A treatise on the theory of Bessel functions, by G. N. Watson.

CHAPTER III

BESSEL FUNCTIONS
3'1. The generalisation of Bessel's differential equation.
The Bessel coefficients, which were discussed in Chapter II, are functions
of two variables, z and n, of which z is unrestricted but n has hitherto been
required to be an integer. We shall now generalise these functions so as to
have functions of two unrestricted (complex) variables.
This generalisation was effected by Lommel*, whose definition of a Bessel
function was effected by a generalisation of Poisson's integral; in the course
of his analysis he shewed that the function, so defined, is a solution of the
linear differential equation which is to be discussed in this section. Lommel's
definition of the Bessel function Jv (z) of argument z and order v wast
J (z)= Fr ( r) r () cos (z cos 0) sin2v OdO,
1 ( +~) i) (D) o
and the integral on the right is convergent for general complex values of v
for which R(v) exceeds -. Lommel apparently contemplated only reAl
values of v, the extension to complex values being effected by Hankel+;
functions of order less than - were defined by Lommel by means of an extension of the recurrence formulae of ~ 2'12.
The reader will observe, on comparing ~ 3'3 with ~ 1'6 that Plana and
Poisson had investigated Bessel functions whose order is half of an odd integer
nearly half a century before the publication of Lommel's treatise.
We shall now replace the integer n which occurs in Bessel's differential
equation by an unrestricted (real or complex) number~ v, and then define a
Bessel function of order v to be a certain solution of this equation; it is of
course desirable to select such a solution as reduces to J, (z) when v assumes
the integral value n.
We shall therefore discuss solutions of the differential equation
(1) 2" d~y dy
~~~~(1) gcz + (Z2 - 2) y = o,
which will be called Bessel's equation for functions of order v.
* Studien fiber die Bessel'schen Functionen (Leipzig, 1868), p. 1.
+ Integrals resembling this (with v not necessarily an integer) were studied by Duhamel, Cours
d'Analyse, n. (Paris, 1840), pp. 118-121.
$ Math. Ann. i. (1869), p. 469.
~ Following Lommel, we use the symbols v, A to denote unrestricted numbers, the symbols
n, nz being reserved for integers. This distinction is customary on the Continent, though it has
not yet come into general use in this country.- It has the obvious advantage of shewing at a
glance whether a result is true for unrestricted functions or for functions of integral order only.