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Table Of Integrals Series And Products

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

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Principles Of Optics Electromagnetic Theory Of Propagation Interference And Diffraction Of Light

Wave propagation and group velocity

Preface Bibliography 1. Interpolation. Introduction The Taylor expansion The finite Taylor series with the remainder term Interpolation by polynomials The remainder of Lagrangian interpolation formula Equidistant interpolation Local and global interpolation Interpolation by central differences Interpolation around the midpoint of the range The Laguerre polynomials Binomial expansions The decisive integral transform Binomial expansions of the hypergeometric type Recurrence relations The Laplace transform The Stirling expansion Operations with the Stirling functions An integral transform of the Fourier type Recurrence relations associated with the Stirling series Interpolation of the Fourier transform The general integral transform associated with the Stirling series Interpolation of the Bessel functions 2. Harmonic Analysis. Introduction The Fourier series for differentiable functions The remainder of the finite Fourier expansion Functions of higher differentiability An alternative method of estimation The Gibbs oscillations of the finite Fourier series The method of the Green's function Non-differentiable functions Dirac's delta function Smoothing of the Gibbs oscillations by Fejer's method The remainder of the arithmetic mean method Differentiation of the Fourier series The method of the sigma factors Local smoothing by integration Smoothing of the Gibbs oscillations by the sigma method Expansion of the delta function The triangular pulse Extension of the class of expandable functions Asymptotic relations for the sigma factors The method of trigonometric interpolation Error bounds for the trigonometric interpolation method Relation between equidistant trigonometric and polynomial interpolations The Fourier series in the curve fitting 3. Matrix Calculus. Introduction Rectangular matrices The basic rules of matrix calculus Principal axis transformation of a symmetric matrix Decomposition of a symmetric matrix Self-adjoint systems Arbitrary n x m systems Solvability of the general n x m system The fundamental decomposition theorem The natural inverse of a matrix General analysis of linear systems Error analysis of linear systems Classification of linear systems Solution of incomplete systems Over-determined systems The method of orthogonalisation The use of over-determined systems The method of successive orthogonalisation The bilinear identity Minimum property of the smallest eigenvalue 4. The Function Space. Introduction The viewpoint of pure and applied mathematics The language of geometry Metrical spaces of infinitely many dimensions The function as a vector The differential operator as a matrix The length of a vector The scalar product of two vectors The closeness of the algebraic approximation The adjoint operator The bilinear identity The extended Green's identity The adjoint boundary conditions Incomplete systems Over-determined systems Compatibility under inhomogeneous boundary conditions Green's identity in the realm of partial differential operators The fundamental field operations of vector analysis Solution of incomplete systems 5. The Green's Function. Introduction The role of the adjoint equation The role of Green's identity The delta function -- The existence of the Green's function Inhomogeneous boundary conditions The Green's vector Self-adjoint systems The calculus of variations The canonical equations of Hamilton The Hamiltonisation of partial operators The reciprocity theorem Self-adjoint problems Symmetry of the Green's function Reciprocity of the Green's vector The superposition principle of linear operators The Green's function in the realm of ordinary differential operators The change of boundary conditions The remainder of the Taylor series The remainder of the Lagrangian interpolation formula

Linear Differential Operators. By C. Lanczos. Pp. 580. 80s. 1961. (Van Nostrand, London)

Introduction and some remarks on generalisations of complex analysis a-holomorphic function theory Electrodynamical models Massive spinor fields Hypercomplex factorization, systems of non-linear partial differential equations generated by Futer-type operators.

Integral Representations for Spatial Models of Mathematical Physics

Applied quaternionic analysis

We consider a recently discovered representation for the general solution of the Sturm–Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm–Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm–Liouville problems, effectively by calculation of the roots of a polynomial. We discuss the examples of the numerical implementation of the SPPS method and show it to be equally applicable to a wide class of singular Sturm–Liouville problems as well as to problems with spectral parameter-dependent boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.

Spectral parameter power series for Sturm–Liouville problems

Introduction.- I. Pseudoanalytic function theory and second-order elliptic equations.- 1. Definitions and results from Bers' theory.- 2. Second order equations.- 3. Formal powers.- 4. Cauchy's integral formula.- 5. Complex Riccati equation.- II. Applications to Sturm-Liouville theory.- 6. Sturm-Liouville equation.- 7. Spectral problems and Darboux transformation.- III. Applications to real first-order systems.- 8. Beltrami fields.- 9. Static Maxwell system.- IV. Hyperbolic pseudoanalytic functions.- 10. Hyperbolic numbers and analytic functions.- 11. Hyperbolic pseudoanalytic functions.- 12. Klein-Gordon equation.- V. Bicomplex and biquaternionic pseudoanalytic functions and applications.- 13. The Dirac equation.- 14. Complex second order elliptic equations and bicomplex pseudoanalytic functions.- 15. Multidimensional second order equations.- Open problems.- Bibliography.- Index.

Applied Pseudoanalytic Function Theory

A representation for the general solution of the equation (pu′)′ + qu = ω2 u in terms of a non-trivial solution of is given. This result is obtained with the aid of the theory of pseudoanalytic functions and their relationship to solutions of the stationary two-dimensional Schrodinger equation. The representation has a simple and easily verifiable form and lends itself to numerical computation. Its applications to spectral problems are discussed.

Vladislav V. Kravchenko

Papers

Integral Representations for Spatial Models of Mathematical Physics

Applied quaternionic analysis

Spectral parameter power series for Sturm–Liouville problems

Applied Pseudoanalytic Function Theory

A representation for solutions of the Sturm–Liouville equation