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

496 THEORY OF BESSEL FUNCTIONS [CHAP. XV
We now consider the four sides of C in turn. It is first to be observed that
on all the sides of C,
HV() (w) = (7w)w iei (W-Vr-i){1 + 1^ (~ )
(tw) = (- e-^-i> —4) = (+w2,v (W{)+
where nl,, v(w) and '2, (w) are 0 (1/w) when w qv I is large.
Now, since the integrand is an odd function*, we have, as B - oo,
1 [-i+17riz (^) J1+l(w) d _ 1 r+2 j+1 (M)
2ril iB+il (v) J, (w) 2r iB-I() J ()
1 [iB+I:il(v)
'-.i = 2 idw -i (v).
2ri J i~B —il(v)
Next take the integral along the upper horizontal side of C; this is equal to
21 iB+,ir i (v) J^+l (w) dw
1= = S {a + 82,V-1(w)}[I + 0 (e} 0 iw)] dw
r J iJB+-iI (v) + 1m+2, v (u)+ j
I2 7 I iB + m m77- + I p/r + 171
=2- ^r" + 177R()+{r+ 2 —i log iB + i(4 +0(1/B)
-m + R(v)+~,
as B - oo.
Similarly the integral along the lower side tends to the same value, and
so the limit of the integral along the three sides now considered is m -+ v +.
Lastly we have to consider the integral along the fourth side, and to do
this we first investigate the difference
J (-_) - tan (w - I r - 7),
which, when Iw is large, is equal to
2~+1 (1)
2w IV w
riB+in17r+1vr+17r
Now J tan (w - - 2vr -- - 7r) dw = O,
-iB+m^r+v7-r+trr
and so
1 iB+fn7r+ivfr+r4 Jv~_(w) d
2 riB+z+m7r+4r+ Jv (W)
= - l fiB+rn;+3+tlrr {2v +1 + o ( d
27r7 -+B+?+v17r+77r +r 2v qS2
-- (2v + 1) + 0 (l/m).
Hence the limit of the integral round the whole rectangle is in + 0 (1/mn).
* Allowance is made for the indentations, just specified, in the first step of the following analysis.