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.

370

Expansion of ambiguous analytic functions.

Bk. IV. ch. III.

1 1
From this equation, substituting = (z - a)L, t = ( - t)m, we
find for the function f(u) the relation:
IIIa. f(u) { (f -- dz
-J tf'(z).zf}
1 v r -1 J 1 1
-l ) () -.-a)m -
(cl) (z- a) m ( — ) m _ — ( ( )' (Cm)
The first integral is to be formed for the external boundary of the
winding surface; the others refer severally to each curve that encloses a
(non-branching) singular point c. The first integral can consequently
1
be expanded by powers of (\ ~)m since U signifies a point within
the curve to which the values of z refer. In any of the other integrals,
z signifies a point on the curve round the point c, and u lies outside
this curve; therefore we have
1 1
i r ^ - y i (z - U)m - (C -i (m i
mod mod t - - - mo -< 1
( - a) (c- ( )
and the quotient
1 1
1: {(z - (u_ -)m )
can be expressed by a series beginning with the terms:
-1 1 1 1 2
1 + (z -C) -(c —")' (Z —) ---(c -- a
-t_ |, I +tz ~=(c + ~ (__.... t
( -- a) -- m l - (C —) a) ( ) u- a)L - (C- a1) 1i
Accordingly, from equation liIa. we have the following expansion:
1 f /f'()dz - 1 f(z) d z
IV. f(,) 2 i (t - I z + ( a) 1
(a) (Z 0)m (a) (Z ) ) "
27
~ f ) _' %-) I
+ 1 f(z)d
2 r Um | 1 - L - r1-T
(u - a) '-_ (c - a) " (z - a) r"
$+. — 1 _ _(_ -_-__ (Cf - c) d1
(M.)".- (C/ _ - c) -t) (C'"