2. 14.1 Line Integral in Complex Plane
• A complex definite integrals are called complex line integrals
• It denotes as
• Here, the integrand is integrated over a given curve .
• The curve is a path of integration with parametric
representation
• Examples
( )
f z
C
3. Line Integral in Complex Plane (cont.)
• We assume the curve is smooth, thus it has continuous and
non-zero derivatives
• The definition of the derivatives
4. Definition of Complex Line Integral
• Suppose that the time in parametric representation is divided
into
• The is indeed partitioned into
where
C
5. Definition of Complex Line Integral (cont.)
• On each subdivision of we choose any point
• The we can form the sum
• The limit of the sum is termed as line integration of over
the path of integration of
• Important definition
C i
( )
f z
C
12. Dependence on Path
• If we integrate from to along different
paths, then the integral in general will have
different values.
• A complex line integral depends not only on
the endpoints of the path, but also on the
path itself.
( )
f z 0
z 1
z
