INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
283 
where C' is a contour which embraces the positive half of the real axis and includes 
within its bulb the circle of convergence of y ( y). 
V. When \x\ is very large, the final integral in the equality just written is equal 
to (—xy~ e J (x), where, if l'>l, | J (x) | tends to zero as \x\ tends to infinity. We 
thus obtain a superior limit to the asymptotic value of (x ; d) when | x | is large. 
The problem of actually obtaining an asymptotic expansion depends upon a knowledge 
of the singularities of y (y) within its circle of convergence. 
Part VII. 
The Functions t X r / 1 + a ^ (0<«<l) and % tfT{l+nO) /y >0 \ 
§ 42. The asymptotic expansions of these two functions are connected with one 
another. The functions do not belong to the categories previously considered; their 
associated functions have not finite radius of convergence. 
We give the results which may be obtained for these two functions, referring the 
reader elsewhere for the detailed analysis. # 
I. If 
r, \ S PT(l+om) -j . f* 
f( x ) =,y r(»+i) ’ we have> lf a<1 > /(*) = J 0 ex p {—y +x y a }dy, 
the integration being taken along the positive half of the real axis. 
Hence, if (x) < 0, we have the asymptotic expansion 
f( x ) = 
1 ; (-)"r((n+l)/q) 
a (—x) Va n= 0 r (n + 1) (—x) n/a ’ 
the principal value of {-x) Va , which is real when x is real and negative, and which has 
a cross-cut along the positive half of the real axis, being taken. 
II. W hen ( x ) > 0 and | arg x \ < (l — a) U 77 -, we have 
1 /• °° 1 
f(x) = (aa-) 1_ “j exp {(a a x) l ~ a (t a —at)} dt. 
By the substitution y = 1 — —\-t we deduce the asymptotic expansion 
a 1 
__ 1 / 9 V / 2 
f(x) = exp^(l-a) a 1 _a £C 1_a } (ax) 2(1_a) 
1 —a 
the c.’s being definite constants. 
r(J)+ 
71=1 
(a.n) (1 a) 
See a paper which will shortly appear in ‘ Cambridge Philosophical Transactions,’ vol. 20. 
2 O 2 
