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

10-42, 10-43]

ASSOCIATED FUNCTIONS

333

We deduce that, when j arg z I< 7r and z I is large,
(_1 Z)v-l rp (l,)m. (S2m): ]
(1) H, (z) = Y, (z) + 2 +t) (2) E-) + 0 (Z-2p)~
r + 2:0 +" 42) in=o 1(.-,).
provided that R (p- v + 2) > 0; but, as in ~ 7'2, this last restriction may be
removed.
This asymptotic expansion may also be written in the form
1 P-1 + 1
(2) H" (z) = Y () + F ( i + -m ) o (~ -2p-p)
wr m=0 F (at + 1 m 7,9.) (~Z)2m —v+l 0 Op
It may be proved without difficulty that, if v is real and z is positive, the
remainder after p terms in the asymptotic expansion is of the same sign
as, and numerically less than the first term neglected, provided that
R (p + - v) > 0. This may be established by the method used in ~ 7'32.
The asymptotic expansion* was given by Rayleigh, Proc. London lMath. Soc. xix. (1888),
p. 504 in the case v=0, by Struve, iein. de l'Acad. Imp. des Sci. de St Petersbourg, (7)
xxx. (1882), no. 8, p. 101, and Ann. der Phys. und Chemie, (3) xvii. (1882), p. 1012 in the
case v=l; the result for general values of v was given by J. Walker, The Analytical
Theory of Light (Cambridge, 1904), pp. 394-395.
If v has any of the values 2, -,..., then (1 + Us2/2)v-i is expressible as a
terminatingseries and Yv (z) is also expressible in a finite form. It follows
that, when v is half of an odd positive integer, H, (z) is expressible in terms
of elementary functions. In particular
Hi (z) = ( (I - cos z),
(:3) 1 2 21 Cs
__ I \ i' f^ f- \ f)- cos\
H ()) 2= J 1 + -)- ( sin +S
10'43. The asymptotic expansion of Struve's functions of large order.
We shall now obtain asymptotic expansions, of a type similar to the
expansions investigated in Chapter viii, which represent Struve's function
Hv (z) when Iv I and I z I are both large.
As usual, we shall write
v = z cosh (a + i/) = z cosh y
and, for simplicity, we shall confine the investigation to the special case in
which cosh y is real and positive. The more general case in which cosh 7 is
complex may be investigated by the methods used in ~ 8'6 and ~ 10'15, but it is
of no great practical importance and it involves some rather intricate analysis.
* For an asymptotic expansion of the associated integral
1 0 e — 2L2+ -1 du,
see Rayleigh, Phil. M11ag. (6) vIII. (1904), pp. 481-487. [Scientific Papers, v. (1912), pp. 206-211.]