MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
497 
The large zeros of P/ (2) are approximately those of cosh 77 v2, and the latter are 
such that, with the usual notation, a„ = n 2 + n + 
We notice that the general form of a n given above shows that the large zeros of 
P p (2) separate and are separated by those of P/ (2), a fact which agrees with the 
extension of Rolle’s theorem. 
§ 94 . From the preceding example it is now evident that we are in a position to 
prove that, for all the types of integral functions of which asymptotic expansions 
have been obtained in this memoir, the order of the function is equal to the order of 
its derivative. And not only so, but we are theoretically in a position to determine 
as nearly as we please a formula for the (large) n th zero of the derivative. It would 
be tedious to consider in turn all cases which can arise.: we will take one or two as 
typical of the rest. 
As an immediate corollary of the preceding example it may be seen that the 
derivative of a simple non-repeated function of order - less than unity with algebraic 
zeros of the type a n = n p -f 9 n p ~ l + . . . is a similar function of equal order, whose 
zeros are typified by b a = n p + (9 -j- p — I) n p ~ l + . . . 
§ 95 . As a suggestion of the possibility of extending the expansions of Parts III. 
and IV. let us next write down the first few terms of the asymptotic expansion of 
P (a + 2) = log n 
1 + 
z + a\ - z _±£ + ... + <-) p C+«) p- | 
a n 
6 an 
pa n i‘ 
, where a is any quantity of finite 
modulus, and a n = n l,p 
h 
1 17/ 1 1 1 l € 2 ' 
The expansion will be (§ 68) 
77* 
00 ( _V ^ 
- (z + a) p — \ log (2 + a) — — log 277 + S' -7-.. ^ , , 
SUITrp v 1 ' 2 &v ’ 2 P * ,=- p s(z + ay \pj 
_ __ u _I_ a) p (1-6l) 4- p (p + 1 ~ 2p€l) b 3 __ 
sin 71-p (1 — e x ) ' ' 2 1 sin irp (1 —26]) 
F (— 
V P / 
(2 + a) p (1_2e A 
sin 7 rpK , », ( — ) s_1 „ ( €,, e 9 . .\ 
-7-77-^—- (2 + a) p(1_e2) + . . . + S -7-r Z Ip, s' J 1 j 
sm 7 rp (1 - e s ) V ’ s =- p s(z + a) s \ \. . ./ 
— Z ( 0 ; 
6 ], e 3 . . . 
\ ’ Kh---I ’ 
and may be transformed into 
77* 1 go ( — V 1 fS 
z p — ^ log 2 — — log 27 t + S' -— F 
5=— J) 
sm 7 r /3 " 0 2 p ~ n " ' sz s 
+ 2 P_1 
+ 
P ( P ~ 1 ) 7r 
sin 77 p 
V - 
a a z' 
p Sill 77 p 
+ (-)■’AAa 2 F(A + (-)>'- I aF( £ -^) z -- 1 
+ • • • 
[Over. 
VOL. CXCIX.-A. 
3 s 
