230 
KADELFINGER. 
>b (x) — 2 f . . 
Xj = 0 Ao = 0 
F ( R , 0) 
Ai 4- Aj . . . 4- An 
Aj + A2 
+ A n 
10 
(e n 
i) 
Ai 4" Aj> 
+ A r 
[(n — 1 ) Ai 4 - (w — 2 ) Aj 4 - 
+ An _ j 
which may be employed in the calculation of the coefficients 
of substitution of a function in the neighborhood of an essen¬ 
tial singularity. 
Second Note .—In his second note G. Mittag-Leffler points 
out that when the development (5) is written in the form 
FA 0) 
A 1 = 0 
F(a) 
Ai 4 Ao ... 4 * A 
Ai 4 * 
it may be regarded as an n times infinite series, which for 
n = 1 is identical with Taylor’s series and for finite values 
of n converges within stars inscribed within the star A. He 
further shows that Taylor’s series has one property not com¬ 
mon, in general, with (10), in that Taylor’s development 
diverges for all values outside of a circle passing through 
the nearest singular point, hence it may be said to have a 
circle of convergence, while the expression (10) for n>3 may 
converge for values outside of the corresponding inscribed 
star, hence cannot be said to have a star of convergence. In 
order to derive an expression having a star of convergence for 
all values of n, it is necessary to place a restriction on the 
coefficients C* in (5). A special case is worked out in which 
the coefficients are made to depend on an arbitrary param¬ 
eter; the resulting expansion has a star of convergence 
and for n — 1 reduces to Taylor’s expansion. 
Third Note .—In his third note Mittag-Leffler derives other 
prolonomial expansions having a star of convergence , by per¬ 
forming a transformation 
Z—a=(x — a)f( u l a ) (11) 
( 9 ) 
n 
■ • + A n 
( 10 ) 
on a Taylor’s development, 
