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

628

THEORY OF BESSEL FUNCTIONS

[CHAP'. XIX

at 0, +1, ~ 2,.. is concerned, we may omit the second term on the right in (1),
and calculate the residues of
pF, ( ()
sin 7rz
where F (z) is defined by the formula
() -- (2) F (lOS 2 I d- tJ e-izv f (tv) dA dt.
(2) = 2) 2r( -^ V)r(-) ()- tJ ) _e. dt.
Again, from. 19'22 (6) and equation (1) of this section we have
'IO (np7 -1 I dt2vt
-,1 O 2r r ( 1- t -v ) ) ) — (
(3) (- V) F() JY o dtL
1n=l1 (n + ~-d- J
In the special case in which v the modified form of (3) shews that an
additionalhe fir) must inserted on the right in (is equ to ( + )f), except when
hen we change the notation to the nota question normally used for Bessel
functions and Struvs functions, t he expansion becomes
(5) (o) (o) - [ + a(- mJ) (+ b (- mx)
wheret
(6) ( am= 1 f —^ r ( i) - t)- [L^ vf f (to)snvdv] dt,
m=l
l bmr(l-^r - ( t)'sy t [I V fi f (tv) sin mvdvj dt.
This is the spgenerali cse in which 0, the moed form of Schmilchs that aexpansion.
19 24. The boundedness of F(z), as! z 1 C.
We shall now prove that, when the. function f (x) is restricted, in. a suitable manner,
ddithe function term f() must b inserted on the value of arg z The reader).
will remember then we changet the assu tion th at (isbounded was malde in 1921 to secure
the converges and Struves functions the contour integral.comes
We take the series of ~ 19-22 (1), by which F(z) was originally defined, namely
o a () + b H, (x)
~(am,~ ~~~~ and1 d
n=1-r(/ l(tv) Cos -v dt,
* When v is negative it is necessary to use a modified expression for the integrals; cf, ~ 19 3.
(6)2
b[ - - 2 f (tv) sin mv dv dt.
This is the generalised form of Sch16milch's expansion.
19'24. The boundedness of F(z), as I z ->.
We shall now prove that, when the function f (x) is restricted, in, a suitable manner,
the function F (z) is bounded when I z o, whatever be the value of arg z. The reader
will remember that the expression for has to be odified by thmade insertion of 19 tm 2(0), insecure
th consequence of the continuity in value of the expression the right of (3).
We take the series of ~ 19'22 (1), by which F(z) was originally defined, namely
~ When v is negative it is necessary to use a modified expression for the integrals' cf, ~:19'3.
t When - = 0, the expression for a0 has to be modified by the insertion of the term 2f (0), in,
consequence of the discontinuity in value of the expression on the right of (3).