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2013 | Buch

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Special Functions of Mathematical (Geo-)Physics

verfasst von: Willi Freeden, Martin Gutting

Verlag: Springer Basel

Buchreihe : Applied and Numerical Harmonic Analysis

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction: Geomathematical Motivation

Abstract

In the first chapter we briefly introduce four fields showing strong geophysical background. Thereby, we are naturally led to differential equations which are closely related to solution systems of special functions. Since the Earth is a ball in first approximation, a spherical coordinate frame and spherical functions play a huge role in geomathematics.

Willi Freeden, Martin Gutting

Auxiliary Functions

Frontmatter

Chapter 2. The Gamma Function

Abstract

In what follows, we introduce the classical Gamma function in Sect. 2.1. It is essentially understood to be a generalized factorial. However, there are many further applications, e.g., as part of probability distributions (see, e.g., Evans et al. 2000). The main properties of the Gamma function are explained in this chapter (for a more detailed discussion the reader is referred to, e.g., Artin (1964), Lebedev (1973), Müller (1998), Nielsen (1906), and Whittaker and Watson (1948) and the references therein).

Willi Freeden, Martin Gutting

Chapter 3. Orthogonal Polynomials

Abstract

In this chapter, we introduce polynomial function systems that are orthogonal with respect to a scalar product characterized by a measure $\mathrm{d} \uplambda$. We start with some very general results from Fourier analysis (see, e.g., Davis 1963; Reed and Simon 1972; Rudin 1991; Yoshida 1980), before we begin to specifically consider polynomials.

Willi Freeden, Martin Gutting

Spherically Oriented Functions

Frontmatter

Chapter 4. Scalar Spherical Harmonics in $$\mathbb{R}^3 $$

Abstract

As we have seen in the geomathematical motivation of Chap. 1, spherical functions are an essential tool for all geosciences. In this chapter we develop orthonormal function systems for scalar functions on spheres in the three-dimensional Euclidean space, namely the scalar spherical harmonics, which are then generalized to vectorial functions in Chap. 5 as well as to systems for scalar functions on spheres in the more general, q-dimensional setting in Chap. 6.

Willi Freeden, Martin Gutting

Chapter 5. Vectorial Spherical Harmonics in $${\mathbb{R}}^{3}$$

Abstract

Various applications imply different formulations of vector spherical harmonics, putting the accent on different issues (see, e.g., Sects. 1.3 and1.4). One important aspect in our understanding is the easy transition from scalar spherical harmonics to the vectorial ones. A simple approach is to formulate the vectorial problem in terms of Cartesian components. However, this procedure leads back to anisotropic scalar component equations, so that the physical relevance usually is difficult to realize, the mathematical formulation is lengthy, and modeling often becomes complicated.

Willi Freeden, Martin Gutting

Chapter 6. Spherical Harmonics in $${\mathbb{R}}^{q}$$

Abstract

The theory of scalar spherical harmonics of Chap. 4can be generalized to spheres in the q-dimensional space, i.e., from ${\mathbb{S}}^{2} \subset {\mathbb{R}}^{3}$ to ${\mathbb{S}}^{q-1} \subset {\mathbb{R}}^{q}$. Obviously, this leads to a more extensive notation and makes some formulas a bit unwieldy. However, many proofs and the whole line of thought of the three-dimensional case carry over to the general setting such that we can skip some details that are analogous.

Willi Freeden, Martin Gutting

Chapter 7. Classical Bessel Functions

Abstract

The classical theory of Bessel functions is closely connected with the investigation of the integral (see(7.3.16))

$$\frac{1} {2\pi }\int \nolimits \nolimits _{0}^{2\pi }\cos (k\sin (u) - ku)\ \mathrm{d}u$$

(7.0.1)

by Bessel(1824). He took k as an integer and obtained many results. After the time of Bessel, investigations of these integrals, which by then bore his name, became numerous.

Willi Freeden, Martin Gutting

Chapter 8. Bessel Functions in $$\mathbb{R}^q$$

Abstract

By virtue of polar coordinates, metaharmonic functions, i.e., the solutions of the Helmholtz equation (Δ+λ)U=0, can be decomposed into a radial and an angular part. The Funk–Hecke formula (see Theorem 6.5.5) serves as the appropriate tool for decomposition. The angular part leads back to spherical harmonics, while the radial part satisfies a characteristic differential equation. Its solutions are the Bessel functions.

Willi Freeden, Martin Gutting

Periodically Oriented Functions

Frontmatter

Chapter 9. Lattice Functions in $$\mathbb{R}$$

Abstract

The Euler summation formula expresses a finite sum of integral points in terms of the integral and derivatives of the function with explicit knowledge of the error in integral form involving the Bernoulli polynomial.

Willi Freeden, Martin Gutting

Chapter 10. Lattice Functions in $$\mathbb{R}^q$$

Abstract

If an attempt is made to generalize the one-dimensional theory to a higher dimensional case, we are confronted with pointwise convergence problems for the bilinear series of the multi-variate counterpart of G(Δ; ⋅). Nonetheless, as we have already seen in the one-dimensional case in Chap.9, we are able to circumvent any possible calamities by paying close attention to the defining constituents. However, the q-dimensional theory remains more complicated, since the characteristic singularity of the lattice function in lattice points becomes much harder to handle with increasing dimension. In conclusion, the proof of the Euler summation formula associated to the Laplace operator as well as the specification of sufficient criteria for validity of the Poisson summation formula is a matter of multi-dimensional potential theory. The results obtained in such a way are applicable in many branches, e.g., the calculation of certain lattice point sums involving charged particles, functional equations of Zeta and Theta functions, etc. Some of the applications are worked into exercises in Sect.10.9, where we also explain the interrelations between Green and spline functions. Our multi-periodic approach is based on the concepts of metaharmonic lattice point theory as presented in Freeden(2011).

Willi Freeden, Martin Gutting

Chapter 11. Concluding Remarks

Abstract

Willi Freeden, Martin Gutting

Backmatter

Titel: Special Functions of Mathematical (Geo-)Physics
verfasst von: Willi Freeden
Martin Gutting
Verlag: Springer Basel
Electronic ISBN: 978-3-0348-0563-6
Print ISBN: 978-3-0348-0562-9
DOI: https://doi.org/10.1007/978-3-0348-0563-6