If and the upper signs are taken, then the restriction on is unnecessary.
The restriction is unnecessary when and is an integer. Special cases are:
the branches being continuous and chosen so that and as . If , are real and positive and , then and are real and nonnegative, and the geometrical relationship is shown in Figure 10.23.1.
The degenerate form of (10.23.8) when is given by
For expansions of products of Bessel functions of the first kind in partial fractions see Rogers (2005).
where is the distance of the nearest singularity of the analytic function from ,
and is Neumann’s polynomial, defined by the generating function:
is a polynomial of degree in and
where is Euler’s constant and (§5.2).
where is as in §10.21(i). If , then
provided that is of bounded variation (§1.4(v)) on an interval with . This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near and .
As an example,
(Note that when the left-hand side is 1 and the right-hand side is 0.)