ANALYTIC REPRESENTATION OF COMPLEX FUNCTIONS. 231 
The generating function / { u l a ) is to a large extent arbi¬ 
trary and depends on an arbitrary constant « the value of 
which determines the domain of convergence of the resulting 
development. Mittag-Leffler confines himself in this paper 
to investigating several cases in which it can be expressed 
as a definite integral and considers in detail the case 
/(“/«) = 
in which 
H 
/:[G±9'-0* < 12 > 
H 
du 
u 
(13) 
Fourth Note .—In his fourth note Mittag-Leffler deduced ex¬ 
pressions for the branch A (#) by the employment of integrals 
around a complex contour analogous to Cauchy’s integral. 
He starts with the integral 
1 /. F[q+(s-q)/ (>/.)] (»\ %+1 dv (14) 
2 * ij y—u \y) 
c 
where 
*=/(“/«),« = /( tt /l), M <R,R>1 
u — o, v = o ; u— 1, i? = 1. 
He shows that by making a suitable choice of / {V / a ) many 
developments analogous to those discussed in his former 
notes may be obtained, and arrives at general results which 
include those of Borel’s first paper as a special case. The 
relation of his w r ork to that of Abel is also shown, thereby 
making this fourth note an appropriate contribution to the 
memoirs published in commemoration of the centennial of 
the birth of Abel. Mittag-LefHer’s first note has attracted 
much attention and his results have been criticised, extended, 
and applied by a large number of mathematicians. Com¬ 
plete references to original sources of publication of these 
