Bessel and Mittag–Leffler functions are prominent within mathematical and scientific fields due to increasing interest in non-conventional models within applied mathematics. Since the analytical solutions of many differential and integral equati
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Yes, you can access From Bessel To Multi-index Mittag-leffler Functions: Enumerable Families, Series In Them And Convergence by Jordanka Paneva-Konovska in PDF and/or ePUB format, as well as other popular books in Biological Sciences & Science General. We have over one million books available in our catalogue for you to explore.
Solving a set of problems in mechanics and mathematical physics is closely related to the Bessel differential equation
The function Jν(z), defined by the equality
is a solution of this equation in the domain C\(−∞, 0] (see e.g. [Watson (1949, 3.1 (8))]). It is called Bessel function of the first kind with an index ν. The function J−ν(z) is also a solution for the above equation. The Bessel functions of the first kind with an integer index are also called Bessel coefficients. They are holomorphic in the whole complex plane.
First, let us consider the case when the parameter ν is not integer. Then the linear combinations (see [Erdélyi et al. (1953, 7.2 (4)–(6))]):
are also solutions of the differential equation (1.1). Yν(z) are called Bessel functions of the second kind, and Hν(1)(z) and Hν(2)(z) are called Bessel functions of the third kind, also known as first and second Hankel functions.
If ν is an integer, then the left-hand side of the equalities (1.3)–(1.5) are not defined. But their limits, when ν → n (n is an integer), exist and they can be used to define the Bessel functions of the second and third kinds with an integer index. In particular, we have Yn(z) = limν→n Yν(z) (see [Erdélyi et al. (1953, 7.2 (28))]), i.e.
The functions Jν(z) and J−ν(z) form a fundamental system of solutions of the differential equation (1.1) iff ν is not an integer (see [Watson (1949, 3.12)]), whereas Jν(z) and Yν(z) always form a fundamental system of solutions of this equation [Watson (1949, 3.63)]. One of the primary motives for introducing the functions Yν(z) is the necessity of a second solution that is linearly independent of Jν(z), when ν = n is a nonnegative integer.
1.2.Modified Bessel Functions
The solving of some problems in mathematical physics is often related to the differential equation
that differs from the Bessel equation by the coefficient of w and it can be obtained from (1.1) by replacing iz instead of z.
The system {Jν(iz), J−ν(iz)} as well as {Jν(iz), Yν(iz)} are fundamental systems of solutions of Eq. (1.7) in the domain z ∈ C\(−∞, 0], but more frequently used are the functions [Watson (1949, 3.7 (2)); Erdélyi et al. (1953, 7.2 (12))]
and I−ν(z). They are called modified Bessel functions of the first kind. The Bessel and modified Bessel functions of the first kind are related by simple dependence in the corresponding domains of the complex plane, namely:
The functions [Erdélyi et al. (1953, 7.2 (13), (36))]
are also solutions of Eq. (1.7). They are called modified Bessel function of the third kind, although the modern definition is given by MacDonald [Erdélyi et al. (1953), 7.2.2].
The functions with an index of the kind n + 1/2 (n = 0, ±1, . . . ), called Bessel functions with a half-integer index or also as spherical Bessel functions, form an interesting class of functions. They can be expressed as rational functions of
, cos z, sin z, and exp z. ...
Table of contents
Cover
Halftitle
Title
Copyright
Dedication
Preface
Acknowledgment
Introduction
Contents
1. Bessel and Associated Functions
2. Generating Functions of Bessel and Associated Bessel Functions
3. Convergence of Series in Bessel Functions
4. Bessel and Neumann Expansions
5. The Completeness of Systems of Bessel and Associated Bessel Functions in Spaces of Holomorphic Functions
6. Multi-index Bessel Functions
7. Mittag-Leffler Type Functions
8. Latest Generalizations of Both the Bessel and Mittag-Leffler Type Functions
