Table Of Content,
This thesis having been approved by the
,
special Faculty Committee is accepted by
the Graduate School of the
,
University of Wyoming
in partial fulfillment of the requirements
Mastir__of__A.rts__
for the degree of 3
________
Dean of the Graduate School.
August $,
Tint# 1 1950
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SOME GENERALIZATIONS OP THE
BESSEL FUNCTION
by
Lawrence J. Prince
A Thesis
Submitted to the Department
of Mathematics and the Graduate School
of the University of Wyoming in Partial
Fulfillment of Requirements for the Degree of
Master of Arts
University of Wyoming
Laramie, Wyoming
August, 1950
LIBRARY
OF THE
UNIVERSITY OF WYOMING
LARAMIE
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UMI Number: EP24226
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ACKNOWLEDGEMENT
The writer wishes to express his gratitude to Doctor
Nathan Schwid for his invaluable guidance and assistance in
the preparation of this thesis.
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TABLE OP CONTENTS
Chapter Pag©
I. . .......................................... 1
Representation of Series Using the Gamma Function • 3
Integral Representation of Jn,m(x).......... 4
Recursion Formulas........................ . 8
The Functions of Integral Orders ............... 11
Zeros of Jn#m(x)....................... . 12
Orthogonality....... • • • • . ............... 18
Evaluation of the Norm. . . . . . . . . ..... 22
Closed Forms of Jn,m(x) ....................... 26
Fourier-Bessel Expansion of Functions . . . . . . . 29
Solution Corresponding to a Bessel Function of
Second Kind. ................................32
II.................. 41
Representation of Series Using the Gamma Function • 43
Integral Representation of Series 44
Recursion Formulas......... 46
Zeros of Kn,m(x). . . . . . . . . . . . . ....... 48
Orthogonality 50
Evaluation of the Norm. . . . . . . . . . . . . . . 54
SELECTED REFERENCES . ................................57
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INTRODUCTION
The purpose of this study is to present two generaliza
tions of the Bessel function. A Bessel function is usually
defined as a particular solution of the linear differential
equation of the second order,
(0.1) X* + X + (X*-/»/yU = 0 ,
d. x dx "
known as Bessel»s equation. In this equation, as we shall
consider it, n is any real number.
When n is neither a positive nor a negative integer, nor
zero, the general solution of Bessel's equation has been found
to be
^ A
J » + 0 £/«(*) 1
a
where A and B are arbitrary constants and Jn(x) is the partic
ular solution found by taking
<30
* m
■4 = o
and determining the coefficients A^ when s = n. J.n(x) is the
particular solution found by taking the above series solution
and determining the coefficients Ajj- when s= -n; it can be
shown that in such cases, Jn(x) and J.n(x) are independent
solutions of the Bessel equation.
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When n is an integer, the general solution of Bessel's
equation is
A 7m(x) &
^ = +
where Yn(x) is a Bessel function of the second kind of order
n and has been determined by various devices, the most widely
used being, the assumption of a logarithmic solution followed
by a determination of coefficients.
In the development of the theory of Bessel functions
credit must be given to many great mathematicians. Before
1824, when Bessel first published the work in which he had
examined in detail the function which now bears his name,
several other men had done work involving these functions. In
1738 Daniel Bernoulli published a number of theorems on the
oscillations of heavy chains. In one of these he presents a
series that is now described as a Bessel function of order
zero and he gives a proof of the fact that this function has
an infinite number of zeros.^
The vibrations of a membrane were investigated by Euler
In 1764. This investigation contains the earliest appearance
of a Bessel coefficient of general integral order.2
In the works of Fourier, appearing in 1822, the Bessel
functions of order zero occurred in the discussion of the
^Watson, G. N., 4 Treatise on the Theory of Bessel
Functions (Cambridge: University Press, 1945), p. 3.
gjbid., pp. 5-6.
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vl
symmetrical motion of heat in a solid circular cylinder.
Fourier also examined the expansion of an arbitrary function
into a series of Bessel functions of order zero and gave the
formula for the general coefficient in the expansion as a
definite integral.3
Other men associated with the Bessel functions before
1824 and whose works are equally as important as those men
tioned above, are: Riccati, Leibniz, James Bernoulli,
Lagrange, Carlini, Laplace, and Poisson.
After the time of Bessel, investigations on the functions
became so numerous that a chronological account of the develop
ment of the theory of Bessel functions is difficult. However,
it may be of interest to mention here that during the latter
part of the nineteenth century and the beginning of the twen
tieth, a great deal of interest arose in connection with the
logarithmic solution or Bessel functions of the second kind.
Some of the more important results during this period were
found by: Hankel, Bocher, Neumann, Schlafli, Weber, Lommel,
and Heine.
In addition to the work that has been done with the
Bessel equation, n a real number, a great deal of research,
with many important results, has been accomplished with the
Bessel equation, n a complex number. For details, the reader
may consult the references listed at the end of this paper.
3Watson, G. N., A Treatise on the Theory of Bessel
Functions (Cambridge: University Press, 1945), pp. 9-10.
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vii
The generalizations of the Bessel function presented in
the following pages were obtained by solving the equations
^ + 0 -*0 * ) X jL jf + (X l-/n - 2/m m )AJ O
--r
where m is any real number. These equations reduce to (0.1)
when m - 0.
The properties of the first generalization have been
worked out in detail and the form of the generalization of a
Bessel function of the second kind has been found. In the
second generalization, only the function corresponding to the
Bessel function of the first kind has been found and some of
the more general and important properties established. It
was determined that all properties holding for the general
Bessel equation hold for the two generalizations containing
the additional parameter. These generalizations, to have
practical applications would necessarily need to be supple
mented by tables similar to those now existing for values of
Jn (x) and Yn(x)• However, the properties of the Bessel func
tions have always held a great deal of Interest on purely
mathematical grounds, as indicated by the enormous amount of
work done by so many different mathematicians in developing
existing theory.
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