1550251515-Classical_Complex_Analysis__Gonzalez_

722 Chapter^9

(with b finite or oo ). In this case a cut of the C* plane is made joining the
points a and b (either directly or by way of oo, depending of the particular
case), so as to render f(z) single-valued in the cut plane. Extreme care
must be taken in specifying the branch chosen for f(z) and in evaluating
f(z) properly, not only on the boundaries of the cut but also elsewhere in
the plane. This is illustrated in detail by the following examples.

Examples 1. To evaluate J~ 1
1
dx/(1 + x^2 )v'l - x^2 • Here, as in what
follows, by· y'C, with c real and positive, is understood the positive value of
the root. Let f(z) = 1/(1 + z^2 )v'l - z^2 and consider the plane slit along

the segment [-1, 1] joining the branch points -1 and +1 Fig. 9.25). In

order to specify a branch for g(z) = v'l - z^2 = (1 - z)(l + z), let

so that

and

z + 1 = r1ei61,

z -1 = r2ei62,

0 ::::; f)i ::::; 27r

0::::; fh ::::; 27r

k = 0,1

By choosing the branch g 0 corresponding to k = 0, we obtain

go(z) = y'r"lr2e(1/2)i(61 +82+11') = iy'r"lr2e(l/2)i(9 1 +o 2 )

For any point z = iy with y > 0 (Fig. 9.26a) we have 81 + 82 = 7r, so that

go(iy) = iy'r"lr2eirr/^2 = -r1

However, if z = iy with y < 0, then f)i + 82 = 37r (Fig. 9.26b), and

go(iy) = iy'r"lr2ei^3 1l'/^2 = +r 1

y

-1 x

Fig. 9.25