(1)
1
+
=
2 + 2
,
( ) =
(2) If is analytic on an open set A and () = (, ) + (, ), then u, v satisfy
Cauchy-Riemann equations:
=
,
=
and u, v are harmonic and v is
called a harmonic conjugate of u. Conversely, if u and v are 1 on A, and they
satisfy Cauc
Formulas for Exam 2
(1) If () = () on an open set containing a curve (), , then
() = () ().
(2) If f is analytic on and inside a simple closed curve with positive orientation
then
(a) () = 0
(b) for 0 inside , (0 ) =
1
2
()
0
, () (0 ) =
!
2
()
(0 ) +1
Theorem If u and v are real-valued differentiable functions on an open subset A of
, and they satisfy the Cauchy-Riemann equations on A, then
is
analytic on A.
Proof. Let
(
)
(
(
)
)
(
)
Since u and v are differentiable,
Let
(
)
(
(
)
(
)
(
)
)
)
(
)
(
)
5.
Taylor and Laurent series
Complex sequences and series
An innite sequence of complex numbers, denoted by cfw_zn, can be
considered as a function dened on a set of positive integers into
n+1
the unextended complex plane. For example, we take zn =
2n
1+i
(1)
1
+
=
2 + 2
,
( ) =
(2) If is analytic on an open set A and () = (, ) + (, ), then u, v satisfy
Cauchy-Riemann equations:
=
,
=
and u, v are harmonic and v is
called a harmonic conjugate of u. Conversely, if u and v are 1 on A, and they
satisfy Cauc
Theorem If u and v are real-valued differentiable functions on an open subset A of
, and they satisfy the Cauchy-Riemann equations on A, then
is
analytic on A.
Proof. Let
(
)
(
(
)
)
(
)
Since u and v are differentiable,
Let
(
)
(
(
)
(
)
(
)
)
)
(
)
(
)
Formulas for Exam 2
(1) If () = () on an open set containing a curve (), , then
() = () ().
(2) If f is analytic on and inside a simple closed curve with positive orientation
then
(a) () = 0
(b) for 0 inside , (0 ) =
1
2
()
0
, () (0 ) =
!
2
()
(0 ) +1