ME. E. W. BAENES ON INTEGEAL FUNCTIONS. 
461 
Now, where \jj ( t ) is a given form, and such that we can integrate expressions like 
xfj ( t ) t s ~ l dt ( s positive or negative, but integral) we need only carry out this process 
and “sum” the ensuing series of positive and negative powers of z to obtain the 
dominant term of the asymptotic expansion of log F (z). If, however, xfj ( t ) is not 
thus formally given, we have to face the difficulty that the lower limits of the 
definite integrals are different quantities. The lower limits, however, corresponding 
to negative values of s, are such as to give rise to zero terms. If, then, we consider 
r<t> (»o dt 
only indefinite integrals of the type I xfj (t) t s —, and take care that in any trans¬ 
formation of these we do not introduce arbitrary additive constants, we may take 
the asymptotic expansion in the form 
Cy o 
log F (z) - - A log z -F J log (f) (n) 
dn — N 
i (-r 1 r 
+f 
S=1 Si 
4 xfr (t) clt 
t 
fVOO d 
dn 
l + W' < ’ + L w ! 
S 55. It is the integral 
8 
= L t f 
i=x J 
yfr (t) dt 
1 + 2 '(— 
s =-A * . 
which gives rise to the dominant term of the asymptotic expansion of log F ( z ). 
. . . . 
This integral is evidently equal to L t 
£•=*> J 
yfr (t) dt 
"(vr-fd-r 
J t 
i+i 
Suppose now that z = re ie and take i/x = log ( — j, the logarithm, when t = r, 
having a cross-cut along the negative half of the real axis, so that 
px = log +7 tl — l0 , where log - is arithmetic. 
Then 
I 
= Lt i f 
k= k> J 
i log \ + v - e . . u sin (k + J ) ll 7 
1 1 y — djji. 
sm i /u. 
Now the form of the dominant term I does not depend on the quantity log 
which vanishes when r is sufficiently large. We have then 
xfr (rtl) 
T T [ n ~ 9 . r r w sill (k + A) Li , 
I = Ln lib {ze t( ' x_,7) } . v - a --- da 
t-=J 1 J smi/x . 
an integral of the type first considered by Dirich let. # 
* v. ‘Crelle,’ vol. 4, pp. 157, et seq. 
