1550251515-Classical_Complex_Analysis__Gonzalez_

492 Chapter^7

within C. Also, 7r lies inside C. Hence, by using (7.21-2) with n = 3,
we get

J t~n: :~ = 2;i !"' ( 7f) = ~i ( - cos 7f) = ~i

c

Exercises 7.4

1. Evaluate each of the following integrals.

( )

J

a z4 + ( z2 + .)3z +^1 d C· =^2 it^0 < t <^2

z-z^3 z,. z e , _ _ 7f

C'

J

e2z

(b) -dz 5 ' C: z = eit '^0 - < t < - 27f

z

a

(

zs

(c) j (z - l) 6 dz, C: z = 2eit, 0 ::=; t ::=; 67f

c

(d) j r cosh4z z ·

3 dz, C: z = e-•t,^0 ::=; t ::=; 47r

c

J

sinz it

( e) ( z 2 +

1

) 2 dz, C: z = 1 + 2e , 0 ::=; t ::=; 27f

c

2. Let f be continuous along the rectifiable arc 'Y, and let zo, z1 be two

points in one of the regions R determined by 1*. Let

F(z) = J J(()d(

(-z

'Y

for z E R, and define

I J f(()d(

F (z1,zo) = ((-zi)(( _ zo)

'Y

to be the mixed derivative of Fat the pair z 0 ,z 1 • Prove that:

(a) F(z1) - F(zo) = (z1 - zo)F'(z1, zo)

(b) F'(zo) = lim F'(z1,zo)

z1 --+-zo

3. Evaluate

J

JdzJ

Jz-aJ^4

c

where C: z = reit, 0 :::; t :::; 211-, and JaJ i= r.