PPT: Complex Integral | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Defining Line Integrals in the Complex Plane
0
az ?
N
bz ?
1
z
2
z
1
?
3
z
2
?
3
?
N
?
1 N
z
?
n
z
n
?
x
y
#. ?
?? ???? ?? ?????????????? ?? ?? -1
?????? ?? ?? #.???????????????? ??h?? ???????? ?? ?? = 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
#. ?????? ??h?? ???????????? ???? ???????????????????????? ?? ? 8,
??????h ??h???? ??? ?? = ?? ?? -?? ?? -1
? 0?????? ???????????? ?? = 
?? ?? ?? ?? ???? = lim
?? ?8
?? ?? = lim
?? ?8
 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
(???????????? ???? ?????????????????????? ???? ??h?? ?????????????? ???? ??h?? ??????h ?????????????????????? )
C
2
Page 2


Defining Line Integrals in the Complex Plane
0
az ?
N
bz ?
1
z
2
z
1
?
3
z
2
?
3
?
N
?
1 N
z
?
n
z
n
?
x
y
#. ?
?? ???? ?? ?????????????? ?? ?? -1
?????? ?? ?? #.???????????????? ??h?? ???????? ?? ?? = 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
#. ?????? ??h?? ???????????? ???? ???????????????????????? ?? ? 8,
??????h ??h???? ??? ?? = ?? ?? -?? ?? -1
? 0?????? ???????????? ?? = 
?? ?? ?? ?? ???? = lim
?? ?8
?? ?? = lim
?? ?8
 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
(???????????? ???? ?????????????????????? ???? ??h?? ?????????????? ???? ??h?? ??????h ?????????????????????? )
C
2
Equivalence Between Complex and Real Line Integrals 
Note that-
So the complex line integral is equivalent to two real line integrals on C.
3
Page 3


Defining Line Integrals in the Complex Plane
0
az ?
N
bz ?
1
z
2
z
1
?
3
z
2
?
3
?
N
?
1 N
z
?
n
z
n
?
x
y
#. ?
?? ???? ?? ?????????????? ?? ?? -1
?????? ?? ?? #.???????????????? ??h?? ???????? ?? ?? = 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
#. ?????? ??h?? ???????????? ???? ???????????????????????? ?? ? 8,
??????h ??h???? ??? ?? = ?? ?? -?? ?? -1
? 0?????? ???????????? ?? = 
?? ?? ?? ?? ???? = lim
?? ?8
?? ?? = lim
?? ?8
 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
(???????????? ???? ?????????????????????? ???? ??h?? ?????????????? ???? ??h?? ??????h ?????????????????????? )
C
2
Equivalence Between Complex and Real Line Integrals 
Note that-
So the complex line integral is equivalent to two real line integrals on C.
3
Review of Line Integral Evaluation 
t
t
-a a
t
-a a
1
t
2
t
3
t
1 N
t
?
C
n
t
x
y
0
t
fN
tt ?
1
t
2
t
3
t
1 N
t
? n
t
… …
0
t
fN
tt ?
t
A line integral written as 
is really a shorthand for:
where t is some parameterization of C :
or
Example- Parameterization of the circle     x
2 
+ y
2
= a
2
1) x= a cos(t), y= a sin(t),      0 ? t(= ?) ? 2 ?
2) x= t,  y= ?? 2
-?? 2
, t
0
= ?? , t
f=-
a
And  x= t,  y=- ?? 2
-?? 2
, t
0
= -?? , t
f=
a
}
(?? ?? ,?? ?? )
4
Page 4


Defining Line Integrals in the Complex Plane
0
az ?
N
bz ?
1
z
2
z
1
?
3
z
2
?
3
?
N
?
1 N
z
?
n
z
n
?
x
y
#. ?
?? ???? ?? ?????????????? ?? ?? -1
?????? ?? ?? #.???????????????? ??h?? ???????? ?? ?? = 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
#. ?????? ??h?? ???????????? ???? ???????????????????????? ?? ? 8,
??????h ??h???? ??? ?? = ?? ?? -?? ?? -1
? 0?????? ???????????? ?? = 
?? ?? ?? ?? ???? = lim
?? ?8
?? ?? = lim
?? ?8
 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
(???????????? ???? ?????????????????????? ???? ??h?? ?????????????? ???? ??h?? ??????h ?????????????????????? )
C
2
Equivalence Between Complex and Real Line Integrals 
Note that-
So the complex line integral is equivalent to two real line integrals on C.
3
Review of Line Integral Evaluation 
t
t
-a a
t
-a a
1
t
2
t
3
t
1 N
t
?
C
n
t
x
y
0
t
fN
tt ?
1
t
2
t
3
t
1 N
t
? n
t
… …
0
t
fN
tt ?
t
A line integral written as 
is really a shorthand for:
where t is some parameterization of C :
or
Example- Parameterization of the circle     x
2 
+ y
2
= a
2
1) x= a cos(t), y= a sin(t),      0 ? t(= ?) ? 2 ?
2) x= t,  y= ?? 2
-?? 2
, t
0
= ?? , t
f=-
a
And  x= t,  y=- ?? 2
-?? 2
, t
0
= -?? , t
f=
a
}
(?? ?? ,?? ?? )
4
Review of Line Integral Evaluation, cont’d 
1
t
2
t
3
t
1 N
t
?
C
n
t
x
y
0
t
fN
tt ?
While it may be possible to parameterize C(piecewise) using x 
and/or y as the independent parameter, it must be remembered 
that the other variable is always a function of that parameter i.e.-
(x is the independent parameter)
(y is the independent parameter)
(?? ?? ,?? ?? )
5
Page 5


Defining Line Integrals in the Complex Plane
0
az ?
N
bz ?
1
z
2
z
1
?
3
z
2
?
3
?
N
?
1 N
z
?
n
z
n
?
x
y
#. ?
?? ???? ?? ?????????????? ?? ?? -1
?????? ?? ?? #.???????????????? ??h?? ???????? ?? ?? = 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
#. ?????? ??h?? ???????????? ???? ???????????????????????? ?? ? 8,
??????h ??h???? ??? ?? = ?? ?? -?? ?? -1
? 0?????? ???????????? ?? = 
?? ?? ?? ?? ???? = lim
?? ?8
?? ?? = lim
?? ?8
 
?? =1
?? ?? ( ?
?? )(?? ?? -?? ?? -1
)
(???????????? ???? ?????????????????????? ???? ??h?? ?????????????? ???? ??h?? ??????h ?????????????????????? )
C
2
Equivalence Between Complex and Real Line Integrals 
Note that-
So the complex line integral is equivalent to two real line integrals on C.
3
Review of Line Integral Evaluation 
t
t
-a a
t
-a a
1
t
2
t
3
t
1 N
t
?
C
n
t
x
y
0
t
fN
tt ?
1
t
2
t
3
t
1 N
t
? n
t
… …
0
t
fN
tt ?
t
A line integral written as 
is really a shorthand for:
where t is some parameterization of C :
or
Example- Parameterization of the circle     x
2 
+ y
2
= a
2
1) x= a cos(t), y= a sin(t),      0 ? t(= ?) ? 2 ?
2) x= t,  y= ?? 2
-?? 2
, t
0
= ?? , t
f=-
a
And  x= t,  y=- ?? 2
-?? 2
, t
0
= -?? , t
f=
a
}
(?? ?? ,?? ?? )
4
Review of Line Integral Evaluation, cont’d 
1
t
2
t
3
t
1 N
t
?
C
n
t
x
y
0
t
fN
tt ?
While it may be possible to parameterize C(piecewise) using x 
and/or y as the independent parameter, it must be remembered 
that the other variable is always a function of that parameter i.e.-
(x is the independent parameter)
(y is the independent parameter)
(?? ?? ,?? ?? )
5
Line Integral Example
Consider-
x
y
C
?
r
z z
It’s a useful result and a 
special case of the 
“residue theorem”
?? .???????????????? 
?? ?? ?? ???? ,??h?????? ?? :?? = ???????? ?? ,?? = ?????????? , 0= ?? = 2??
? ?? = ?????????? +???????????? = ?? ?? ????
,
? ???? = ???? ?? ????
???? 6