1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

232 CHAPTER, 6 • COMPLEX INTEGRATION

Theorem 6.9 gives an important method for evaluating definite integrals

when the integrand is an analytic function in a simply connected domain. In

essence, it permits you to use all the rules of integration that you learned in

calculus. When the conditions of Theorem 6.9 are met, applying it is generally

much easier than parametrizing a contour.

• EXAMPLE 6.17 Show that

l

dz.

- , = l+i,

c 2z•

where zi is the principal branch of the square root function and C is the line

segment joining 4 to 8 + 6i.

S olutio n We showed in Chapter 3 that if F (z) = z1, then F' (z) = --1y-, where
2z2
the principal branch of the square root function is used in both the formulas for F
and F '. We note that C is contained in the simply connected domain D 4 ( 6 + 3i),
which is the open disk of radius 4 centered at the midpoint of the segment C.
Since f (z) = --1y-is analytic in D 4 (6 + 3i), Theorem 6.9 guarantees that
2.~

• EXAMPLE 6.18 Show that fccosz dz= - sinl +isinhl, where C is the
  line segment between 1 and i.

Solutio n An antiderivat ive of J(z) = cosz is F(z) = sinz. Because Fis

entire, we use Theorem 6.9 to conclude that

fa cosz dz= Ji cosz dz= sin i - sin 1 =- sin1 + isinb i.