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

564

THEORY OF BESSEL FUNCTIONS

[CHAP. XVII

and that
e e- s fin2-#d#= o —sh cosh g0. cos2"-1 Od0
0o e-~ sins- ~i pd~ cosh ~7r7Jo
- X 2+ 121} ~2 + 32}... {I2 +(2m- 1)2'
By substitution in (5) and (6) and writing 2' for ' in (5) we at once infer the
truth of (1) and (2) when'R (v) > 0; and the mode of extending the results to
all other values' of X has already been explained. The required generalisations
of Meissel's expansions are therefore completely established.
17'32. The expansion of zn into a Kapteyn series.
With the aid of Meissel's generalised formula it is easy to obtain the
expansion of any integral power of z in the form of a Kapteyn series. It is
convenient to consider even powers and odd powers separately.
In the case of an even power, z2n, we take the equation given by ~ 1731 (1)
in the form
() 1 r2F(n+1+~i)r(n+l-i ) E J,(2mz)
2= n- r( + i r(nli(n+1 -i z_ II_2+ 2 ~
27rzi r =( +1 + i') r (m I -i)0
where the contour of integration is the circle i | = n +. Since both series
converge uniformly on the circle, when z lies in the domain K, term-by-term
integrations are permissible.
Consider now the value of
1r_ (12 + '2) (22 + 2)... (n2 + "2)d
27r7-i 1l=n+- 2n-i (MW2 + 2) b
When m - n, there are no poles outside the contour, and so the contour may
be deformed into an infinitely great circle, and the expression is seen to be
equal to unity; but when m > n, the poles + im are outside the circle and the
expression is equal to unity minus the sum of the residues of the integrand
at these two poles, i.e. to
(m + n)!
172n+1i (m- n — I)!'
The expression on the left of (1) is therefore equal to
2 J (2z)-2 Z (m +n)! Jm (2mz)
mr=l 1m=n+ 'm+l. ( n - -1)!
Next we evaluate
1 r (n + 1 + i ) r (n + 1 - i') d'
27riJ I s f+2 r (m + 1 + iC) r (m + 1 - id) 2n-2m+l