1550251515-Classical_Complex_Analysis__Gonzalez_

470 Chapter^7

the value of the integral along a closed contour C 1 around the origin, de-
scribed once in the positive direction, is 27ri. Hence if C1 is made a part of
"f, Oc 1 (0) = k, and 'Yl is a direct path from 1 to z (Fig. 7.19), so that

'Y = kCt + 'Y1

we have

Fi(z) =(-Yi) - + 2hi

l

z d(

1 (

(7.16-5)

By a "direct" path from 1 to z we mean any path in G that does not
wind about the origin. In order to determine precisely the value of

(7.16-6)

we need only to remove from G some ray

L = {z: z = eio:t, a f. 2n7r,O < t < oo}

and to restrict 'Yi to the simply connected region G - L. So restricted
'Yi will not wind about the origin, and for a given z E G - L the value
of (7.16-6) will be independent of the choice of 'Yl· In particular, if we take

the cut to be Lo= {z: z = -t,O < t < oo}, i.e., the negative real axis, the

single-valued function

Fi(z) =(-Yi) -

l

z d(

1 (

is easily seen to be no other than Logz (the principal branch oflogz). In
fact, on G - Lo we have

F{(z) = ~ and (Logz)' = ~

z z

y

z

Lo

0 x

Fig. 7.l.9