1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

312 CHAPTER 8 • RESIDUE THEORY

(8-12)

(8-13}

The proof of Theorem 8.4 is similar to the proof of Theorem 8.3. Before

tuu~ing to the proof, we illustrate how to use Theorem 8.4.

• EXAMPLE 8.17 Evaluate P.V. f~ 00 x~q~/".

Solu tion T he function f in Equation (8-12) is f (z) = •x.~!t, which has a
simple pole at the point 2i in the upper half-plane. Calculating the residue yields

Res[/, 2 i) = Jim exp (iz~ z = 2ie~

2

= 2-.

•-2; z + 2i 4i 2e2

Using Equation (8-14) gives

1

(^00) x~x~. •
P.V. 2 =2•Re(Res[/,2i])= 2.
_ 00 x + 4 e

• EXAMPLE 8.18 Evaluate P.V. f~ 00 c°:i"'+~x.

S olution The function fin Equation (8-12) is f (z) = ·~~¥~^1 , which has simple

poles at the points z 1 = 1 + i and z2 = - 1 + i in the upper half-plane. We get