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

6-12]

INTEGRAL REPRESENTATIONS

167

As in the corresponding analysis of ~ 6'1, the ranges of validity of (4) and
(5) may be extended by swinging round the contours and using the theory of
analytic continuation.
Thus, if - 2r < ) < TVr, we have
r rr < w (1_r).vv~J (I
(6 (I) — _ (12-V) r 2 r eizt (t2 — 1)v-2 dt;
(6 H7 i (2) Xooiexp(-io)
while, if - 37r < co < 1r, we have
(7) H(2) (Z)= 2-) (2) f (izt (t2 - 1)- dt,
7r- - (-2) cep (-I__
2700 i exp(-w)
provided that, in both (6) and (7), the phase of z lies between - 7r +C and
~7r + Co.
Representations are thus obtained of H,(1) (z) when arg z has any value
between -r7 and 27r, and of HV(2) (z) when arg z has any value between
- 27r and 7-.
If co be increased beyond the limits stated, it is necessary to make the contours coil
round the singular points of the integrand, and numerical errors are liable to occur
in the interpretation of the integrals unless great care is taken. Weber, however, has
adopted this procedure, Math. Ann. xxxvii. (1890), pp. 411-412, to determine the formulae of ~ 3-62 connecting H,(1) (- z), H(2) (- z) with I(1) (z), H (2) (z).
NOTE. The formula 2iYv (z)=H V(1) (z)- (2) (z) makes it possible to express Yv (z) in
terms of loop integrals, and in this manner Hankel obtained the series of ~ 3'52 for
Yn(z); this investigation will not be reproduced in view of the greater simplicity of
Hankel's other method which has been described in ~ 3'52.
6*12. Integral representations of functions of the third kind.
In the formula ~ 6-11 (6) suppose that the phase of z has any given value
between - r and 27r, and define 13 by the equation
arg z = o + /,
so that - 7r< </3 < 7tr.
Then we shall write
t - 1 = e-"i z- (- U),
so that the phase of - u increases from - r + / to vr + / as t describes the
contour; and it follows immediately that
i /(1 - ) e(Z-V-IT) (0+) ( i \ V -
(1) gI(1) (z) = -2 — f e-(-u)v-1(l+ ) du,
rV/(2z).' co exp i 2z
where the phase of 1 + iu/z has its principal value. Again, if /3 be a given
acute angle (positive or negative), this formula affords a representation of
Hv,,( (z) valid over the sector of the z-plane in which
- -7r + / < arg z < 37vr + /.