Freeden W., Gutting M. Special Functions of Mathematical (Geo-)Physics

Springer Basel, 2013. XV, 501 p. 37 illus., 15 illus. in color. — ISBN: 978-3-0348-0562-9, ISBN: 978-3-0348-0563-6 (eBook), DOI 10.1007/978-3-0348-0563-6.Presents special functions as essential tools contributing to solutions for geoscientific problems.
Attractive textbook for the education in geomathematics.
Addresses mathematicians, physicists, geo-engineers and geoscientists.
Special functions enable us to formulate a scientific problem by reduction such that a new, more concrete problem can be attacked within a well-structured framework, usually in the context of differential equations. A good understanding of special functions provides the capacity to recognize the causality between the abstractness of the mathematical concept and both the impact on and cross-sectional importance to the scientific reality.The special functions to be discussed in this monograph vary greatly, depending on the measurement parameters examined (gravitation, electric and magnetic fields, deformation, climate observables, fluid flow, etc.) and on the respective field characteristic (potential field, diffusion field, wave field). The differential equation under consideration determines the type of special functions that are needed in the desired reduction process.Each chapter closes with exercises that reflect significant topics, mostly in computational applications. As a result, readers are not only directly confronted with the specific contents of each chapter, but also with additional knowledge on mathematical fields of research, where special functions are essential to application. All in all, the book is an equally valuable resource for education in geomathematics and the study of applied and harmonic analysis.Students who wish to continue with further studies should consult the literature given as supplements for each topic covered in the exercises.
Content Level » Upper undergraduateKeywords » Cauchy–Navier and Navier-Stokes equation - Laplace and Poisson equation -Maxwell equation - constructive approximation by function systems - spherically and periodically oriented functions - spheroidization and periodization
Related subjects » Birkhäuser Geoscience - Birkhäuser Mathematics.Introduction: Geomathematical Motivation
Example: Gravitation (Laplace and Poisson Equation)
Example: Geomagnetism (Maxwell’s Equations)
Example: Fluid Flow (Navier–Stokes Equation)
Example: Elastic Field (Cauchy–Navier Equation)Auxiliary Functions
The Gamma Function
Definition and Functional Equation
Euler’s Beta Function
Stirling’s Formula
Pochhammer’s Factorial
Exercises (Incomplete Gamma and Beta Function, Applications in Statistics)
Orthogonal Polynomials
Properties of Orthogonal Polynomials
Quadrature Rules and Orthogonal Polynomials
The Jacobi Polynomials
Ultraspherical Polynomials
Application of the Legendre Polynomials in Electrostatics
Hermite Polynomials and Applications
Laguerre Polynomials and Applications
Exercises (Gaussian Integration, Legendre Series, Kernel Expansions)Spherically Oriented Functions
Scalar Spherical Harmonics in R³
Basic Notation
Orthogonal Invariance
Homogenous Polynomials on theUnit Sphere in R³
Closure and Completeness of Spherical Harmonics
The Funk–Hecke Formula and the Irreducibility of Scalar Harmonics
Green’s Function with Respect to the Beltrami Operator
The Hydrogen Atom
Exercises (Low Discrepancy Method, Locally Supported Wavelets, Up Function, Anharmonic
Functions for the Ball, Fast Multipole Method,Wigner Matrices, Quaternionic Generation of Spherical Harmonics)
Vectorial Spherical Harmonics in R³
Basic Notation
Definition of Vector Spherical Harmonics
The Helmholtz Decomposition Theorem
Closure and Completeness of Vector Spherical Harmonics
Homogeneous Harmonic Vector Polynomials
Vectorial Beltrami Operator
Vectorial Addition Theorem
Vectorial Funk–Hecke Formulas
Vectorial Counterparts of the Legendre Polynomial
Application to Elastic Fields
Exercises (Uncertainty Principle, Classification of Zonal Functions, Coupling Integrals and Navier–Stokes Equation)
Spherical Harmonic in R^q
Nomenclature and Basics
Integral Theorems for the Laplace Operator
Integral Theorems for the Laplace–Beltrami Operator
Homogeneous Harmonic Polynomials
Spherical Harmonic of Dimension q
Integral Theorems for the Helmholtz–Beltrami Operator
Exercises (Cartesian Generation of Spherical Harmonics, Best Approximations)
Classical Bessel Functions
Derivation and Definition of Bessel Functions
Orthogonality Relations
Bessel Functions with Integer Index
Exercises (Bessel Function Expansions, Hankel Transform and Discontinuous Integrals)
Bessel Functions in R^q
Regular Bessel Functions
Modified Bessel Functions
Hankel Functions
Kelvin Functions
Expansion Theorems
Exercises (Helmholtz Equation, Entire Solutions, Bessel Function Like Asymptotics)Periodically Oriented Functions
Lattice Functions in R
Bernoulli Polynomials
Periodic Polynomials
Lattice Functions
Euler Summation Formula
Riemann Zeta Function
Poisson Summation Formula for the Laplace Operator
Theta Function
Exercises (Trapezoidal Rule, Periodic Sobolev Spaces, Projection Method)
Lattice Functions in R^q
Lattices in Euclidean Spaces
Periodic Polynomials
Lattice Function for the Laplace Operator
Euler Summation Formula for the Laplace Operator
Zeta Functions
ntegral Asymptotics for Lattice Functions
Poisson Summation Formula
Theta Functions
Exercises (Algebraic, Periodic, and Spherical Splines, Lattice Point Sums, Lattice Point Distributions)
Concluding Remarks