458 
MR. E. \Y. BARNES ON INTEGRAL FUNCTIONS. 
Now, by the usual theory of Fourier’s series, p + S' - 7 er m =- qp 
* t , (~> 5 
s=-k 
-si9 _ 
77 
provided — tt < 9 < tt. 
1 
s + -- 
p 
sin 
7r 
Therefore 
/( 0 )= - + 
7T 
7T 
(? P 
Sill 
so that 
f(t) 
iO i 
Sill - 
77 
cp p -f t — zd — p 
Hence 
4 , (-)*- 1 ^ 
7T ZP 
s= — k S (pS -f 1) 3 s .7 T m 
r Sill — 
1 mP 
+ lo g a: - p- 
The second series S' is at once seen to be equal to — \ log ~ . 
s=—k * 2 
And, since the term by term differential of a summable divergent series is equal 
to the differential of its sum, the third series vanishes identically for all positive 
integral values of r. 
1 
Thus, when 1 2 : | p lies between m and m — 1, or possibly is equal to the latter 
quantity, we have the asymptotic expansion, while — tt < arg 2 < it. 
log P p ( 2 ) = (m — 1) log z — p (m — T) log m -f- pm — p log \/2 n 
+- z p + m log 
. 7 r 1 0 
Sill 
p 
! 1 m p , 4 (~ y 1 F (ps) 
pm — ^ log — + S 
S=1 
or, 
77 l -p]og^¥- - - <--Al F W 
log P p (:) = — zp — p log \ log 2 + 2 
sin- 5=1 
P 
st 
Thus, when 1 2 1 has any large value, and — 7r < arg 2 < tt, we have the arithmetically 
asymptotic expansion 
n 
n= 1 
^ q/p 
gsinir/p " 
+ 
| (-)* 1 
S=1 
F (ps) 
§ 53. The approximation represents an arithmetic not a functional equality. It 
does not vary with the argument of 2 , and it exists everywhere in the neighbourhood 
of infinity except at points on or near the line of zeros of the function. Not only 
so, but at points on the line of zeros of P p ( 2 ) which are not in the immediate vicinity 
of one of its zeros, both the function and the asymptotic series have arithmetic 
continuity, and therefore the equality will hold at such points. These results accord 
with the general theory developed in Part II. 
