1
Lecture 7
Cauchy Theorem
:
Let
f
be analytic inside and on a simple, closed, piecewise
smooth curve C. Then,
( )
0
C
f
z dz
.
Definitions:
Let
( ),
z t
a
t
b
, be parametric representation of
the curve C.
Simple Curve
: The curve C is said to be
simple,
if it does not
have any self
‐
intersections
(i.e.
1
2
(
)
(
)
z t
z t
whenever
1
2
1
2
(
,
)
t
t
a
t
t
b
).
Closed Curve
: The curve C is said to be
Closed,
if end point of the
curve is the same as its initial point
(i.e.
( )
( )
z a
z b
).
Piece
‐
wise smooth Curve
: The curve C is said to be
Piece
‐
wise
smooth,
if
( )
z t
is piece
‐
wise differentiable (i.e. differentiable for
all except finitely many
t
) and
( ) (
( ))
d
z t
denoted as z t
dt
is piece
‐
wise continuous in the interval
[ , ]
a b

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