Let us establish a few important properties of , starting by its behavior under the inversion of the sign of . We start with the analogous property of , for which we have, using the Maclaurin series for these functions [3], which converges over the whole complex plane,
(5) |
Using this in the expression for we get
(6) |
that is, is an odd function of , for all and all .
Next we show that has a simple pole at . Since , and are analytical functions over the whole complex- plane, it follows that is analytical over the whole plane except for those points where the denominator vanishes, where it has poles. These are the origin and the zeros of the Bessel function in the denominator. Note that while for non-integer and the functions involved have branching points at , the function never does. In order to determine the residue of at we consider the limit
(7) |
where is the gamma function and we used once more the Maclaurin series for . Since the limit is finite and non-zero, it follows that has a simple pole at , and that is the corresponding residue. Turning to the poles at , since is a simple zero of , at which its derivative is different from zero, it follows that has a Taylor expansion around this point, with the form
(8) |
for certain finite coefficients . In order to determine the residue of at we consider then the limit
(9) |
where we used this Taylor expansion. Since the derivative is finite and non-zero at , this limit also is finite and non-zero, and hence it follows that has a simple pole at , and that is the corresponding residue. We can simplify this expression using the well-known identity [4]
(10) |
which applied at , since , results in
(11) |
It follows therefore that we have for the residues of the poles at ,
(12) |
We will now establish the behavior of the absolute value of the ratio of Bessel functions which is contained in the expression of in Eq. (3), for large values of . In order to do this we use the asymptotic expansion of the Bessel functions [5], valid in the whole complex plane so long as , written in terms of and , to the lowest orders, and with the trigonometric functions expressed as complex exponentials,
where and are certain limited functions of and is a certain real number, given by
(14) |
The behavior of the expression in Eq. (13) for large values of depends on the sign of , and the particular case has to be examined separately. In this particular case we have
(15) |
where all the functions involved are now limited, so that for large values of we have for the dominant part of ,
(16) |
Note now that the points where are the zeros of , expressed in the asymptotic limit. We will now choose a way to take the limit such that these zeros are avoided. We may simply chose for the passage of the circuit across the real axis that point between two zeros where and . Since for we have
(17) |
and we must have for some integer , we conclude that Eq. (4) holds, which will cause the crossing of the circuit and the real axis to avoid the zeros. This defines the limit in full detail. It follows that for our purposes here we may write the asymptotic expansion in the case as
(18) |
In the case we put the dominant real exponential in evidence and obtain
(19) |
where all the functions within the brackets are now limited or go to zero in the limit. Finally, we do the same thing for the case , obtaining
(20) |
where once more all the functions within the brackets are now limited or go to zero in the limit. We are now in a position to analyze the behavior of the absolute value of the ratio of two Bessel functions which appears in the definition of . The factors which do not depend on are common to the numerator and denominator, and cancel out. In the case we get
(21) |
so that in the limit we get
(22) |
It is not difficult to verify that for both the case and the case we get this same value for this limit. We see therefore that the limit of the absolute value of this ratio is simply , for all values of in .