1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

8.6 • INTEGRANDS WITH BRANCH POINTS 325

Using Theorem 8.7, we have

dx= Res -1 = -

1

(^00) x" 2 7ri 27ri ( e;,,,, )

0 x (x + 1) 1-ei^02 " (!, J 1 - efo2" - 1

= ei<Ur -e-f~lt' = sin aJr

2i

We can apply the preceding ideas to other multivalued functions.

•EXAMPLE 8.24 Evaluate P.V. J;' .,1~: 2 dx, where a> O.

Jog_ltz

Solution We use the function f (z) = ~· Recall that

log~ z = lnlzl +iarg~ z = lnr +iO,

where z = rei^9 :/: 0 and -~ < 0 :::;^3 ;. The path C of integration will consist of

the segments [-R, - r] and [r, R] of the x-axis together with the upper semicircles

Gr: z = r ei^9 and GR: z = Rei^9 , for 0:::; e:::; 7r, as shown in Figure 8.8.

We chose the branch log_ " because it is analytic on G and its interior-
hence so is the function f. ThiS choice enables us to apply the residue theorem
properly (see the hypotheses of Theorem 8.1), and we get

1

7r In a 7r^2

f (z) dz= 2 7riRes [f,ai] = --+ i -

2

.

c a a

y

Figure 8.8

log_,,,z

The contour C for the integrand f ( z) = 2 2 2.

z +a