Chapter 6: Integrals and Complex
variable function series
6.5 The complex variable function series
6.6 Classification of anomalies


The complex variable function series
• A series is the sum of the terms of a sequence.

• For any sequence 𝑧𝑚 of real numbers, complex numbers, functions, etc., the associated series is defined
as the ordered formal sum
∞

𝑧𝑚 = 𝑧1 + 𝑧2 + …
𝑚=1

• The sum of the first n terms 𝑠𝑛 = 𝑧1 + 𝑧2 + ⋯ + 𝑧𝑛 is called the nth partial sum of the series.
• A convergent series is one whose sequence of partial sums converges
∞

lim 𝑠𝑛 = 𝑠

𝑛→∞

Then we write

𝑠=

𝑧𝑚 = 𝑧1 + 𝑧2 + ⋯
𝑚=1

and call s the sum or value of the series.

• A series is not convergent is called a divergent series.


The complex variable function series
• Given a sequence of complex variable function: 𝑢1 𝑧 , 𝑢2 𝑧 , … , 𝑢𝑛 𝑧 , … , in a region G, the
series of complex variable functions is

∞

𝑢𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 𝑧 + ⋯

(1)

𝑛=1
• The sum of the first n terms

𝑆𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 (𝑧) is called the nth partial sum of the series.

• If the series is convergent with 𝒛 = 𝒛𝟎 then 𝒛𝟎 is called the point of convergence.
• If the series is divergent with 𝒛 = 𝒛𝟎 then 𝒛𝟎 is called the point of divergence.


Uniform Convergence
∞

𝑓 𝑧 =

𝑢𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 𝑧 + ⋯


(1)

𝑛=1

• Convergence test: The series of functions (1) is convergent if and only if for every given 𝜺 > 𝟎 (no matter
how small) we can find a real number 𝑵(𝜺, 𝒛) such that: If 𝒏 > 𝑵 𝜺, 𝒛 then 𝒇 𝒛 − 𝑺𝒏 (𝒛) < 𝜺
• Uniform Convergence: The series of functions (1) is called uniformly convergent in a region G if for
every given 𝜺 > 𝟎 we can find an 𝑵 = 𝑵(𝜺), not depending on 𝑧, such that

𝑓 𝑧 − 𝑆𝑛 (𝑧) < 𝜀 for all 𝑛 > 𝑁 𝜀 and all 𝑧 in G
• Test for Uniform Convergent (Weierstrass M-Test): if 𝒖𝒏 𝒛
∞
𝒏=𝟏 𝒂𝒏

< 𝒂𝒏 for all 𝑧 in G and the series

converges then the series of functions (1) is uniformly convergent in G.


Uniform Convergence
Properties
∞

𝑓 𝑧 =

𝑢𝑛 𝑧 = 𝑢1 𝑧 + 𝑢2 𝑧 + ⋯ + 𝑢𝑛 𝑧 + ⋯

(1)


𝑛=1

1. Continuity of the Sum
Let the series (1) be uniformly convergent in a region G. Let 𝑓(𝑧) be its sum. Then if each term 𝑢𝑛 𝑧 is

continuous at a point 𝑧1 in G, the function 𝑓(𝑧) is continuous at 𝑧1 (See the proof in Page 700).
2. Termwise Integration
Let the series (1) be a uniformly convergent series of continuous functions in a region G. Let C be any path in
G. Then the series
∞

𝑓 𝑧 𝑑𝑧 =
𝐶

is convergent and has the sum

𝑢𝑛 𝑧 𝑑𝑧 =
𝑛=1 𝐶

𝐶

𝑢1 𝑧 𝑑𝑧 +
𝐶

𝑓 𝑧 𝑑𝑧 ( See the proof in Page 702).

𝑢2 𝑧 𝑑𝑧 + ⋯
𝐶



Power Series
• A power series in powers of 𝑧 − 𝑧0 is a series of the form
∞

𝑎𝑛 (𝑧 − 𝑧0 )𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯

(2)

𝑛=0

where

𝑧 is a complex variable,
𝑎0 , 𝑎1 , … are complex (or real) constants, called the coefficients of the series,
𝑧0 is a complex (or real) constant, called the center of the series.

• If 𝑧0 = 0, we obtain as a particular case a power series in powers of 𝑧:
∞

𝑎𝑛 𝑧 𝑛 = 𝑎0 + 𝑎1 𝑧 + 𝑎2 𝑧 2 + ⋯
𝑛=0

(3)


Convergence of a Power Series
∞

𝑎𝑛 (𝑧 − 𝑧0 )𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯


(2)

𝑛=0

1. Every power series (2) converges at the center 𝑧0
2. If (2) converges at a point 𝑧 = 𝑧1 ≠ 𝑧0 , it converges absolutely for every 𝑧 closer to 𝑧0 than 𝑧1 , that is,
𝑧 − 𝑧0 < 𝑧1 − 𝑧0 . See the Figure.
3. If (2) diverges at 𝑧 = 𝑧2 , it diverges for every 𝑧 farther away from 𝑧0 than 𝑧2 . See the Figure.


Radius of Convergence of a Power Series
∞

𝑎𝑛 (𝑧 − 𝑧0 )𝑛 = 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯

(2)

𝑛=0

• We consider the smallest circle with center 𝑧0 that includes all the points at which a given power series (2)
converges. Let 𝑅 denote its radius.
• The circle 𝑧 − 𝑧0 = 𝑅 is called the circle of convergence and its radius 𝑅 is the radius of convergence of
(2)

• Determination of the Radius of Convergence from the Coefficients:
𝑎𝑛
1
𝑅 = lim
𝑜𝑟 𝑅 = lim 𝑛
𝑛→∞ 𝑎𝑛+1

𝑛→∞
𝑎𝑛


Taylor Series
The Taylor Series of a function 𝑓(𝑧), the complex analog of the real Taylor series is
∞

𝑓 𝑧 =

𝑎𝑛 (𝑧 − 𝑧0 )
𝑛=1

𝑛

1
𝑤ℎ𝑒𝑟𝑒 𝑎𝑛 = 𝑓
𝑛!

𝑛

𝑧0

1
=
2𝜋𝑖

𝑓(𝑧 ∗ )
𝑑𝑧 ∗
∗

𝑛+1
(𝑧 − 𝑧0 )

(4)

𝑧 ∗ is the variable of integration
We can write

𝑓 𝑧 = 𝑓 𝑧0

𝑧 − 𝑧0 ′
(𝑧 − 𝑧0 )2 ′′
𝑧 − 𝑧0
+
𝑓 𝑧0 +
𝑓 𝑧0 + ⋯ +
1!
2!
𝑛!

Where 𝑅𝑛 (𝑧) is called the remainder of the Taylor series after the term 𝑎𝑛 (𝑧 − 𝑧0 )𝑛
(𝑧 − 𝑧0 )𝑛+1
𝑅𝑛 𝑧 =
2𝜋𝑖

𝑓(𝑧 ∗ )
𝑑𝑧 ∗
∗
𝑛+1
∗

𝑧 − 𝑧0
(𝑧 − 𝑧)

 Taylor series are Power series
 A Maclaurin series is a Taylor series with center 𝑧0 = 0

𝑛

𝑓

𝑛

𝑧0 + 𝑅𝑛 (𝑧)


Important Special Taylor Series
∞
𝑧

𝑒 =
𝑛=0

𝑧𝑛
𝑧
𝑧
=1+ + +⋯
𝑛!
1! 2!

∞


sin 𝑧 =

2𝑛+1
3
5
7
𝑧
𝑧
𝑧
𝑧
(−1)𝑛
=𝑧− + − +⋯
2𝑛 + 1 !
3! 5! 7!

𝑛=0

∞

cos 𝑧 =

(−1)𝑛
𝑛=0

𝑧 2𝑛
𝑧2 𝑧4 𝑧6
= 1− + − +⋯
2𝑛 !
2! 4! 6!


𝑧2 𝑧3 𝑧4
𝐿𝑛 1 + 𝑧 = z − + − + ⋯
2
3
4
𝑧3 𝑧5
𝑎𝑟𝑐𝑡𝑔𝑧 = 𝑧 − + − ⋯
3
5


Laurent Series
Let 𝑓 𝑧 be analytic in a domain containing two concentric circles 𝐶1 and 𝐶2 with center 𝑧0 and the annulus
between them. Then 𝑓(𝑧) can be represented by the Laurent series
∞

∞
𝑛

𝑓 𝑧 =

𝑎𝑛 (𝑧 − 𝑧0 ) +
𝑛=0

𝑛=1

𝑏1
𝑏2
𝑏𝑛

2
= 𝑎0 + 𝑎1 𝑧 − 𝑧0 + 𝑎2 (𝑧 − 𝑧0 ) + ⋯ +
+
𝑛
𝑧
−
𝑧
𝑧 − 𝑧0
(𝑧 − 𝑧0 )
0

2

+⋯

consisting of nonnegative and negative powers. The coefficient of this Laurent series are given by the integrals
1
𝑎𝑛 =
2𝜋𝑖

𝑓(𝑧 ∗ )
∗
𝑑𝑧
∗
𝑛+1
𝐶 (𝑧 − 𝑧0 )

1
𝑏𝑛 =
2𝜋𝑖


𝑧 ∗ − 𝑧0
𝐶

taken counterclockwise around any simple closed path 𝐶 that lies in the
annulus and encircles the inner circle.
∞

𝑛=1

𝑏𝑛
is called the principal part of 𝑓(𝑧)
(𝑧 − 𝑧0 )𝑛

𝑛−1 𝑓

𝑧 ∗ 𝑑𝑧 ∗

(5)


Classifying singularities
• A singular point of an analytic function 𝑓(𝑧) is a 𝑧0 at which 𝑓 𝑧 is not analytic, but every
neighborhood of 𝑧 = 𝑧0 contains points at which 𝑓 𝑧 is analytic.
• A zero is a 𝑧 at which 𝑓 𝑧 = 0
• A isolated singularity of 𝑓(𝑧) is 𝑧 = 𝑧0 if it has a neighborhood without further singularities of 𝑓 𝑧
𝜋
2

Ex: tan 𝑧 has isolated singularities at ± , ±


3𝜋
,…
2
𝑏

• In Laurent series, if the principal part has only finitely many terms 𝑧−𝑧1 + ⋯ +
0

𝑏𝑚
𝑧−𝑧0 𝑚

(𝑏𝑚 ≠ 0) then

the singularity of 𝑓(𝑧) at 𝑧 = 𝑧0 is called a pole, and 𝑚 is called is order.
• If the principal part has infinitely many terms, we say that 𝑓(𝑧) has at 𝑧 = 𝑧0 an isolated essential
singularity.
• We say that a function 𝑓(𝑧) has a removable singularity at 𝑧 = 𝑧0 if 𝑓(𝑧) is not analytic at 𝑧 = 𝑧0 but

can be made analytic there by assigning a suitable value 𝑓 𝑧0
Ex: 𝑓 𝑧 =

sin 𝑧
𝑧

become analytic at 𝑧 = 0 if we define 𝑓 0 = 1