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

524

THEORY OF BESSEL FUNCTIONS

[CHAP. XVI

16'12. Neumann's* analogue of Laurent's theorem.
Let f(z) be a function of z which is analytic and one-valued in the ringshaped region defined by the inequalities
r z < R.
Let C and c be the contours formed by the circles
Iz= R, z =r,
both contours being taken counter-clockwise; then, if z be a point of the region
between the circles, we have
f(Y =1 I f(t)t + 1 If(t)dt
2 ciriJc t -Z 2wi-C Z-t
= ' J (z) f (t) (t) dt + 2i n (z) f (t) Jn (t) dt.
n=O 2t -c n=0 07 J
Consequently f(z) is expansible in the form
(1) f(z)= anJn (z) + an On (Z)
n=(0 n=0
where
(2) ao = 2J f (t) Ol (t) dt, an' = i f(t) J (t dt.
If the Laurent expansion off(z) in the annulus is
0b
f(z)= bn"Z+: bn'
n=O n=1 zn
we have, as in ~16'11,
aa0 = bo,
an —3n C n-21n (n
m=O m
~(3) l>~an= n Y 2 nm bn-2m, (n > 1).
(4) an' = ( )n+2M b + +
m-0 2n+-O m! (n + m)!n+m
16'13. Gegenbauer's generalisation of Neumann's expansion.
By using the polynomial An,,v(t) defined in ~9'2, Gegenbauert has
generalised the formula given in ~ 16'11.
If f(z) is analytic inside and on the circle z I= R, and if C denotes the
contour formed by this circle, we have
zf(z) =2if zvf(t) dt
)2-ij t-z
2 f {S J^+~, (z) An (t) f (t) dt,
27ri C (n=0 )
and so
(1) ZVf (z) = an J.+n (Z),
-00
* Theorie der Bessel'schen Functionen (Leipzig, 1867), pp. 36-39.
t Wiener Sitzungsberichte, LXXIV. (2), (1877), pp. 124-130. See Wiener Denkschriften, XLVIII.
(1884), pp. 293-316 for some special cases of the expansion.