An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 196. 197.

In a domain containing singular points.

371

1 12
+ iu;"L)m — (C-|-" (/ ) ( — ) t | +
1(~ - (u, n-, ) m-(%-,) (% 0,:) "2.
+( t 1 1 3 f-(z) -d
(U (-c - CO ) '- -. —)? k)
+ t 1 f r fzi (Ck
$T ) - JL -1- /- (' -~
(Ua(Uy/ {S-a<)111
___ ______ P f( z ) I - )
The singular point ci is non-essential or essential, according as
the expansion relative to it is finite or not. In the former case an
integer n can be assigned such that the product:
_1 ___ 1 n
f(3u) tt - a - c, - a )
is finite for It = c-i.
197. When the ramification a in which m leaves of the function
are cyclically connected is also a singular point, the point t = 0 is
likewise singular in the function cp( ) that results from f(z) by the
1
substitution = -(z - a)m. Hence the expansion (~ 189) within a
domain in which there are no further singular points, is:
P(t) =- dg + t f? () dg + t?) dc +..
(0) (0) tu)
+ 2ain T{f(~)d~ + t-J /P(~)dcg + jy(~)d +d *-
(0) (0) (0)
To this corresponds for f(u) the formula:
V. f() - { + ( - a) - t -
(a) ( ) (z _ — ) m
2
(a) (z a) "'
( - )m (Z -) t" (i -.()" -- ) "2
1.+ Az)~_ { J+ f z)z +.
~+ - ~~3 Z-: m
"( _ ao)m(,, (Z-ac) t2
24t