of polarization mode dispersion(PMD) in optical
fibers. It introduces a unified notation and methodology to
link the various views and concepts in jones space and strokes
space. The discussion includes the relation between Jones vectors
and Strokes vectors and how they are used in formulating the
jones matrix by the unitary system matrix.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
Successive approximation of neutral stochastic functional differential equati...Editor IJCATR
We establish results concerning the existence and uniqueness of solutions to neutral stochastic functional differential
equations with infinite delay and Poisson jumps in the phase space C((-∞,0];Rd) under non-Lipschitz condition with Lipschitz
condition being considered as a special case and a weakened linear growth condition on the coefficients by means of the successive
approximation. Compared with the previous results, the results obtained in this paper is based on a other proof and our results can
complement the earlier publications in the existing literatures.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
This document provides historical context on key concepts in Schwartz space and test functions. It discusses how Laurent Schwartz defined the Schwartz space in 1947-1948 to consist of infinitely differentiable functions that, along with their derivatives, decrease faster than any polynomial. Test functions, a subset of Schwartz space, have compact support. Joseph-Louis Lagrange and Norbert Wiener helped develop the method of multiplying a function by a test function and integrating, which is fundamental to distribution theory. The term "mollifier" for test functions was coined by Kurt Friedrichs in 1944, although Sergei Sobolev had previously used them. Many mathematicians, including Leray, Sobolev, Courant, Hilbert, and Weyl,
Redundancy in robot manipulators and multi robot systemsSpringer
This document summarizes how variational methods have been used to analyze three classes of snakelike robots: (1) hyper-redundant manipulators guided by backbone curves, (2) flexible steerable needles, and (3) concentric tube continuum robots. For hyper-redundant manipulators, variational methods are used to constrain degrees of freedom and provide redundancy resolution. For steerable needles, they generate optimal open-loop plans and model needle-tissue interactions. For concentric tube robots, they determine equilibrium conformations dictated by elastic mechanics principles. The document reviews these applications and illustrates how variational methods provide a natural analysis tool for various snakelike robots.
I am Charles S. I am a Design & Analysis of Algorithms Assignments Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, York University, Canada. I have been helping students with their assignments for the past 15 years. I solve assignments related to the Design & Analysis Of Algorithms.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with the Design & Analysis Of Algorithms Assignments.
Marginal Deformations and Non-IntegrabilityRene Kotze
1) The document discusses integrability in quantum field theories and string theories, specifically N=4 supersymmetric Yang-Mills theory (SYM).
2) It was previously shown that N=4 SYM is integrable in the planar limit through a mapping to spin chains. Deformations of N=4 SYM called Leigh-Strassler deformations were also studied.
3) The document analyzes string motion on the gravity dual background known as the Lunin-Maldacena geometry, which corresponds to Leigh-Strassler deformations. Through analytic and numerical methods, it is shown that string motion on this background is non-integrable for complex deformations, indicating
This document discusses four ways to represent functions: verbally, numerically, visually through graphs, and algebraically through explicit formulas. It provides examples of each type of representation and discusses key properties of functions like domain, range, and the vertical line test. The document also covers piecewise-defined functions and functions with symmetry properties like even functions whose graphs are symmetric about the y-axis.
This document discusses different representations of curves and surfaces in computer graphics, including explicit, implicit, and parametric representations. Explicit representations express one variable as a function of others, but cannot represent all shapes. Implicit representations use equations like f(x,y)=0 but can be difficult to render points on. Parametric representations express each coordinate as a function of one or more parameters and have advantages over explicit and implicit forms like versatility and ease of rendering. The document focuses on using rational parametric polynomials, as they are simple yet powerful representations.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
Let f(x) =
and g(x) =
1 x
x
that fog is not defined.
19.
If f : A B is bijective, then write n(A) in terms of n(B).
20.
If f : A B is onto and n(A) = 5, n(B) = 3. Then write n(A).
SHORT ANSWER TYPE QUESTIONS (2 MARKS)
1.
Define reflexive, symmetric
This document summarizes a research article that introduces a new iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions to a variational inequality problem, and the set of solutions to an equilibrium problem in a Hilbert space. The authors provide background on existing methods for related problems and note the need to construct an iteration process that can solve this more general problem. They then introduce their new monotone hybrid iterative scheme and state that they will establish strong convergence theorems for their proposed iteration method.
Newspaper Report Writing - Examples, Format, Pdf ExaCheryl Brown
The document provides instructions for requesting writing assistance from HelpWriting.net, including creating an account, completing an order form with instructions and deadline, and reviewing writer bids before selecting one and placing a deposit to start the assignment. Customers can then review the completed paper and request revisions if needed, with the company promising original, high-quality content and refunds for plagiarized work.
How To Use A Kindle Paperwhite On And Off -Cheryl Brown
The Great Depression of the 1930s was the worst economic crisis in US history. It had worldwide impacts and raised questions about how to recover. Franklin Roosevelt was elected in 1932 based on his New Deal programs to combat the depression through relief, recovery, and reform. The Social Security Act had the most significant impact by establishing retirement benefits and unemployment insurance, laying the foundation for the modern US social welfare system.
Mais conteúdo relacionado
Semelhante a Application Of The Theory Of Linear Singular Integral Equations And Contour Integration To Riemann Hilbert Problems For Determination Of New Decoupled Expressions
The document discusses the mathematical background of the finite element method. It begins with an overview of concepts from functional analysis including vector spaces, scalar products, linear functionals, and Riesz representation theorem. Green's functions are introduced as the Riesz elements representing linear functionals. Green's identities are then used to derive classical influence functions for displacements and forces. Physically, Green's functions represent monopoles and dipoles with infinite energy, placing them outside the theory of weak boundary value problems. In nonlinear problems, Green's functions can be viewed as Lagrange multipliers at the linearization point.
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
This document summarizes and reviews results on contraction and extension maps in Hilbert spaces. It discusses important theorems related to representing linear operators using normal operators like self-adjoint and unitary operators. It also covers Neumark's theorem which shows that every generalized spectral family can be represented by the projection of an ordinary spectral family in an extended space. The document applies these results to problems in elasticity theory.
This document discusses equations of motion for higher-order Legendre transforms in quantum field theory. It begins by introducing Legendre transforms and their use in expressing potentials in terms of Green's functions. It then:
1) Derives Schwinger's equation and constraint equations for the generating functional of Green's functions. These determine the Green's functions uniquely.
2) Shows that iterating the equations expresses the generating functional as the sum of all vacuum loops, demonstrating the connected Green's functions are generated.
3) Notes that generalizing this proof to higher-order Legendre transforms would demonstrate their m-irreducibility properties, in line with previous work.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Interpreting Multiple Regressionvia an Ellipse Inscribed in a Square Extensi...Toshiyuki Shimono
The document discusses interpreting multiple regression results geometrically using ellipses. It presents 3 theorems:
1) The multiple correlation coefficient is the ratio of line segments from the origin to a point P inside the ellipse and its projection P' onto the ellipse.
2) Regression coefficients are values of linear scalar fields defined on the axes at point P.
3) Partial correlation coefficients are values of affine functions defined on the longest line segments through P parallel to the axes.
This paper reviews the fundamental concepts and
the basic theory of polarization mode dispersion(PMD) in optical
fibers. It introduces a unified notation and methodology to
link the various views and concepts in jones space and strokes
space. The discussion includes the relation between Jones vectors
and Strokes vectors and how they are used in formulating the
jones matrix by the unitary system matrix.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
Successive approximation of neutral stochastic functional differential equati...Editor IJCATR
We establish results concerning the existence and uniqueness of solutions to neutral stochastic functional differential
equations with infinite delay and Poisson jumps in the phase space C((-∞,0];Rd) under non-Lipschitz condition with Lipschitz
condition being considered as a special case and a weakened linear growth condition on the coefficients by means of the successive
approximation. Compared with the previous results, the results obtained in this paper is based on a other proof and our results can
complement the earlier publications in the existing literatures.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
This document provides historical context on key concepts in Schwartz space and test functions. It discusses how Laurent Schwartz defined the Schwartz space in 1947-1948 to consist of infinitely differentiable functions that, along with their derivatives, decrease faster than any polynomial. Test functions, a subset of Schwartz space, have compact support. Joseph-Louis Lagrange and Norbert Wiener helped develop the method of multiplying a function by a test function and integrating, which is fundamental to distribution theory. The term "mollifier" for test functions was coined by Kurt Friedrichs in 1944, although Sergei Sobolev had previously used them. Many mathematicians, including Leray, Sobolev, Courant, Hilbert, and Weyl,
Redundancy in robot manipulators and multi robot systemsSpringer
This document summarizes how variational methods have been used to analyze three classes of snakelike robots: (1) hyper-redundant manipulators guided by backbone curves, (2) flexible steerable needles, and (3) concentric tube continuum robots. For hyper-redundant manipulators, variational methods are used to constrain degrees of freedom and provide redundancy resolution. For steerable needles, they generate optimal open-loop plans and model needle-tissue interactions. For concentric tube robots, they determine equilibrium conformations dictated by elastic mechanics principles. The document reviews these applications and illustrates how variational methods provide a natural analysis tool for various snakelike robots.
I am Charles S. I am a Design & Analysis of Algorithms Assignments Expert at computernetworkassignmenthelp.com. I hold a Master's in Computer Science from, York University, Canada. I have been helping students with their assignments for the past 15 years. I solve assignments related to the Design & Analysis Of Algorithms.
Visit computernetworkassignmenthelp.com or email support@computernetworkassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with the Design & Analysis Of Algorithms Assignments.
Marginal Deformations and Non-IntegrabilityRene Kotze
1) The document discusses integrability in quantum field theories and string theories, specifically N=4 supersymmetric Yang-Mills theory (SYM).
2) It was previously shown that N=4 SYM is integrable in the planar limit through a mapping to spin chains. Deformations of N=4 SYM called Leigh-Strassler deformations were also studied.
3) The document analyzes string motion on the gravity dual background known as the Lunin-Maldacena geometry, which corresponds to Leigh-Strassler deformations. Through analytic and numerical methods, it is shown that string motion on this background is non-integrable for complex deformations, indicating
This document discusses four ways to represent functions: verbally, numerically, visually through graphs, and algebraically through explicit formulas. It provides examples of each type of representation and discusses key properties of functions like domain, range, and the vertical line test. The document also covers piecewise-defined functions and functions with symmetry properties like even functions whose graphs are symmetric about the y-axis.
This document discusses different representations of curves and surfaces in computer graphics, including explicit, implicit, and parametric representations. Explicit representations express one variable as a function of others, but cannot represent all shapes. Implicit representations use equations like f(x,y)=0 but can be difficult to render points on. Parametric representations express each coordinate as a function of one or more parameters and have advantages over explicit and implicit forms like versatility and ease of rendering. The document focuses on using rational parametric polynomials, as they are simple yet powerful representations.
This document outlines the contents of a Mathematics II course, including five units: vector calculus, Fourier series and Fourier transforms, interpolation and curve fitting, solutions to algebraic/transcendental equations and linear systems of equations, and numerical integration and solutions to differential equations. It lists three textbooks and four references used in the course. It then provides examples and explanations of key concepts from the first two units, including vector differential operators, gradient, divergence, curl, and Fourier series representations of functions.
Let f(x) =
and g(x) =
1 x
x
that fog is not defined.
19.
If f : A B is bijective, then write n(A) in terms of n(B).
20.
If f : A B is onto and n(A) = 5, n(B) = 3. Then write n(A).
SHORT ANSWER TYPE QUESTIONS (2 MARKS)
1.
Define reflexive, symmetric
This document summarizes a research article that introduces a new iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions to a variational inequality problem, and the set of solutions to an equilibrium problem in a Hilbert space. The authors provide background on existing methods for related problems and note the need to construct an iteration process that can solve this more general problem. They then introduce their new monotone hybrid iterative scheme and state that they will establish strong convergence theorems for their proposed iteration method.
Semelhante a Application Of The Theory Of Linear Singular Integral Equations And Contour Integration To Riemann Hilbert Problems For Determination Of New Decoupled Expressions (20)
Newspaper Report Writing - Examples, Format, Pdf ExaCheryl Brown
The document provides instructions for requesting writing assistance from HelpWriting.net, including creating an account, completing an order form with instructions and deadline, and reviewing writer bids before selecting one and placing a deposit to start the assignment. Customers can then review the completed paper and request revisions if needed, with the company promising original, high-quality content and refunds for plagiarized work.
How To Use A Kindle Paperwhite On And Off -Cheryl Brown
The Great Depression of the 1930s was the worst economic crisis in US history. It had worldwide impacts and raised questions about how to recover. Franklin Roosevelt was elected in 1932 based on his New Deal programs to combat the depression through relief, recovery, and reform. The Social Security Act had the most significant impact by establishing retirement benefits and unemployment insurance, laying the foundation for the modern US social welfare system.
Bat Writing Template Bat Template, Writing TemplatCheryl Brown
The document provides instructions for creating an account on the HelpWriting.net site and submitting requests for paper writing assistance. It outlines a 5-step process: 1) Create an account with an email and password. 2) Complete a form with paper details, sources, and deadline. 3) Review bids from writers and select one. 4) Review the completed paper and authorize payment. 5) Request revisions until satisfied. The site uses a bidding system and promises original, high-quality work or a full refund.
The document provides guidance on how to teach first grade students to rationally count from 1 to 15. It explains that rational counting requires understanding one-to-one correspondence, stable order, order irrelevance, and cardinality. The teacher would ensure mastery by modeling counting aloud and having students repeat, using physical objects to demonstrate one-to-one correspondence, and assessing students. Accommodations for English learners and students with learning differences are also discussed.
Family Tree Introduction Essay. My Family History ECheryl Brown
The document discusses how Sophocles uses diction in the play Oedipus the King to characterize Oedipus and further the plot. Specifically, it analyzes Oedipus' speech and word choices as he discovers the horrifying truth about his past. The diction reveals Oedipus' emotional state and downfall as his high status is destroyed by his tragic realization.
Thesis Statement For Research Paper On DreaCheryl Brown
The document provides a comparison analysis of the plays "Life Sucks" and "Uncle Vanya". Both plays are tragicomedies that deal with themes of dissatisfaction, regret over life choices, and being stuck. While they use similar themes to criticize society, the adaptation "Life Sucks" examines the characters in greater depth, directly addresses the audience, and aims to empower the viewers more so than the original "Uncle Vanya".
Good Short Stories To Write Textual Analysis On - HerolasCheryl Brown
J.R. Batliboi defines the ledger as the chief book of accounts where all business transactions are ultimately classified and recorded under their respective accounts. A ledger can be maintained as a bound register, loose-leaf binder, or cards. Transactions recorded in journals or subsidiary books are posted to the relevant accounts in the ledger. The ledger ensures each account contains all transactions related to it in one place. Debit and credit rules are followed when posting transactions, with the words "To" and "By" used to indicate debit and credit sides respectively.
Where Can I Buy Parchment Writing PapeCheryl Brown
The document discusses Thomas Hobbes and Jean-Jacques Rousseau's social contract theories. Both philosophers theorized that individuals in a state of nature came together and agreed to form societies for protection and fulfillment. However, they differed in their views of human nature - Hobbes saw humans as competitive while Rousseau saw them as cooperative. The document goes on to provide more details on their social contract theories.
Pin By Emily Harris On Teach McTeachersonCheryl Brown
The document outlines the steps to get writing help from HelpWriting.net:
1. Create an account with a password and email.
2. Complete a 10-minute order form providing instructions, sources, deadline and sample work.
3. Choose a writer based on their bid, qualifications, history and feedback. Place a deposit to start.
4. Review the paper and authorize full payment if pleased, or request revisions for free. HelpWriting ensures original, high-quality work or a full refund.
My Favourite Author Worksheet (Teacher Made)Cheryl Brown
The document provides a SWOT analysis of the digital marketing agency Add3. It notes that Add3 has been in business since 2010-2011 and offers various services including on-page SEO, content marketing, social media management, and pay-per-click campaigns. The analysis highlights Add3's strengths as a certified Google partner that serves over 370 million Google ads per month, but notes it is unclear if they offer video SEO services.
Dialogue Essay Example For 4 Person - Interpreting SuccessCheryl Brown
The document discusses the functions of the Earth's climate system. It outlines the key components that interact to determine climate, including the sun, oceans, atmosphere, biosphere, and energy cycle. It defines climate versus weather and explains that climate is long-term while weather is short-term. The layers of the atmosphere, including the troposphere and stratosphere, are identified along with the hydrosphere and biosphere.
The document provides steps for requesting writing assistance from HelpWriting.net, including creating an account, completing an order form with instructions and deadline, and reviewing bids from writers to select one. It notes the site uses a bidding system and stands by providing original, high-quality content or offering a refund if plagiarized. Customers can request revisions to ensure satisfaction.
Essay-Writing.Org Discount Code 2021 Coupons FresCheryl Brown
This document discusses two authors, C.S. Lewis and Robert Frost, and their perspectives on humility during the Postmodern era. Lewis believed some Christians pretend to be humble but are actually proud. He warned against worshipping an "imaginary God" to gain pride over others. Frost's poem suggests humility is difficult and people should be wary of intentions before God. Both authors offered insights on humility for Christians facing changes in the Postmodern world.
Images For Graffiti Words On Paper Graffiti WordsCheryl Brown
The document discusses the film The Invention of Lying and how it relates to the concept of El Dorado. El Dorado represents a utopian society without crime or need for government, relying on the virtue of its citizens. Similarly, in the film everyone tells the truth, trusting each other to do the same, resembling an El Dorado-like society. For such a place to exist, humans would need to demonstrate the highest virtues like honesty, courage, and justice at all times. The film imagines a world where lying does not exist, somewhat capturing the ideal nature of life in El Dorado.
The document provides instructions for requesting and obtaining writing assistance from HelpWriting.net. It outlines a 5-step process: 1) Create an account with a password and email. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Choose a bid from qualified writers and place a deposit to start the assignment. 4) Review the completed paper and authorize final payment if satisfied. 5) Request revisions to ensure satisfaction, and HelpWriting.net guarantees original, high-quality content or a full refund.
Reasons For Attending College Or University. ReasCheryl Brown
The document provides instructions for using the HelpWriting.net service to get writing assistance. It outlines a 5 step process: 1) Create an account, 2) Complete an order form with instructions and deadline, 3) Review bids from writers and select one, 4) Review the completed paper and authorize payment, 5) Request revisions if needed and know plagiarized work will be refunded. The service uses a bidding system and promises original, high-quality content.
Sample College Paper Format ~ Writing An EssaCheryl Brown
The document provides instructions for requesting an assignment writing service from HelpWriting.net. It outlines a 5-step process: 1) Create an account with a password and email. 2) Complete an order form with instructions, sources, and deadline. 3) Review bids from writers and choose one. 4) Review the completed paper and authorize payment. 5) Request revisions to ensure satisfaction, with a refund offered for plagiarized work.
SUNFLOWERS Personalised Writing Paper Set Of 20 PersCheryl Brown
The book Scorched by Mari Mancusi follows a girl named Trinity and the last dragon Emmy as they form a bond to prevent the Scorch. Trinity protects Emmy from those who want to use her, as Emmy protects Trinity. They rely on each other as Trinity tries to change the future and stop the Scorch from happening. The book explores themes of friendship, trust, and escaping those who are after them.
The document provides instructions for requesting writing assistance from HelpWriting.net. It outlines a 5-step process: 1) Create an account with a password and email. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and select one. 4) Review the completed paper and authorize payment. 5) Request revisions to ensure satisfaction, with the option of a full refund for plagiarized work.
Can AI Read And Rate College Essays More Fairly Than HuCheryl Brown
This document discusses the benefits of youth sports participation. It begins by defining sports as activities involving physical exertion and competition. The document then states that getting kids involved in sports at a young age is important for parents to do, even if the kids don't stick with a particular sport long-term. It notes that youth sports participation has decreased by 8% according to recent data. The document promises to explain the positive benefits of youth sports involvement.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the body’s response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Setup Warehouse & Location in Odoo 17 Inventory
Application Of The Theory Of Linear Singular Integral Equations And Contour Integration To Riemann Hilbert Problems For Determination Of New Decoupled Expressions
1. 1
1
Application of the Theory of Linear Singular Integral Equations to a
Riemann Hilbert Problem for a New Expression of Chandrasekhar’s H-
function in Radiative Transfer.
Rabindra Nath Das*
Visiting Faculty Member , Department of Mathematics , Heritage Institute of
Technology , Chowbaga Road , Anandapur, P.O. East Kolkata Township ,
Kolkata- 700 107 , West Bengal , India.
Email address : rndas1946@yahoo.com
Abstract: The linear non homogeneous singular integral equation( LNSIE)
derived from the non-linear non homogeneous integral equation (NNIE) of
Chandrasekhar’s H –functions is considered here to develop a new form of H-
functions .The Plemelj’s formulae are applied to that equation to determine a new
linear non homogeneous integral equation (LNIE) for H- functions in complex
plane. The analytic properties of this new linear integral equation are assessed and
compared with the known linear integral equations satisfied by H-functions. The
Cauchy integral formulae in complex plane are used to obtain this form of H-
functions not dependent on H- function in the integral. This new form of H-
function is represented as a simple integral in terms of known functions both for
conservative and non-conservative cases. This is identical with the form of H-
functions derived by this author by application of Wiener – Hopf technique to
LNIE . The equivalence of application of the theory of linear singular integral
equation in Riemann Hilbert Problem and of the technique of Wiener- Hopf in
linear integral equation in representing the H -functions is therefore established .
This new form may be used for solving the problems of radiative transfer in
anisotropic and non coherent scattering using the method Laplace transform and
Wiener-Hopf technique.
Key words : Radiative transfer , Singular integral equation , Riemann Hilbert
Problem , Cauchy Integral formulae.
*Permanent address: KB-9, Flat –7, Sector III , Saltlake City (Bidhan nagar) ,
Kolkata-700098 , West Bengal , India.
2. 2
2
1 . Introduction :
The H- functions, extensively developed by Chandrasekhar [1] play an
important role in the study of radiative transfer in homogeneous plane parallel
scattering atmosphere of finite and infinite thickness .. The detailed application
of the mathematical theory to the equations of radiative transfer or neutron
transport depends , however upon the availability of the H- function in sufficient
closed form. This requirement has been partly met by Chandrasekhar [1] .
Mathematical framework and the numerical evaluation of these functions have
been made by him extensively by his own method. There is still a need for
further detailed theoretical and numerical work on these functions , particularly
to have a closed form by application of some other methods . Accordingly , we
have re -examined the question of mathematical formulation of a new form of
the H-functions from a different angle. The object of this is to providing current
and future requirements of numerical value of these functions in tabular form for
the case of isotropic scattering for any value of particle albedo both in
conservative and non conservative cases of physical interest . The work presented
in this paper is an extensive formulation of mathematical theory to build up a new
form of the H-functions for atmosphere with coherent scattering.
The mathematical form of H functions presented by different authors
Chandrasekhar [1] , Kourganoff [2] , Busbridge [3] , Fox [4] , Dasgupta [5],[6] ,
Das [7] , by different methods are stated to be exact for numerical integration
with some suitable constraints. We believe that the forms of H- functions
derived so far, as a solution of NNIE or LNIE for numerical computation are
in fact dependent on these H- function itself within the integral. However ,we
believe that those are worthwhile to frame a basis for a new functional
representation of H – functions as a solution of the LNIE of H- functions .
The forms of H-functions derived by Fox[4], Zelazny [8] , Mullikin[9] ,
Zweifel[10],Ivanov[11] ,Siewert [12], Das [13] are considered to be important
from theoretical and numerical point of view . Those solutions are not dependent
on the H-function itself.
The forms of H – functions derived by Siewert [14] , Garcia and Siewert
[15],Sulties and Hill [16] , Barichello and Siewert [17 , 18] , Bergeat and Rutily
[ 19,20] in terms of new functions from different standpoints are still considered
to be useful for numerical and theoretical point of view.
3. 3
3
Further more we believe that the result obtained by this method of application
of the theory of singular integral equation in this work to a LNSIE of H functions
for solving a Riemann Hilbert problem is a new and sufficiently different from
others in the literature of H- functions to warrant its communication .
Consequently we hope that this functional representation of H – functions
extends the result of Fox [4] for better understanding of the method used by him .
We hope that these results identify the equivalence of the application of Wiener-
Hopf technique to LNIE of H-functions Das [13] and the application of theory
of linear singular integral equation to LNSIE of H- functions so as to have the
same form of H – function by two different approaches . The existence and
uniqueness of those functions are achieved already in Das[13]. It is therefore
shown that it is now safe to handle the LNSIE or LNIE of H- functions with due
respect to the concluding comment of Busbridge [21].
2. Mathematical analysis :
The integro-differential equations of radiative transfer have been solved by
different authors to obtain the intensity of radiation at any optical depth. and at the
boundary. They finally derived solutions in terms of Chandrasekhar [1] H-
functions .
The H –functions satisfy NNIE as
H(x)= 1 + x H(x)
1
0
U(u) H(u) d u / (u + x)
, 0≤ x ≤1 , (1)
where U(u) is a known characteristic function . In physical context
to solve equation (1), certain restrictions on U(u) are necessary :
i) U0 =
1
0
U (u) d u = ½
(2)
where U0 = ½ refers to conservative case and U0 < ½ refers to non
conservative case ;
ii) U(u) is continuous in the interval (0,1) and satisfy
Holder conditions , viz . ,
I U(u2) – U(u1) I < K I u2- u 1 I α
(3 )
where u2 , u1 lie in the interval (0,1) ( u1 is in the neighbourhood of u2 ) , where
K and α are constants and 0 < α ≤ 1 .
If z = x +iy , a complex variable in the complex plane cut along (-1,1) then
Chandrasekhar [1] proved that
( H (z) H (-z) ) –1
= 1 - 2 z 2 1
0
U(u) d u / ( z 2
– u2
) = T(z) , (4)
and he determined H – function to satisfy the integral solution as
4. 4
4
i ∞
log H(z) = ( 2πi)-1
∫ z log ( T(w) ) d w / ( w2
- z2
) (5)
-i ∞
where T(z) has the properties :
i)T( z) is analytic in the complex z plane cut along (-1,1); ii) it is an even function
of z ;iii) it has two logarithmic branch points at –1 and +1 ; iv) T(z) 1 as 1 z 1 0 ,
v) it has only two simple zeros at infinity when U0 = ½ ; vi) only two
real simple zeros at z = +1/k and z = -1/k where k is real , 0< k ≤ 1 when
U0 < ½ ;vii) T(z) - C/ z2
as z ∞ when U0= ½ , C = 2 U2 , a real
positive constant ;viii)T (z) D = 1- 2 U0 as z ∞ when U0 < 1/2 ,
D is a real positive constant ; ix) it is bounded on the entire imaginary
x)T ( ∞) = D = 1- 2 U0 when U0 <1/2
= ( 1 -
1
0
H(x) U (x) dx ) 2
= 1/ ( H ( ∞ ) ) 2
where Ur =
1
0
u r
U(u) d u, r = 1,2,3…. (7)
Fox (1961) determined that, equation (1) is equivalent to LNIE in the complex z
plane cut along (-1.1)
T(z) H(z) = 1 + z
1
0
U(u) H(u) d u / ( u – z ) = 1/H(-z) (8)
with constraints
i) when U0 <1/2
1 +
1
0
U(u) H(u) d u / ( u k – 1 ) = 1/H(-1/k) = 0 (9)
1 +
1
0
U(u) H(u) d u / ( u k+ 1 ) = 1/H(1/k) = 0 (10)
and ii) when U0 =1/2
1 -
1
0
U(u) H(u) d u = 0 (11)
and to LNSIE if real ( z ) = x lies in (-1,1)
T0(x) H(x) = 1 + P x
1
0
U(u) H(u) d u / ( u – x ) (12)
5. 5
5
where P represents the Cauchy Principal value of the integral and .
T0 (t) = 1 - t
1
0
U (x) d x / (x + t) +
t
1
0
( U(x)–U(t)) dx /( x – t)+ t U(t) ln( ( 1-t ) / t ) . (13)
In this paper, we shall consider the LNSI Eq (12) of H-
functions to solve it by the theory of linear singular integral equation fully
described in the standard work of Muskhelishvilli [22] associated with the theory
of analytic continuation and Cauchy theory on analytic function in complex
plane . We derive a unique solution which is obtained by Das[23] by using
Wiener Hopf technique along with theory of analytic continuation and Cauchy
theory of analytic function in complex plane . We shall show the equivalence of
Riemann Hilbert Theory for solving LNSIE and the Wiener Hopf technique for
solving LNIE so far as it relates to determination of a new form for H- functions
Let us consider the function
Φ(z)= ψ(t )d t / ( t – z) , (14)
L
where L is some contour in the complex z plane with a cut along (0,1) and z
does not lie on L . Here Φ (z) be an analytic function of z in the complex plane
cut along (0,1) and be O ( 1/z) when z ∞ . If z is on L , then the integral for
Φ(z) is taken to be a Cauchy principal value and the curve L becomes a line of
discontinuity for the function Φ(z)., The function Φ(z ) must pass through a
discontinuity when z crosses L. Let A and B be the end points of the arc L at z
= 0 and z = 1 respectively .
We want to determine the value of Φ (z) as z t0 from both sides of the
contour where t0 lie on L and t0 is not an end point of L. We define the region D+
as the region to the left of the contour( if we look to the left moving counter
clockwise along the contour) and D-
as the region to the right .
6. 6
6
.Let us define
Φ+ or –
(t0) = lim Φ(z) , z belongs to D+ or –
(15)
z t
0
We also define Cauchy principal value integral
Φ (t0) = P ∫ ψ(t) d t / ( t – t0 ) (16)
L
= lim ∫ ψ (t) d t / ( t – t0 ) (17)
є 0 L-Lє
where Lє is that part of L contained within a circle of radius є with center at t= t0
and where L – Lє is that part of L contained without it .
We assume ψ(t) is analytic at t = t0 and continuous every where, hence by
analytic continuation ψ(t) is analytic in a small neighbourhood of t0 and this
continuation can be extended to the whole complex plane.
We therefore using the definition in equations ( 16 ) and (17)
Φ+
(t0 ) = lim Φ (z) , z lie within D+
z t
0
є0
= Φ (t0) + i π ψ (t0) ( 18)
Φ-
(t0 ) = lim Φ (z) , z lie within D-
z t
0
є0
Φ
(t0 ) = Φ (t0) - iπ ψ (t0) ( 19)
Equation ( 18 ) and (19) are referred to as the Plemelj formulae .They may be
equivalently written as
+
(t0) - Φ-
(t0) = 2πi ψ (t0 ) (20)
7. 7
7
Φ+
(t0) + Φ-
(t0) = 2 P ∫ ψ(t) d t / ( t – t0 ) (21)
The end points of the contour L cause considerable difficulty . A complicated
analysis in Muskhelishvilli (1953) shows that at the end points tA and tB of L ,
Φ(t) will behave as
1 Φ (t) 1 ≤ KA / 1 t – t A 1 α
A , 0 ≤ α A < 1 (22)
1 Φ (t) 1 ≤ KB / 1 t – t B 1 α
B , 0 ≤ α B < 1 (23)
where KA , KB , α A and α B are constants .
3. Determination of a new LNIE for H-functions :
we shall now introduce a function Y(z) of the complex variable z
Y( z) =
1
0
U(t) H(t) d t / ( t – z ) (24)
which has the properties :
i)Y(z) is analytic in the complex plane cut along (0,1) ; ii) Y(z) = O (1/z) when
z ∞ ; iii)1 Y (z) 1 ≤ K0 / 1 z 1 α
0 , 0 ≤ α0 < 1 ; iv) 1 Y (z) 1 ≤ K1 / 1
1-z 1 α
1 , 0 ≤ α1 < 1; v) U(t) H(t) is continuous in the interval (0,1) and satisfy
equivalent Holder conditions stated in equation (3) and K0, K1 , α0 and α1 are
constants .
We shall now apply the Plemelj’s formulae (20 ) and (21) to the equation (12) to
have
Y+
(x) = g(x)Y-
(x) + 2πi U(x) / T-
(x) , 0≤ x ≤1 (25)
where Y+
(x) , Y-
(x) and g(x) are complex numbers and g(x) is related to
T +
(x)= T0(x) + i π x U(x) (26)
T-
(x) = T0(x) – i π x U(x) (27)
g(x) = (T0(x) + i π x U(x) ) / ( T0(x) – i π x U(x) )
= T+
(x)/T-
(x) (28)
where T0 (x) is given by equation ( 13 ).
From equation (25) we have to determine the unknown function Y(z) for being
analytic in the complex plane cut along (0,1) having properties outlined above
and g(x) and U(x) are known functions. This is known as non homogeneous
Riemann Hilbert problem.
8. 8
8
We shall first determine the solution of homogeneous Riemann Hilbert problem
obtained from equation (25) .
It is that of finding a new analytic function X(z) for which
X+
(x) = g(x) X-
(x) , 0≤x≤1 (29)
where X(z) is to satisfy the following properties :
i) X(z) is analytic in the complex plane cut along (0,1) ; ii) X(z) is not zero
for all z in complex plane cut along (0 ,1) ; iii) O(1/z) when z ∞ ;iv) 1 X (z) 1
≤ K2 / 1 z 1 α
2 , 0 ≤ α2 < 1 ;and v)1 X (z) 1 ≤ K3 / 1 1 -z 1 α
3 , 0 ≤ α3 <
1 , where K2, K3 , α 2 and α 3 are constants .
We now assume that for 0 ≤ x ≤ 1, U(x) is real , positive , single
valued and satisfy Holder condition in (0,1) . We also assume that T0(x) is not
equal to zero both in the conservative case U0 = ½ and in non conservative
cases U0 < ½ . As T0(x) is dependent on U(x) it can be proved that T0(x) is real
,one valued and satisfy Holder conditions in (0,1) and modulus of ( T0(x) +i π x
U(x) ) , ( T0(x) – i π x U(x) ) are not equal to zero, T0(0) = 1 and T0(1) - ∞ as
z 1 from within the interval (0,1) .
Taking logarithm to equation (29)and using (28) we find that
log( X+
(x) ) – log (X-
(x)) =log g(x) = 2 i θ (x) (30)
where
tan θ(x) = π x U (x) / T0(x) ; –π/2 < θ (x) < π/2 (31)
θ(0)=0, (32)
and θ(x) is assumed single valued .
We shall evaluate X(z).Using Plemelj’s formulae (20) to equation (30) we get
log X(z) = (2πi)-1 1
0
log g(u) du /(u – z) + Pn (z) (33)
where Pn(z) is an arbitrary polynomial in z of degree n. in the complex plane cut
along (0,1) and it is continuous in the complex plane across a cut along (0,1)
because it is analytic there. Thus equation (33) does also satisfy the first of
Plemelj’s formulae(20).
We set
L(z) = (2πi)-1 1
0
log g(u) d u / ( u – z) (34)
9. 9
9
Equation (33) with equation (34) then yields
X(z) = X0 (z) exp ( L(z) ) (35)
where X0(z) = exp ( Pn(z) ) is analytic for all z in the complex plane and is to be
determined such that X(z) given by equation (35) satisfy all properties outlined
above and the use of the Plemelj’s formulae is not invalidated .
We have to determine the value of X0(z) at the end points: Equation (34) may be
written as
L(z) = (2πi ) -1
log g(z)
1
0
d u /( u-z)
+ (2πi)-1 1
0
( log g(u) – log g(z) ) d u / (u – z) (36)
At the end point z = 0 , equation (36) can be written as
L(z) = = - (log g(z) / 2πi ) log 1z1 + q 0 (z) (37)
where q0(z) incorporates the second integral of equation (36) and the value of
first term at z = 1 . Here q0(z) is clearly a bounded function .We therefore find
that , in the neighborhood of the end point at z= 0 ,the behavior of X(z) is
dominated by
X(z) X0 (z) 1z1- (log g(z) / 2πi )
(38)
Near the end point z=0 for the Plemelj’ formulae to be valid , X0(z) and X(z)
must behave in the neighborhood of z = 0 as
X0(z) 1z1 m
0 (39)
X(z) 1z1 (m
0
- (log g(z) / 2πi ) )
(40)
We must have the constant m0 to satisfy there
0 ≤ Real ( log g(z) / 2πi ) - m 0 < 1 (41)
As X0(z) will be analytic and continuous across the cut
in complex z plane along (0,1) we must have using eqs.(30),(32)
m0 = Real ( log g(0) / 2πi) = θ(0 ) / π = 0 (42)
At the end point z = 1 , equation (36) can be written as
L(z) = = (log g(z) / 2πi ) log 11- z1 + q 1 (z) (43)
10. 10
10
where q1(z) incorporates the second integral of equation (36) and the value of
first term at z = 0 . Here q1(z) is clearly a bounded function . Hence, in the
neighborhood of the end point at z= 1 ,the behavior of X(z) will be dominated
by X0(z) as
X(z) X0 (z) 11-z1 (log g(z) / 2πi )
(44)
and for the Plemelj’ formulae to be valid there , X0(z) and X(z) must also
behave in the neighborhood of z = 1 as
X0(z ) 1 1-z1 m
1 (45)
X(z) 11-z1 (m
1
+ (log g(z) / 2πi ) )
(46)
We must have the constant m1 to satisfy there
0 ≤ - Real ( log g(z) / 2πi ) - m 1 < 1 (47)
As X0(z) will be analytic and continuous across the cut in complex z plane
along (0,1) hence using eq.(30)
m1 = - Real ( log g(1) / 2πi) = - θ(1 ) / π (48)
But we know that
θ(1) = r π , r = 1,2,3 etc (49)
so m1 = -1 for r =1 = ½ the zeros of T(z) in the complex
plane cut along(-1,1) .Hence on consideration of the properties of X0(z) at both
end points we have
X0(z) = (1-z) –1
(50) .
Hence equation (35) using equations(50) , yields
X(z) = ( 1 –z )-1
exp(L(z) ) (51).
This is the solution of homogeneous Hilbert problem mentioned in equation
(29) in the complex z plane cut along (0,1) .
This can be written in explicit form when z is in complex in z plane cut
along(0,1) using eqs.(34) &(30)
X(z) = (1-z)-1
exp( (2πi)-1 1
0
log g(u) d u / (u – z) ) (52)
11. 11
11
= (1-z ) –1
exp ( π -1 1
0
θ(u) d u / ( u – z) ) (53)
where θ(u) is given by using equation (31) as
θ(u) = tan-1
( π u U(u) / T0(u) ) (54)
We shall determine a solution of non homogeneous Hilbert problem.
The equation (25) may be changed by first writing
X+
(u) = g(u) X-
(u) , 0<u<1 (55)
when u lie on cut along (0,1),X+
(u),g(u) and X-
(u)are all nonzero .Substitution of
equation (55) in equation (25) we get
Y+
(u) / X+
(u)-Y -
(u) / X-
(u) = 2πi U(u) / X+
(u) T-
(u) , (56)
when u lie in the interval (0,1). Using Plemelj’s formulae in equation(20) to
equation (56) we get for z in complex plane cut along (0,1),
Y(z) = X(z)
1
0
U(u) d u / (X+
(u) T-
(u) ( u – z) )
+ Pm(z) X(z) (57)
where Pm(z) is an arbitrary polynomial in z of degree m and is continuous in the
complex plane across a cut along (0,1 ).
Thus equation (57) does satisfy the first of Plemelj’s formulae as in. equation
(20) The arbitrariness of Pm(z) is removed usually by examining the behaviour
of Y(z) and X(z) at infinity and /or the end points of the cut along (0,1) .
Knowledge of X (z) then completes the solution for Y (z). Y(z) will then be the
sum of solution of homogeneous Hilbert problem in equation (29) and the
particular solution of the equation (56).
We now return our attention to the evaluation of the particular solution of
equation (56)for Y(z) from equation (57) and in particular , to the
determination of the polynomial Pm(z) . Since X(z) is equivalent to exp(L(z) ) ,
the term appearing on the RHS of equation (57) is a polynomial of degree m .
However, if we use the properties of X(z) and Y(z) when z ∞, we must
have
Pm(z) = 0 (58)
Hence we get the particular solution Yp(z) of equation (56)in complex z plane
cut along (0,1)) for Y(z) as
12. 12
12
Yp(z) = X(z)
1
0
U(u) d u / (X+
(u) T-
(z) ( u – z) ) (59).
We shall now represent Yp(z) in terms of equation X(z) in the complex z plane
cut along (0,1) by using Cauchy’s integral theorems
We set
V(z) =(2πi)-1 1
0
U(u)du /(X+
(u)T-
(u) (u-z) ) (60)
where V(z) is analytic in the complex z plane cut along (0 , 1).
We consider two contours as follows:
F(z) = (2πi)-1
1
C
F(u) d u / (u-z) ) (61)
where C1 = L1 U L2 . Here contour L 1 is a sufficiently large in the complex z
plane to contain the cut along (0,1) within and L2 is a circle with center at the
origin and of very large radius R and L1 lies interior to this L2 and z lies
outside L1 but inside L2 . Both L1 and L2 are taken in counter clock wise sense.
The function X(z) and 1/X(z) are analytic and nonzero in the annulus C1 . We
shall now apply the Cauchy Integral theorem on the function F(z) =1/ z X(z)
around C1 to obtain
1/ z X(z) = (2πi)-1
2
L
d w / ( X(w) w (w-z) )
- (2πi)-1
1
L
dw / ( X(w) w (w-z) ) (62)
Using equation (51) we can write
1/X(w) = ( 1 – w) exp ( - L (w) ) (63)
and when w= u , u real and 0 < u < 1
we can write from eq.(63)
1/X +
(u) = ( 1 – u) exp ( - L+
(u) ) (64)
1/X-
(u) = ( 1 – u) exp ( - L-
(u) ) (65)
13. 13
13
where the superscript ‘+’ denotes the value from above the cut along (0,1) and
the superscript ‘-’ denotes the value from below the cut along (0,1) of the
respective functions in the complex z plane and
L+
(u) = ( π -1
P
1
0
θ(t) d t / ( t – u) + i θ (u) ) (66)
L-
(u) = ( π -1
P
1
0
θ(t) d t / ( t – u) - i θ (u) ) ( 67)
Using eqs. (66) & (67) in equations (64) and(65) we can write that
1/X +
(u) - 1/X -
(u) = ( 1 - exp( 2iθ(u) ) / X +
(u) (68).
Using equation (29 ) and ( 30 ) we get
exp (2iθ(u) ) = T+
(u) / T-
(u) (69).
Using equations (69), (26) & (27) in equation (68) we get
1 / X +
(u) – 1 / X -
(u) = - 2πi u U(u) / T-
(u) X +
(u) (70)
To make the contour L1 to be well defined we shall shrink the contour L1 to i) a
circle C0 around the origin of the complex plane in counter clockwise sense
such that w= r exp (i α), 0≤ α ≤ 2π ; ii) a line CD below the cut along (0,1) from
r to 1-r where w = u, real , u varies from 0 to 1; iii) a circle C2 counter clockwise
sense around w=1 such that w = 1 + r exp(i β ) , -π ≤ β≤ π and iv) a line BA
above the cut along (0,1) from 1-r to r on which w = u , real , u varies from 1
to 0 . . The value of the integral , on the circle C0 , in the limit r0 becomes
(2πi)-1
0
C
dw/(X(w)w(w-z) )= - ( z X(0) )-1
(71)
The value of the integral , on the circle C2, in the limit r0 becomes
(2πi)-1
2
C
d w / ( X(w) w (w-z) ) = 0 (72)
The value of the integral ,on the line CD , in the limit r0 becomes
(2πi)-1
CD
d w / ( X(w) w (w-z)) = (2πi)-1 1
0
d u / ( X-
(u) u (u-z) ) (73)
The value of the integral , on the line BA , in the limit r0 becomes
14. 14
14
(2πi)-1
BA
d w / ( X(w) w (w-z)) = (2πi)-1 0
1
d u / ( X+
(u) u (u-z) ) (74)
Hence the integral , on the contour L1 in eq(62), becomes using eqs.(70-74)
(2πi)-1
1
L
d w /(X(w) w (w-z) ) = - (2πi)-1 1
0
(1/ X +
(u) – 1/X -
(u) )d u /(u (u-z) )
- 1 / ( z X(0) ) (75)
=
1
0
U (u) d u /( X+
(u)T-
(u) (u-z) )
- 1 / ( z X(0) ) (76)
The integral , on the contour L2 , when R ∞ becomes
(2πi)-1
2
L
d w /(X(w) w (w-z) ) = - 1 (77 )
as z X(z) = - 1 when z ∞ (78)
Hence equation (60) with equations (62), (76) and (77) gives
V(z) = ( z X(0) )-1
- 1 – (z X(z))-1
(79)
Hence the particular solution Yp(z) of equation ( 56) gives
Yp(z) = X(z) V(z) = X(z) / (z X(0) ) - X(z) - 1/ z (80)
Hence the general solution of the equation (25) in the complex z plane , is
Y(z) = A X(z) + Y p (z) (81)
where A is a constant yet to be determined by using the pole of Y(z), if any, at
the zero of T(z) and X(z) , Yp(z) are determined by equations(51) and (80)
respectively .This Y(z) in Equation (81) will help in representing 1/H(-z) in
equation (8).
We shall now show that H(z) in equation (8) is continuous across the cut along
(0,1).. Let H(x) be any real valued solution of NNIE at equation ( 1 ) for 0≤ x≤ 1
then it can be proved that H(z) defines in equation (8) is the meromorphic
extension of H(x) , 0 ≤ x≤ 1 to the complex domain 1z1>0 . In addition H(x) will
satisfy the LNSIE given by equation ( 12 ) .By using the Plemelj’s first formulae
as in equations(18) &(19) to equation (8) when z approaching the cut along
(0,1) from above and below respectively :
15. 15
15
T+
(x) H+
(x) = 1 + x P
1
0
H(t) U(t) dt / (t-x) + i π H(x) U(x), 0≤ x≤ 1 (82)
T-
(x) H-
(x) = 1 + x P
1
0
H(t) U(t) dt / (t-x) - i π H(x) U(x), 0≤ x≤ 1 (83)
where superscript ‘+’ and ‘-‘ denotes the value as z approaches from above and
below to the cut along (0,1) . Using the values of T+
(x) and T-
(x) from equations
(26) , (27) to equations (82) and (83) and using the equation (12) therein we get
H+
(x) = H(x) = H-
(x) (84)
This shows that H(z) defined by equation (8) is continuous across the cut along
(0,1) and real valued on (0,1) Hence it indeed defines a meromorphic extension
of H(x) at least in the region 0 <Re(z) <1 , Im(z) > 0 and Im (z) < 0 and T(1) 0
,and it can be said that z =1 is a removable singularity. We shall now frame a new
LNIE from the theory of linear singular integral equations. Using equations
(8),(26),(27),(81) ,(82) ,(83)& (84) in equation (24), we get ,
(T+
(x) H(x) - x A X+
(x) - x Yp
+
(x))
– ( T-
(x) H(x - x X-
(x) - x Yp
-
(x ) ) = 0 (85)
Using first of Plemelj’s formulae as in equation (20) to equation (85) we get
T(z) H(z) - z A X(z) - zYp(z) = Pp(z) (86)
where Pp(z) an arbitrary polynomial is to be determined from the properties of
X(z) , Yp(z) ,T(z) and H(z) at the origin..
Since H(z) and T(z) both 1 when z0 , equation (86) gives
Pp (z) = 1 (87)
Equation (86) with equation (87) gives a new representation of LNIE in the
complex z plane cut along (0,1)from the theory of linear singular integral equation
and by analytic continuation to complex plane cut along (-1,1) to
T(z) H(z) = 1+ z A X(z) + z Yp (z) (88) .
4. Determination of constant :
16. 16
16
Equation (88) with equation (8) and (81) gives
1/H(-z) = 1+ z A X(z) + z Yp (z) (89)
= 1 + z Y(z) (90)
We have now to determine the constant A in equation (88).We know that T(z)
has two zeros at z = 1/k or –1/k , 0≤ k ≤1 . 1/H(-z) has a zero at z=1/k . Equation
(89) , on substitution of z = 1/k , gives
1+ A X(1/k) / k + Yp(1/k) / k =0 (91).
Using equation (80) for z =1/k we get
Yp(1/k) = k ( X(1/k)/X(0) – X(1/k)/k –1) (92).
Equation (91) and (92) we get
1+ A X(1/k) /k + X(1/k) / X(0) – X(1/k) /k –1 =0 (93)
This equation (93) on simplification gives
A = 1 – k / X(0) (94)
We shall have to determine X(0). Substituting z=0 in equation ( 53 ) we get
X(0)=exp( π-1 1
0
θ(t)d t/t ) (95)
We can determine other form of A for a new form of X(0). In equation (24),(51)
& (59)if we make z ∞ , then
Y(z) = - β0 / z1
- β1 / z2
- β2 / z3
- - (96)
X(z) = - 1/ z - O(1/z2
) ; ( 97)
Yp(z) = O(1/z2
) ; (98)
where β r =
1
0
x r
U (x) H(x) d x , r = 0 ,1,2,3.. (99)
Equating the like power of z –1
from both sides
of equation ( 81) for large z we get after manipulation with eq(6)
A = β0 =
1
0
U (x) H (x) d x = 1 – ( D)1/2
(100)
From equations (94),(100) , we get
17. 17
17
X(0) = k / D1/2
(101)
From eqs.(101), (95) we can determine
1
= π-1 1
0
θ(t)d t/t = ln ( k / D1/2
) (102)
5. Determination of new form of H- function :
We shall now determine the new form of H(z).On substitution of the values of
A , X(0) and Yp(z) from eqs.(94),(101) and(80)respectivelyin equation (89)we
get in the complex plane cut along (0,1)
1/ H(-z) = ( 1 – k z ) X(z) (D)1/2
/ k , (103)
where X(z) is given by equation (53)
Hence in the complex z plane cut along (-1,0) we get
H(z) = k /( X(-z)( 1 + k z ) ( D)½
). (104)
Using the values of X(z) and X(-z) in equations (103) and (104)
we get the new form of H(z) and H(-z) as
H(z) = k ( D)- ½
(1+z) exp ( - π -1 1
0
θ(u) d u / ( u + z) ) / (1 + k z ). (105)
H(-z) = k ( D)- ½
(1-z)exp ( - π -1 1
0
θ(u) d u / ( u - z) ) / (1 –k z ). (106)
These are the same form derived by Das [13] using Wiener Hopf technique to
linear non homogeneous integral equation in Eq.(8).
6. New form of H- function in Cauchy Principal value sense:
We have to find a new form for H(x) in Cauchy’s principal value sense by
application of Plemelj’s formula.We know from equation ( 81 ) using eq.(94)
that
Y(z) = X(z) (1-kz) / ( z X(0) ) (107)
Using first of the Plemelj’s formulae (20) to equation (107) we get
H(x) = ( Y+
(x) – Y-
(x) ) / ( 2πi U(x) ) (108)
18. 18
18
= ( 1- k x ) ( X+
(x) – X-
(x) ) / (2πi x U(x) X(0) ) . (109)
But
X+
(x) – X-
(x) = 2i sin ( θ(x) ) exp( P π-1 1
0
θ(t) d t / (t – x) ) / (1-x) ,(110)
sin (θ(x) ) = π x U(x) /( T0 (x) + π 2
x2
U 2
(x) )1/2
(111)
X(0) = k D-1/2
(112)
D = ( 1 – 2 Uo ) (113)
Equation (109) with equations (110-113) gives
H(x) = (1 – k x) D1/2
exp( - π-1
P
1
0
θ(t) dt / (t – x) ) X
( T0 (x) + π 2
x2
U 2
(x) ) -1/ 2
( k (1-x) )-1
(114)
This is the new explicit form of H(x) different from Fox’s [4]solution . The
integral is Cauchy principal Value sense.
7. Removal of Cauchy Principal Value sense from H- function:
We shall now show that the representation of H(x) in eq.(114) is nothing but the
representation of H(x) in equation .(105). It can be done by way of removal of
singular part from equation (114).
Let M(x) = π-1
P
1
0
θ(t) dt / (t – x) , 0x1 (115)
N(x) = π-1 1
0
θ(t) dt / (t + x ) ,0x1 (116)
If z approaches from above to the cut along (0,1) to x , 0≤x≤1 we get
H+
(x) H+
(-x) = 1/ T+
(x) (117)
Here H+
(x) = H(x) as H(x ) is continuous across the cut along (0,1) but H+
(-x) is
not continuous there . Therefore we get H+
(-x) , H+
(x) from equation (106) &
(105) using equations (115) & (116) that
H+
(-x) = (1-x ) k ( D)-1/2
exp ( - M(x) - iπ θ (x) ) / (1-kx) (118)
H+
(x) = (1-x ) k ( D)-1/2
exp ( - N(x) ) / (1+kx) (119)
19. 19
19
T+
(x) = T0(x) + i π x U(x) (120)
Using equations (118-120) in equation (117) we get
( 1 – x2
) k2
D-1
( 1 – k2
x2
)-1
exp(-M(x) - N(x) ) ( cos θ(x) - i sin θ(x) )
= ( T0 (x) – iπ x U(x) }/ ( T0 (x) + π 2
x2
U 2
(x) ) (121)
Equating the real and imaginary parts of equation (141) we get
( 1 – x2
) k2
D-1
( 1 – k2
x2
)-1
exp(-M(x) - N(x) ) cos θ(x)
= T0 (x) / ( T0 (x) + π 2
x2
U 2
(x) ) (122)
( 1 – x2
) k2
D-1
( 1 – k2
x2
)-1
exp(-M(x) - N(x) ) sin θ(x)
= π x U(x) }/ ( T0 (x) + π 2
x2
U 2
(x) ) . (123)
But using equations (111), (115) and (116) in equation (123) we get
( 1 – x2
) k2
D-1
( 1 – k2
x2
)-1
exp(-M(x) - N(x) )
= 1 / ( T0 (x) + π 2
x2
U 2
(x) ) 1/2
. (124)
Now we use equation (124) to equation (114) in order to eliminate the Principal
part of the integral in equation (114) and we get the same form of H(x) ( as in
equation (105) ) with constraints (9) & (10) in non conservative cases and
constraint (11) in conservative cases
Using equations (101) and (95) to equation (105) we get the form outlined by
Mullikin [9].in non conservative cases:
H(z) = (1+z) exp ( π -1 1
0
θ(u) d u / u( u + z) ) / ( 1 + k z ). (125)
8. Determination of H- function in conservative cases :
From equation (105) we can derive the representation of H(z) for
conservative cases . In conservative cases U0 =1/2 , T(z) will have zeros at
infinity . Hence k = 0. When k=0, U0 =1/2 , D = 0 by equation( 113 ) .
Hence when k0 , k D-1/2
( 2 U2 )-1/2
, in conservative case , H(z) in equation
(105) will take the form
H(z) = ( 2 U2 )-1/2
(1+z) exp ( - π -1 1
0
θ(u) d u / ( u + z) ) . (126)
with constraint in equation (11) .
20. 20
20
9. Admissible solutions:
Chandrasekhar [1], Busbridge [3] ,Dasgupta [23] , Abhyankar [24] proved that ,
in non conservative cases , these two constraints(9) &(10) will provide two
different unique solutions but in conservative cases those two solutions will be
identical to form one unique solution with constraint(11) . In non conservative
cases , they also derived the form of constraints from equations (9) and (10)
respectively to
1
0
U(u) H(u) d u = 1 – D1/2
(127)
1
0
U(u) H(u) d u = 1 + D1/2
(128)
They took one meaningful solution of H(z) of equation (8) for physical
context with constraint (127) . The other solution H1(z) with constraints (128)
was
H1(z) = (1+kz) H (z) /(1- kz) (129)
We therefore, resolve that, in non conservative cases, equation (105) for H(z)
with constraints (9) or (127) will provide meaningful unique solution of equation
(12) of physical interest and the other unique solution H1(z) will be defined by
equation (129) with H(z) from equation (105) and with constraints (10) or (128).
10.Conclusion :
The basic approach in this paper is to place a new method to extend the only
available solution of Fox[4] for the H-functions from LNSIE as solution of a
Riemann Hilbert problem. The representation of H- functions obtained from
LNSIE and from LNIE has become the same . The equivalence between
application of the theory of linear singular integral equation and application of
Wiener-Hopf technique to the linear integral equations is proved to be true so far
it relates to the H-functions .The numerical evaluation of these H-functions from
this new form is awaiting for communication . This new method may be applied
in anisotropic line transfer problems in non coherent scattering , in problems of
multiple scattering and also in time dependent problems of radiative transfer to
determine the H – function related to those problems .
21. 21
21
Acknowledgement:
I express my sincere thanks to the Department of Mathematics , Heritage Institute
of Technology , Anandpur , West Bengal , India for the support extended.
References :
[1] Chandrasekhar , S . Radiative transfer , Dover , New York , 1950.
[2] Kourganoff,V.A. Basic methods in transfer problems. Clarendon Press,
Oxford., 1952.
[3].Busbridge,I.W. The mathematics of radiative transfer , Cambridge Tracts, No
50 , Cambridge, 1960.
[4]Fox ,S.C. A solution of Chandrasekhar’s integral equation , , Trans. Amer.
Math. Soc. 1961, 99 , 285-291.
[5].Dasagupta ,S.R.On H-functions of radiative transfer , Astrophysics and
space science ,1974 , 30,327-342.
[6].Dasagupta ,S.R.A new representation of the H-function of radiative transfer ,
Astrophysics and space science ,1977,50, 187-203. .
[7]Das ,R.N. Exact solution of the transport equation for radiative transfer with
scattering albedo w0 <1 using Laplace transform and Wiener hopf technique and
a expression of H- function, Astrophysics and space science ,60 ,(1979), 49-58.
[8].Zelazny,R. Exact solution of a critical problem for a slab , J. Math.Phys. , .
1961, 2,538-542.
[9] Mullikin , T.W.Chandrasekhar’s X and Y equations ( 1
)Trans. Amer. Math.
Soc. 1964,113 , 316-332.
[10] P.F. Zweifel ,P.F. Recent applications of neutron transport theory , Micigan
Memorial phoenix project report, university of Michigan, 1964.
[11].Ivanov ,V.V. Transfer of radiation in spectral lines, U.S Department of
commerce, National Bureau of Standards special publication 385, 1973.
[12] Siewert,C.E. On computing eigenvalues in radiative transfer, J. Math. Phys.
1980,21,2468-2470. .
[13] Das ,R.N. Application of the theory of Linear singular Integral equation to a
Riemann Hilbert Problem for a new representation of Chandrasekhar’s H- in
radiative transfer . submitted in 2006 for publication to the Journal of
Mathematical Analysis and Applications .(JMAA-06-2670 ).
[14] Siewert.C.E , Journal of Mathematical Physics , 1980, 21, 2468.
[15] Garcia , R.D.M and Siewert , C.E . Transport Theory and Statistical Physics
,1987, 14 , 437.
[16] Sulties ,J.K and Hill,T.H . Nuclear Science and Engineering , 1976, 59, 53.
[17] Barichello,L.B and Siewert , C.E . J of Quantitative Spectroscopy and
Radiative Transfer , 1998, 60, 261.
[18] Barichello,L.B and Siewert , C.E . J of Quantitative Spectroscopy and
Radiative Transfer , 1999, 62, 665.
22. 22
22
[19] Bergeat ,J and Rutily ,B. Auxiliary functions for Radiative transfer problems
in plane parallel geometry - I – the infinite medium , J of Quantitative
Spectroscopy and Radiative Transfer , 1996, 52, 857-886.
[20] Bergeat ,J and Rutily ,B. Auxiliary functions for Radiative transfer problems
in plane parallel geometry - II – the semi- infinite medium , J of Quantitative
Spectroscopy and Radiative Transfer , 1998, 60, 1033-1051.
[21]Busbridge, I.W. On solutions of Chandrasekhar’s Integral equations, Trans.
Amer. Math. Soc. 1962 , 105 ,112-117.
[22]Muskhelishvili,N.I. Singular integral equations , Noordhoff, Groningen,1953.
[23] Dasgupta,S.R. Doctoral thesis, Radiation problems in Astrophysics, Calcutta
University ,1962.
[24].Abhyankar,K.D and Fymat ,A.L. On the solution H1(u) of Ambarzumian –
Chandrasekhar H- equation, J. Quant. Spectrosc.Radiat.Transfer, 1969, 9 , 1563-
1566.
-----------------------------------------------