1550251515-Classical_Complex_Analysis__Gonzalez_

480 Chapter^7

the local interior angle at z 0 , denoted >.(z 0 ), is defined to be 7r (at such a

point Chas a unique tangent). On the other hand, if zo = (k for some k,

we define >.(zo) = 2rr -wk, where Wk = f:h -<Xk, <Xk and f3k being the incli-
nation angles of the semitangents T 1 , T 2 at (k taken in the order induced
by the orientation of C (Fig. 7.24).
To specify the way in which the limit at z 0 E C will be taken, consider a
small circle 1(r): (-z 0 = reu, 0 :$ t :$ 2rr, with center z 0 and radius r, and
let C 1 (r) be the part of C lying outside 1(r) (with the same orientation
as C).

Definition 7.10 The Cauchy principal value of

is defined by

f

J(()d(

(-zo

c

(zo EC)

(Pv)j f(()d( = lim f f(()d(

c (-zo r-^0 lc1(r) (-zo

provided that the limit on the right-hand side exists.

Theorem 7 .26 With the conditions imposed above on the contour C,
the principal value of the integral exists and we have

(PV) ~ j f( () d( = J_ >.(zo)f(zo)

2rrz ( - z 0 2rr

(7.18-1)

c

where zo E C and f is analytic on and within C.

Proof By choosing r small enough the circle 'Y( r) will intersect C* in just
two points z 1 (r) and z 2 (r), z 1 denoting the initial point and z 2 the terminal

point of the arc of 'Y( r) whose interior lies in Int C*. Call that arc r(r ).

c

Fig. 7.24