Complex Analysis
Slide 9: Power Series
MA201 Mathematics III
Department of Mathematics
IIT Guwahati
August 2015
Complex Analysis Slide 9: Power Series
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Learning Outcome of this Lecture
We learn
Sequence of Complex Numbers and Series of Complex Numbers
Sequences & Series of Functions: Pointwise, Absolute, Uniform
Convergence
Power Series
Taylors Theorem / Taylor Series
Laurent Theorem/ Laurent Series
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Sequence of Complex Numbers
A sequence of complex numbers is a map from a : N → C given by
a(n) = an for n ∈ N. It is written as {an } or (an ) or < an >.
Definition
Let {an } be a sequence of complex numbers. If there exists a complex
number a∗ such that for each > 0, there exists a natural number N0
such that
|an − a∗ | < for all n ≥ N0
then we say that {an } converges to a∗ .
a∗ is called the limit of the sequence {an }.
We write it as {an } → a∗ as n → ∞ or lim an = a∗ .
n→∞
Examples:
{an = (1/n) + 2i} converges to 2i.
{an = n(1/n) + i ((n + 1)/n)} converges to 1 + i.
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Results
If {an } converges then the limit of {an } is unique.
If {an } converges then the set S = {an : n ∈ N} is bounded.
If {an } converges then {|an |} converges. But converse is NOT
true.
{an = xn + i yn } converges to a∗ = x∗ + i y ∗ if and only if
{xn } → x∗ and {yn } → y ∗ . That is,
{an } → a∗ if and only if {<(an )} → <(a∗ ) and {=(an )} → =(a∗ ) .
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Series of Complex Numbers
∞
X
an = a0 + a1 + a2 + · · · is called an (infinite) series of complex
n=0
numbers.
Definition
∞
X
Let
an be a series of complex numbers. Define the sequence of
n=0
partial sums by s0 = a0 and sn =
number s such that
n
X
ak . If there
k=0
the sequence {sn } of partial
∞
X
then we say the series
exists a complex
sums converges to s
an converges to s and we write it as
n=0
∞
X
an = s .
n=0
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If the sequence of partial sums does not converge then we say that the
∞
X
series
an diverges.
n=0
Examples:
P
Let {an = (1/n2 ) + i(1/2)n } for n ∈ N. Then
an P
converges.
n
Let {an = (1/n!) + i(1/2) } for n = 0, 1, · · · . Then
an converges
e + 2i.
P
Let {an = (1/n) + i(1/2)n } for n ∈ N. Then
an diverges.
P
P
We say that the series an converges absolutely if |an | converges.
Results:
P
If
an converges then {an } → 0 as n → ∞.
P
P
If
an converges absolutely then
an converges. But converse
is NOT true.
Similarly, we can define Sequence of Complex Functions and Series of
Complex Functions.
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Sequence of Functions: Pointwise Convergence
Let fn : D ⊆ C → C, for n = 0, 1, · · · be the functions defined on a set
D. We say that the sequence {fn (z)} of functions converges
(pointwise) to a function f (z) in D, if for each point z0 ∈ D and for each
> 0, there exists a natural number N0 that may depend on both and
the point z0 such that
|fn (z0 ) − f (z0 )| < for all n ≥ N0 .
In this case, we write it as lim fn (z) = f (z) for z ∈ D.
n→∞
If for some point z0 ∈ D, the sequence {fn (z0 )} does not converge or
tends to ∞ then we say that the sequence {fn (z)} diverges at the
point z = z0 .
Example: Let fn (z) = z n for z ∈ D = {z ∈ C : |z| < 1} where n ∈ N.
Let f (z) = 0 for all z ∈ D. Then, {fn (z)} converges pointwise to f (z) in
D.
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Series of Functions: Pointwise Convergence
Definition
Let fn : D ⊆ C → C, for n = 0, 1, · · · be the functions defined on a set
∞
X
D. The series
fn (z) of functions converges (pointwise) to a
n=0
(
)
n
X
function S(z) in D if the sequence Sn (z) =
fk (z) of partial sums
k=0
converges (pointwise) to the function S(z) in D.
∞
X
In this case, we write it as S(z) =
fn (z) for z ∈ D.
n=0
Example: Let fn (z) = z n for z ∈ D = {z P
∈ C : |z| < 1} where n ∈ N.
Let S(z) = 1/(1 − z) for all z ∈ D. Then,
fn (z) converges pointwise
to S(z) in D.
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Absolute Convergence
Let fn : D ⊆ C → C, for n = 0, 1, · · · be the functions defined on a set
D.
Definition
We say that the sequence {fn (z)} of functions converges absolutely to
a function g(z) in D, if for each point z0 ∈ D, the sequence {|fn (z0 )|}
converges (pointwise) to g(z0 ).
Definition
∞
X
The series
fn (z) converges absolutely to a function T (z) in D if the
n=0
(
)
n
X
sequence Sn (z) =
|fk (z)| converges (pointwise) to the function
k=0
T (z) in D.
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Uniform Convergence
Let fn : D ⊆ C → C, for n = 0, 1, · · · be the functions defined on a set
D.
Definition
We say that the sequence {fn (z)} of functions converges uniformly to
a function f (z) in the set D, if for each > 0, there exists a natural
number N (that may depend only on ) such that
|fn (z) − f (z)| < for all n ≥ N and for all z ∈ D .
Definition
∞
X
The series
fn (z) converges uniformly to a function S(z) in D if the
n=0
(
)
n
X
sequence Sn (z) =
fk (z) of partial sums converges uniformly to
k=0
the function S(z) in D.
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We now introduce a special type of series of functions, namely, power
series.
Power Series
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Power Series
Definition
A power series about a point z0 is an infinite series of the form
∞
X
an (z − z0 )n .
n=0
Example-1: The geometric series
∞
X
z n is one of the easiest
n=0
examples of a power series.
∞
X
zn
Example-2:
is another example of a power series.
n
n=1
∞
X
(z − 3)n
Example-3:
is another example of a power series.
4n
n=1
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Convergence of Power Series
For which values of z does the geometric series
∞
X
z n converge?
n=0
It is easily seen that 1 − z n+1 = (1 − z)(1 + z + z 2 + · · · + z n ) so that
1 + z + · · · + zn =
1 − z n+1
.
1−z
If |z| < 1 then lim z n = 0 and so the geometric series is convergent with
∞
X
n=0
zn =
1
.
1−z
If |z| > 1 then lim z n = ∞ and the series diverges.
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Recall: Limit Superior of Real Sequences
Let {an } be a sequence of real numbers.
lim sup an = lim (sup{an , an+1 , · · · }) .
n→∞
n→∞
lim inf an = lim (inf{an , an+1 , · · · }) .
n→∞
n→∞
Other Notation: lim sup is also denoted by lim. Further these
concepts lim sup and lim inf are defined only for real sequences and
NOT for complex sequences.
Results:
For a real sequence, lim sup an and lim inf an always exist and it
may be +∞ or −∞ also.
Always
lim inf an ≤ lim sup an .
If {an } converges then
lim inf an = lim an = lim sup an .
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Basic Result on Convergence of Power Series
Theorem
For a given power series
∞
X
an (z − z0 )n define the number R,
n=0
0 ≤ R ≤ ∞, by
1
1
= lim sup |an | n
R
n→∞
(Cauchy-Hadamard Formula)
then:
1
if |z − z0 | < R, the series converges absolutely;
2
if |z − z0 | > R, the series diverges;
3
if 0 < r < R, the series converges uniformly on {z : |z| ≤ r}.
Moreover, the number R is the only number having the above said
three properties.
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Continuation of Previous Slide
In the previous theorem:
The number R is called the radius of convergence of the power
series.
The circle |z − z0 | = R is called the circle of the convergence of
the series.
The open disk |z − z0 | < R is called the domain of convergence or
disk of convergence of the series.
Examples:
∞
X
The power series
k n z n has radius of convergence R = 1/|k|.
The power series
The power series
n=0
∞
X
n=0
∞
X
zn
has radius of convergence R = ∞.
n!
2
5n z n has radius of convergence R = 0.
n=0
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Radius of Convergence as the limit of Ratios of
Coefficients
The radius of the convergence of a power series can be calculated
sometimes from the ratio of the coefficients as follows.
Theorem
∞
X
If
an (z − z0 )n is a given power series with radius of convergence R,
n=0
then
an R = lim n→∞ an+1 if this limit exists (including the limit tending to +∞ in the extended real
number system).
Example:
The power series
∞
X
zn
n=0
n!
has radius of convergence R = ∞.
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On Circle of Convergence - What happens?
On the circle of convergence C : |z − z0 | = R, power series may
converge on C, diverge on C, or converge on some part of C and
diverge on the remaining part.
∞
X
The power series
z n diverges at all points on the circle of
n=0
convergence |z| = 1, since |z n | does not tend to 0 as n → ∞.
∞
X
zn
The power series
series diverges at the point z = 1 and
n
n=1
converges at the point z = −1. One can show that this power
series converges at all points on the circle |z| = 1 except at the
point z = 1.
∞
X
zn
converges at all points on the circle of
The power series
n2
n=1
X z n X 1
≤
convergence |z| = 1, since
< ∞.
n2 n2
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Theorem
P
Picard’s Theorem: Consider the power series an z n and suppose
that:
1
The coefficients an are real nonnegative numbers.
2
an ≥ an+1 for n = 1, 2, 3, · · · .
{an } → 0 as n → ∞.
P
Then the power series
an z n converges at all points of the circle
|z| = 1, except possibly at z = 1, so its radius of convergence is at
least 1.
P1
Using the above theorem and using the fact
n diverges, one can
∞
n
Xz
conclude that
converges at all points on the circle |z| = 1 except
n
n=1
at the point z = 1.
3
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Properties
Let
∞
X
n
an (z − z0 ) and
n=0
∞
X
bn (z − z0 )n be power series with radius of
n=0
convergence R1 and R2 respectively. Then,
∞
X
Sum:
(an + bn )(z − z0 )n has the radius of convergence
n=0
R ≥ min(R1 , R2 ).
Scalar Multiplication:
∞
X
λan (z − z0 )n where λ 6= 0 has the radius
n=0
of convergence R = R1 .
∞
n
X
X
Product:
ak bn−k (Cauchy Product)
cn (z − z0 )n where cn =
n=0
k=0
has the radius of convergence R ≥ min(R1 , R2 ).
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Properties (Continuation of Previous Slide)
Product Coordinatewise:
convergence R ≥ R1 R2 .
∞
X
an bn (z − z0 )n has the radius of
n=0
Division Coordinatewise: If bn 6= 0 for all n then
∞
X
an
n=0
bn
(z − z0 )n
has the radius of convergence R ≥ R1 /R2 .
Division of Two Series: If r is the largest real numberPsuch that
P
an (z − z0 )n
bn (z − z0 )n 6= 0 for all z ∈ {z : |z − z0 | < r} then P
bn (z − z0 )n
has the radius of convergence R ≥ min(r, R1 , R2 ).
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What can we say about the sum function of a Power
Series?
P
Let
an (z − z0 )n have the radius of convergence R > 0. Let us
denote
the sum function of this series by f (z). That is,
P
an (z − z0 )n =: f (z) for z ∈ BR (z0 ).
Questions:
Is the sum function f (z) differentiable/ analytic in BR (z0 )?
P
Is the series formed by termwise differentiation nan (z − z0 )n−1
convergent? If so, what is its radius of convergence R∗ ?
∞
X
Set g(z) :=
nan (z − z0 )n−1 for z ∈ BR∗ (z0 ). Is g(z) = f 0 (z) for
n=1
z ∈ BR∗ (z0 ).
What about k-times differentiated series
∞
X
n(n − 1) · · · (n − k + 1)an (z − z0 )n−k ?
n=k
Is there any relation between the coefficients an ’s and f (z)?
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Sum function of a Power Series is analytic
Theorem: Let
∞
X
an (z − z0 )n have radius of convergence R > 0.
n=0
Then,
The function defined by f (z) :=
∞
X
an (z − z0 )n is analytic in
n=0
BR (z0 ) = {z ∈ C : |z − z0 | < R}.
∞
X
n(n − 1) · · · (n − k + 1)an (z − z0 )n−k has the
For each k ≥ 1,
n=k
radius of convergence R.
∞
X
f (k) (z) =
n(n − 1) · · · (n − k + 1)an (z − z0 )n−k for z ∈ BR (z0 ).
n=k
For n = 0, 1, · · · , the coefficient an =
f (n) (z0 )
.
n!
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Example for Previous Theorem
The power series
R = ∞.
∞
X
(−1)n z 2n+1
n=0
(2n + 1)!
The function f (z) = sin z =
has the radius of convergence
∞
X
(−1)n z 2n+1
n=0
(2n + 1)!
is analytic in C.
∞
X
d
(−1)n z 2n
(sin z) = cos z =
and this series also
dz
(2n)!
n=0
has the radius of convergence R = ∞.
Observe that
Note: In previous theorem, like termwise differentiation of power
series, termwise integration (indefinite integral) is also valid for power
series.
For example, by doing termwise integration of power series of cos z, we
can get the power series of sin z.
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To think
Last theorem says, Given power series, its sum function is analytic
(and hence infinitely many times differentiable) in the disk/domain of
convergence of the power series.
Now, think about converse of above statement. Is it true statement?
Question: Let D be an open set and let z0 ∈ D. Given that f (z) is
analytic in D.
Whether f can have power series representation about z0 ?
∞
X
That is, whether f (z) =
an (z − z0 )n for z ∈ BR (z0 ) ⊆ D for some
n=0
R > 0?
If the answer is YES, then is there more than one such power series
possible?
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Analytic function has a Power Series Representation
Taylor Theorem: Let f (z) be analytic in
BR (z0 ) = {z ∈ C : |z − z0 | < R}. Then, f (z) has a power series
expansion around z0 given by
f (z) =
∞
X
an (z − z0 )n
for z ∈ BR (z0 )
n=0
Z
f (w) dw
f (n) (z0 )
1
=
for n = 0, 1, 2, · · · where
n!
2πi Cr (w − z0 )n+1
Cr = {z ∈ C : |z − z0 | = r} for any r with 0 < r < R. This series is
called the Taylor series of f about the point z0 and has radius of
convergence ≥ R.
Further, the Taylor series of f about that point z0 is unique.
where an =
Proof: Worked out on the board.
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Example for Taylor Theorem
Example: Find the power series of f (z) = ez about the point z0 = i.
Observe that for each n ∈ N, f (n) (z) = ez for z ∈ C.
This gives that f (n) (i) = ei for n = 1, 2, · · · .
For each n = 0, 1, 2, · · · ,
an =
ei
f (n) (i)
=
.
n!
n!
Therefore, the Taylor series of ez about the point z0 = i is given by
ez =
∞
X
ei
(z − i)n
n!
n=0
and it has radius of convergence R = ∞.
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Remarks
The Taylor series of f (z) about the point z0 = 0 is called the
Maclaurin series of f .
If f (z) is analytic in |z − z0 | < R for some R > 0, then by Taylor
theorem, f (z) can be approximated with arbitrarily high precision
by a polynomial Pn (z) of sufficiently high degree.
Alternative Way to find Radius of Convergence of Taylor Series:
If f (z) is analytic at z0 , then the radius of convergence R of the
Taylor series of f (z) about z = z0 is the distance from z0 to the
point (singularity) nearest to z0 at which f (z) fails to be analytic.
That is,
R = |z0 − z ∗ | where z ∗ is the singularity of f nearest to z0 .
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Comparison between Real Functions and Complex
Functions
By Taylor theorem, if a function f (z) is infinitely differentiable in an
open set D, then f (z) can be expanded in power series in D. This
result is not true in case of real valued functions of a real variable.
For example, the function f (x) = exp(−1/x2 ) for x ∈ R \ {0} and
f (0) = 0 is infinitely many times differentiable in the neighborhood
of x0 = 0, but f (x) can not be represented by a power series
about the point x0 = 0.
A real Taylor series of a real valued function f of a real variable
converges if and only if the Taylor remainder term goes to zero. In
a complex Taylor series, the remainder term is irrelevant; the
Taylor series will converge to in the largest disk that one can fit
inside the domain of analyticity of f .
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Analytic at ∞
We say that the function f (z) is analytic at z = ∞ if the function
g(w) = f (1/w) is analytic at w = 0.
Thus, we make the change of variable w = 1/z, and we study the
behaviour of f (z) at z = ∞ by studying the behaviour f (1/w) at w = 0.
Examples:
f (z) = 1/z 2 is analytic at ∞.
f (z) = e1/z is analytic at ∞.
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Power series of f (z) about z = ∞
If f (z) is analytic at z = ∞ then the function g(w) = f (1/w) is analytic
at w = 0 and hence
∞
X
g(w) =
cn wn for |w| < r for some r > 0 .
n=0
Thus, fP
(z) can be represented by a power series as
cn
f (z) = ∞
for
|z| > R = 1r and it is the power series
n=0 z n
expansion of f (z) about the point z = ∞.
Example:
1/z
e
=
∞
X
n=0
1
n! z n
for |z| > 0 .
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Is power series of f possible at singular points?
We now wish to investigate the possibility of representing a function by
a power series near singular points.
A singular point z0 is said to be an isolated singular point of f (z) if f (z)
is analytic in the punctured disk 0 < |z − z0 | < r for some r > 0.
Then, the function f (z) can be represented by a power series about an
isolated singular points. But, in this case the power series of f (z)
contains the negative powers of (z − z0 ) also.
Similarly, if f (z) is analytic in the annular region r1 < |z − z0 | < r2 and
f (z) need not be analytic in the region |z − z0 | < r1 then also f (z) can
be represented by a power series in the annular region
0 < r1 < |z − z0 | < r2 .
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Laurent Theorem
Laurent Theorem: Let f be analytic in the annular region
r1 < |z − z0 | < r2 . Then f has a series representation given by
f (z) =
∞
X
n=1
a−n (z − z0 )−n +
∞
X
an (z − z0 )n
for r1 < |z − z0 | < r2
n=0
Z
1
f (w) dw
where the coefficients an =
for any r with
2πi |z−z0 |=r (w − z0 )n+1
r1 < r < r2 .
The above series is called the Laurent Series and converges
absolutely in r1 < |z − z0 | < r2 .
Further, it converges uniformly in R1 ≤ |z − z0 | ≤ R2 where
r1 < R1 < R2 < r2 .
Moreover this series is unique.
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The Laurent series for f in the annular region is usually abbreviated
∞
X
an (z − z0 )n .
n=−∞
∞
X
In the Laurent series
an (z − z0 )n , the series containing the
n=−∞
negative powers of (z − z0 ), namely,
−1
X
an (z − z0 )n =
n=−∞
∞
X
a−n (z − z0 )−n
n=1
is called the principal part of the Laurent series.
The series containing the non-negative powers of (z − z0 ), namely,
∞
X
an (z − z0 )n
n=0
is called the regular part of the Laurent series.
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Taylor Theorem is a special case of Laurent Theorem
If f (z) is analytic at the point z0 then the Laurent series of f (z) about
the point z = z0 does not contain any negative powers of (z − z0 ).
That is, the Laurent series of f (z) has no principal part.
Hence the Laurent series reduces to the Taylor series of f (z) about the
point z = z0 in this case.
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Worked out Example 1
1
.
(z − 1)(z − 2)
Find the Laurent series of f about the point z = 1 (OR) Find the
power series expansion of f in the region 0 < |z − 1| < 1.
Let f (z) =
1
2
Find the Laurent series of f about the point z = 2 (OR) Find the
power series expansion of f in the region 0 < |z − 2| < 1.
3
Find the Laurent series of f about the point z = 0 (OR) Find the
power series expansion of f in the region |z| < 1.
4
Find the Laurent series of f about the point z = ∞ (OR) Find the
power series expansion of f in the region |z| > 2.
5
Find the Laurent series of f in the annular region 1 < |z| < 2.
Details are Worked Out on the Board.
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Worked out Example 2
Expand e1/z in the Laurent series about the point z = 0.
Details are Worked Out on the Board.
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