Chap 5' Series

1
Chap 5. Series Series representations
of analytic functions
43. Convergence of Sequences and Series
An infinite sequence ??
of complex numbers has a limit z if, for each
positive ?, there exists a positive integer n0
such that
2
The limit z is unique if it exists. (Exercise
6). When the limit exists, the
sequence is said to converge to z.
Otherwise, it diverges.
Thm 1.
3
An infinite series
4
A necessary condition for the convergence of
series (6) is that
The terms of a convergent series of complex
numbers are, therefore, bounded,
Absolute convergence
Absolute convergence of a series of complex
numbers implies convergence of that series.
5
44. Taylor Series
Thm. Suppose that a function f is analytic
throughout an open disk
Then at each point z in that disk, f(z) has the
series
representation
That is, the power series here converges to f(z)
6
Maclaurin series.
z00?case
Positively oriented
within
and z is interior to it.
7
The Cauchy integral formula applies
8
(No Transcript)
9
?? ??
(b) For arbitrary z0
composite function
must be analytic when
10
The analyticity of g(z) in the disk
ensures
the existence of a Maclaurin series
representation
11
45 Examples
Ex1.
It has a Maclaurin series representation which is
valid for all z.
12
Ex2. Find Maclaurin series representation of
Ex3.
13
Ex4.
14
Ex5.
?Laurent series ??
15
46. Laurent Series
If a function f fails to be analytic at a point
z0, we can not apply Taylors theorem at that
point. However, we can find a series
representation for f(z) involving both positive
and negative powers of (z-z0). Thm. Suppose that
a function f is analytic in a domain
and let C denote any positively oriented
simple closed contour around z0 and lying in that
domain. Then at each z in the domain
16
where
Pf see textbook.
17
47. Examples
The coefficients in a Laurent series are
generally found by means other than by appealing
directly to their integral representation.
Ex1.
Alterative way to calculate
18
Ex2.
19
Ex3.
has two singular points z1 and z2, and is
analytic in the domains
Recall that
(a) f(z) in D1
20
(b) f(z) in D2
21
(c) f(z) in D3
22
48. Absolute and uniform convergence of power
series
Thm1.
(1)
23

• The greatest circle centered at z0 such that
  series (1) converges at each point inside is
  called the circle of convergence of series (1).
• The series CANNOT converge at any point z2
  outside that circle, according to the theorem
  otherwise circle of convergence is bigger.

24
(No Transcript)
25
then that series is uniformly convergent in the
closed disk
Corollary.
26
49. Integration and Differentiation of power
series
Have just seen that a power series
represents continuous function at each point
interior to its circle of convergence. We state
in this section that the sum S(z) is actually
analytic within the circle.
Thm1. Let C denote any contour interior to the
circle of convergence of the power series (1),
and let g(z) be any function that is continuous
on C. The series formed by multiplying each term
of the power series by g(z) can be integrated
term by term over C that is,
27
Corollary. The sum S(z) of power series (1) is
analytic at each point z interior to the circle
of convergence of that series.
Ex1.
is entire
But series (4) clearly converges to f(0) when
z0. Hence f(z) is an entire function.
28
Thm2.
The power series (1) can be differentiated term
by term. That is, at each point z interior to the
circle of convergence of that series,
Ex2.
Diff.
29
50. Uniqueness of series representation
Thm 1. If a series
Thm 2. If a series
converges to f(z) at all points in some annular
domain about z0, then it is the Laurent series
expansion for f in powers of for
that domain.
30
51. Multiplication and Division of Power Series
Suppose
then f(z) and g(z) are analytic functions in

has a Taylor series expansion
Their product
31
Ex1.
The Maclaurin series for is valid in
disk
Ex2.
Zero of the entire function sinh z