Introduction to Bessel Functions

Frank Bowman

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486604626
• eBook ISBN: 9780486152998
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

Physicists, mathematics students, designers of vibrating systems, mechanical, electrical, and civil engineers, aero- and hydro-dynamicists, and teachers of mathematics, physics, and engineering will find this book a full, clear introduction to the properties and applications of Bessel functions. Practically all of the advanced mathematics needed is developed in the text. Students who have had some calculus and the elements of differential equations should find this work completely self-contained and suitable for self-study, classroom use, or as a supplement to more general texts.
Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order. More than 200 problems throughout the book enable students to test and extend their understanding of the theory and applications of Bessel functions.

CHAPTER I

BESSEL FUNCTIONS OF ZERO ORDER

§ 1. Bessel’s function of zero order.

The function known as Bessel’s function of zero order, and denoted by J₀(x), may be defined by the infinite power-series

(1.1)

If u_r denotes the rth term of this series, we have

which → 0 when r → ∞, whatever the value of x. Consequently, the series converges for all values of x, and since it is a power-series, the function J₀(x) and all its derivatives are continuous for all values of x, real or complex.

§ 2. Bessel’s function of order n, when n is a positive integer.

The function J_n(x), known as Bessel’s function of order n, may be defined, when n is a positive integer, by the infinite power-series

(1.2)

which converges for all values of x, real or complex.

In particular, when n = 1 we have

(1.3)

and when n = 2

(1.4)

We note that J_n(x) is an even function of x when n is even, odd when n is odd.

The graphs of J₀(x), J₁(x) are indicated in Fig. 1.

FIG. 1.

Extensive tables of values of J_n(x), especially of J₀(x) and J₁(x), have been calculated on account of their applications to physical problems.¹

§ 3. Bessel’s equation of zero order.

By differentiating the series for J₀(x) and comparing the result with the series for J₁(x), we find ²

(1.5)

Again, after multiplying t...

Table of contents

• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• CHAPTER I - BESSEL FUNCTIONS OF ZERO ORDER
• CHAPTER II - APPLICATIONS
• CHAPTER III - MODIFIED BESSEL FUNCTIONS
• CHAPTER IV - DEFINITE INTEGRALS
• CHAPTER V - ASYMPTOTIC EXPANSIONS
• CHAPTER VI - BESSEL FUNCTIONS OF ANY REAL ORDER
• CHAPTER VII - APPLICATIONS
• INDEX.