∮
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)(푧− 2 푖)
푑푧
퐶 1
=− 2 휋푖×푔
1
(푧
0
)=− 2 휋푖(푔
1
(푧
0
))
=− 2 휋푖×
2 (− 2 푖
3
)+(− 2 푖
2
)+ 4
(− 2 푖
2
)((− 2 푖)− 2 푖)
=− 2 휋푖×
2 ( 8 푖)+(− 4 )+ 4
(− 4 푖)(− 4 푖)
=− 2 휋푖
16 푖
16 푖
=− 2 휋푖
Kemudian pada kurva 퐶
2
dengan 푧
0
= 0 :
∮
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)(푧− 2 푖)
푑푧
퐶 2
,푧
0
= 0 (orde 2) ; 푔
2
(푧)=
2 푧
3
+푧
2
+ 4
(푧
2
+ 4 )
Maka hasil integralnya (searah jarum jam):
∮
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)(푧− 2 푖)
푑푧
퐶 2
=
− 2 휋푖
( 2 − 1 )!
×푔
2
(
2 − 1
)
(푧
0
)=− 2 휋푖(푔
2
(푧
0
))
=− 2 휋푖[
( 6 푧
2
+ 2 푧)
(
푧
2
+ 4
)
−( 2 푧)( 2 푧
3
+푧
2
+ 4 )
(푧
2
+ 4 )
2
]
=− 2 휋푖[
( 6 ( 0
2
)+ 2 ( 0 ))( 0
2
+ 4 )−( 2 ( 0 ))( 2 ( 0
3
)+ 0
2
+ 4 )
( 0
2
+ 4 )
2
]
=− 2 휋푖[
( 0 )( 4 )−( 0 )( 4 )
( 4
2
)
]= 0
Berikutnya pada kurva 퐶
3
dengan 푧
0
= 2 푖 :
∮
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)(푧− 2 푖)
푑푧
퐶 3
,푧
0
= 2 푖 (orde 1) ; 푔
3
(
푧
)
=
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)
Maka hasil integralnya (searah jarum jam):
∮
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)(푧− 2 푖)
푑푧
퐶 3
=− 2 휋푖×푔
3
(푧
0
)=− 2 휋푖(푔
3
(푧
0
))
=− 2 휋푖[
2 ( 2 푖
3
)+( 2 푖
2
)+ 4
( 2 푖
2
)( 2 푖+ 2 푖)
]
=− 2 휋푖[
2 (− 8 푖)+(− 4 )+ 4
(− 4 )( 4 푖)
]=− 2 휋푖
− 16 푖
− 16 푖
=− 2 휋푖
Maka :
∮
2 푧
3
+푧
2
+ 4
푧
2
(푧+ 2 푖)(푧− 2 푖)
푑푧
푐
=− 2 휋푖+ 0 − 2 휋푖=− 4 휋푖
Latihan Soal
