462 
ME. F, W. BARNES OX INTEGRAL FUNCTIONS. 
The theory previously developed in Part II. tells us that this integral must, qua 
function of z, be independent of 6; in other words, when — tt < 6 < n, that 
L t 
k = :o 
( of/ {z e L 
' n-e 
(fi-TT) 
} 
sin (k + f) fju 
sin h p 
dp — 0. 
But this is precisely Dirich let’s result : we thus have a valuable verification of 
our theory. 
Finally, then, the dominant term of the asymptotic expansion of log F (z) is the 
function 
f(z) - L t f ixfj{z& 
l=X, J 
(^)} — dp. 
sin \ p 
Since we may evidently change the sign of i without altering the value of f (z) we 
have 
f(,\ _ ij - jr {se~‘(*-/*> } sin (k + }) p ^ 
_ j 2 l sin b u ^' 
Now \Jj is the function inverse to <£. If, then, we suppose that £ and rj are 
determined from the relation ze‘ (,r_M) — (f> (£ -f- ip), principal values of inverse 
expressions being taken, and £ and p being functions of z and p, we shall have finally 
sin (k + \) p , 
sin h /x 
There is no doubt that it is possible to construct functions cf) (n) for which the 
preceding analysis will not hold good. # It would ajDpear, however, to be applicable 
to most of the types of functions which would ordinarily arise, and a more accurate 
investigation will need the exquisite finesse of certain developments of the theory of 
functions of a real variable. 
Note that, for the case in which ze‘ (,r ' M> = (£ — ip) p , we have established that 
7T L 
—— zp . 
. 7T 
sin — 
P 
/x, as the simplest form in which we may write f (z). 
f(z) = Lt 
/:= oo 
(}) (n)_ 
§ 56. The dominant term f(z) of the asymptotic expansion of log n 1 + 
n= 1 
takes a very simple form for the case in which \Jj (t) can, when t is large, be expanded 
in descending powers of t in the form 
xjj(t) = tP 
+ 7 + f + 
w 
/here p > 1. 
We have the asymptotic expansion 
m— l 
* We have assumed, for instance, that we can apply the Maclaurin sum formula to - ^ (m), and, 
71 = 1 
therefore, that the conditions of § 41 are satisfied. 
