Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Zeros and Poles of Analytic Functions
Bernd Schröder
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Introduction
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Introduction
1. If f is analytic and has a zero at z0 , then
vice versa.
Bernd Schröder
Zeros and Poles of Analytic Functions
1
f
has a pole there and
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Introduction
1. If f is analytic and has a zero at z0 , then 1f has a pole there and
vice versa.
2. So it makes sense to analyze zeros and poles at the same time.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Introduction
1. If f is analytic and has a zero at z0 , then 1f has a pole there and
vice versa.
2. So it makes sense to analyze zeros and poles at the same time.
3. One important goal are results that allow us to efficiently
compute residues.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Introduction
1. If f is analytic and has a zero at z0 , then 1f has a pole there and
vice versa.
2. So it makes sense to analyze zeros and poles at the same time.
3. One important goal are results that allow us to efficiently
compute residues. (Or, if we don’t memorize the results, practice
with methods to compute residues.)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Introduction
1. If f is analytic and has a zero at z0 , then 1f has a pole there and
vice versa.
2. So it makes sense to analyze zeros and poles at the same time.
3. One important goal are results that allow us to efficiently
compute residues. (Or, if we don’t memorize the results, practice
with methods to compute residues.)
4. We will also be able to derive some results that show that
analytic functions are uniquely determined by a surprisingly
small (but still infinite) number of values.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 . So
∞
f (z) =
∑ an (z − z0 )n
n=m
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 . So
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k .
n=m
Bernd Schröder
Zeros and Poles of Analytic Functions
k=0
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 . So
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Conversely, if
n=m
k=0
f (z) = (z − z0 )m g(z) for |z − z0 | < r for some r > 0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 . So
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Conversely, if
n=m
k=0
f (z) = (z − z0 )m g(z) for |z − z0 | < r for some r > 0, then
∞
f (z) = (z − z0 )m ∑ cn (z − z0 )n
n=0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 . So
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Conversely, if
n=m
k=0
f (z) = (z − z0 )m g(z) for |z − z0 | < r for some r > 0, then
∞
f (z) = (z − z0 )m ∑ cn (z − z0 )n =
n=0
Bernd Schröder
Zeros and Poles of Analytic Functions
∞
∑ ck−m (z − z0 )k .
k=m
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Definition. If f is analytic in a neighborhood of z0 , and f (z0 ) = 0,
then the smallest m > 0 so that f (m) (z0 ) 6= 0 is called the order of the
zero z0 .
Theorem. Let f be analytic in a neighborhood of z0 . Then f has a
zero of order m at z0 if and only if there is a function g that is analytic
in a neighborhood of z0 , g(z0 ) 6= 0 and f (z) = g(z)(z − z0 )m for all z in
that neighborhood.
∞
Proof. Because f is analytic, f (z) =
∑ an (z − z0 )n for |z − z0 | < r for
n=0
some r > 0. Now, if f has a zero of order m at z0 , then
0 = a0 = · · · = am−1 . So
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Conversely, if
n=m
k=0
f (z) = (z − z0 )m g(z) for |z − z0 | < r for some r > 0, then
∞
f (z) = (z − z0 )m ∑ cn (z − z0 )n =
n=0
Bernd Schröder
Zeros and Poles of Analytic Functions
∞
∑ ck−m (z − z0 )k .
k=m
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = sin(z) has a zero of order 1 at z = 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = sin(z) has a zero of order 1 at z = 0.
∞
sin(z) =
(−1)j
∑ (2j + 1)! z2j+1
j=0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = sin(z) has a zero of order 1 at z = 0.
∞
sin(z) =
(−1)j
∑ (2j + 1)! z2j+1
j=0
∞
(−1)j 2j
z
j=0 (2j + 1)!
= z∑
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = 1 − cos(z) has a zero of order 2 at
z = 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = 1 − cos(z) has a zero of order 2 at
z = 0.
∞
(−1)j 2j
1 − cos(z) = 1 − ∑
z
j=0 (2j)!
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = 1 − cos(z) has a zero of order 2 at
z = 0.
∞
(−1)j 2j
1 − cos(z) = 1 − ∑
z
j=0 (2j)!
(−1)j 2j
z
j=1 (2j)!
∞
= −∑
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = 1 − cos(z) has a zero of order 2 at
z = 0.
∞
(−1)j 2j
1 − cos(z) = 1 − ∑
z
j=0 (2j)!
(−1)j 2j
z
j=1 (2j)!
∞
= −∑
∞
= −∑
k=0
Bernd Schröder
Zeros and Poles of Analytic Functions
(−1)k+1 2k+2
z
2(k + 1) !
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The function f (z) = 1 − cos(z) has a zero of order 2 at
z = 0.
∞
(−1)j 2j
1 − cos(z) = 1 − ∑
z
j=0 (2j)!
(−1)j 2j
z
j=1 (2j)!
∞
= −∑
∞
= −∑
k=0
(−1)k+1 2k+2
z
2(k + 1) !
∞
= z2 − ∑
k=0
Bernd Schröder
Zeros and Poles of Analytic Functions
(−1)k+1 2k
z
2(k + 1) !
!
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
about z0 is not equal to zero.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity) we have
∞
f (z) =
∑ an (z − z0 )n
n=m
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity) we have
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k .
n=m
Bernd Schröder
Zeros and Poles of Analytic Functions
k=0
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity) we have
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Because power
n=m
k=0
∞
series are continuous, there is a δ > 0 so that
∑ ak+m (z − z0 )k 6= 0 for
k=0
|z − z0 | < δ .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity) we have
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Because power
n=m
k=0
∞
series are continuous, there is a δ > 0 so that
∑ ak+m (z − z0 )k 6= 0 for
k=0
|z − z0 | < δ . Moreover, (z − z0 )m 6= 0 for 0 < |z − z0 | < δ .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity) we have
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Because power
n=m
k=0
∞
series are continuous, there is a δ > 0 so that
∑ ak+m (z − z0 )k 6= 0 for
k=0
|z − z0 | < δ . Moreover, (z − z0 )m 6= 0 for 0 < |z − z0 | < δ . Thus
f (z) 6= 0 for 0 < |z − z0 | < δ .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a neighborhood of z0 , f (z0 ) = 0 and let
f not be constant (and thus not equal to zero) in any neighborhood of
z0 . Then there is a deleted neighborhood 0 < |z − z0 | < δ of z0 on
which f is not equal to zero.
Proof. Because f is not equal to zero, the power series expansion of f
∞
about z0 is not equal to zero. That is, f (z) =
∑ an (z − z0 )n for
n=0
|z − z0 | < r for some r > 0, and, because f (z0 ) = 0 we have a0 = 0.
Hence, with m the order of the zero z0 (which is not infinity) we have
∞
f (z) =
∞
∑ an (z − z0 )n = (z − z0 )m ∑ ak+m (z − z0 )k . Because power
n=m
k=0
∞
series are continuous, there is a δ > 0 so that
∑ ak+m (z − z0 )k 6= 0 for
k=0
|z − z0 | < δ . Moreover, (z − z0 )m 6= 0 for 0 < |z − z0 | < δ . Thus
f (z) 6= 0 for 0 < |z − z0 | < δ .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Bernd Schröder
Zeros and Poles of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ].
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
r
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
r
γ(δ )
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
'$
r
γ(δ )
&%
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
'$
r
r γ(δ )
&%
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
z0
u
'$
r
r γ(δ )
&%
γ(δ + δ̃ )
C
u
z1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood |z − z̃| < r̃ of z̃.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood
|z − z̃| < r̃ of z̃. But
then there must be a δ̃ so that f γ(t) = 0 on the interval [0, δ + δ̃ ).
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood
|z − z̃| < r̃ of z̃. But
then there must be a δ̃ so that f γ(t) = 0 on the interval [0, δ + δ̃ ).
And that contradicts the choice of δ .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood
|z − z̃| < r̃ of z̃. But
then there must be a δ̃ so that f γ(t) = 0 on the interval [0, δ + δ̃ ).
And that contradicts the choice of δ . Thus δ = 1
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood
|z − z̃| < r̃ of z̃. But
then there must be a δ̃ so that f γ(t) = 0 on the interval [0, δ + δ̃ ).
And that contradicts the choice of δ . Thus δ = 1 and hence f (z1 ) = 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood
|z − z̃| < r̃ of z̃. But
then there must be a δ̃ so that f γ(t) = 0 on the interval [0, δ + δ̃ ).
And that contradicts the choice of δ . Thus δ = 1 and hence f (z1 ) = 0.
Because z1 was arbitrary, f must be equal to zero in D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a domain D and suppose that f is equal
to zero on some domain or line segment that contains z0 . Then f must
be equal to zero in D.
Proof. First, by the preceding theorem, f must be zero on some
neighborhood |z − z0 | < r0 of z0 . Now let z1 be any point in the
domain D. Then there is a path C from z0 to z1 . Let γ : [0, 1] → C be a
parametrization of the path. Then f γ(t) = 0 on some interval [0, δ ).
By continuity of f ◦ γ, we must have f γ(t) = 0 on some interval
[0, δ ]. Choose δ as large as possible and suppose for a contradiction
that δ < 1. Let z̃ := γ(δ ) . Then by the same argument as we applied
at z0 , f must be equal to zero on a neighborhood
|z − z̃| < r̃ of z̃. But
then there must be a δ̃ so that f γ(t) = 0 on the interval [0, δ + δ̃ ).
And that contradicts the choice of δ . Thus δ = 1 and hence f (z1 ) = 0.
Because z1 was arbitrary, f must be equal to zero in D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0, that is,
f = g.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0, that is,
f = g.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0, that is,
f = g.
This result means that if a function f is analytic in some domain
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0, that is,
f = g.
This result means that if a function f is analytic in some domain and it
can be extended to an analytic function on a larger domain
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0, that is,
f = g.
This result means that if a function f is analytic in some domain and it
can be extended to an analytic function on a larger domain, then this
extension is unique.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let f be analytic in a domain D. Then f is uniquely
determined by its values in a domain or a line segment contained in D.
Proof. Suppose f and g are analytic and they are equal in a domain or
a line segment contained in D. Then f − g is zero in a domain or a line
segment contained in D. By the preceding result, f − g = 0, that is,
f = g.
This result means that if a function f is analytic in some domain and it
can be extended to an analytic function on a larger domain, then this
1
extension is unique. So, for example, the function f (z) =
is the
1−z
∞
only way to extend the power series
∑ zk to the whole complex plane
k=0
except z = 1.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. (Schwarz’ Reflection Principle.)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. (Schwarz’ Reflection Principle.) Let f be analytic in the
domain D that contains a segment S of the x axis and which is
symmetric with respect to the x-axis. Then f (z) = f (z) for all z in the
domain if and only if f is real valued for each point on S.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. (Schwarz’ Reflection Principle.) Let f be analytic in the
domain D that contains a segment S of the x axis and which is
symmetric with respect to the x-axis. Then f (z) = f (z) for all z in the
domain if and only if f is real valued for each point on S.
ℑ(z)
6
D
ℜ(z)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. (Schwarz’ Reflection Principle.) Let f be analytic in the
domain D that contains a segment S of the x axis and which is
symmetric with respect to the x-axis. Then f (z) = f (z) for all z in the
domain if and only if f is real valued for each point on S.
ℑ(z)
6
D
ℜ(z)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. (Schwarz’ Reflection Principle.) Let f be analytic in the
domain D that contains a segment S of the x axis and which is
symmetric with respect to the x-axis. Then f (z) = f (z) for all z in the
domain if and only if f is real valued for each point on S.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z).
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
g=f
S
Bernd Schröder
Zeros and Poles of Analytic Functions
ℜ(z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
ℑ(z)
6
D
g=f
S
ℜ(z)
g(z) = g (z)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero:
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
ℑ(z)
6
D
ℜ(z)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
ℑ(z)
6
D
ℜ(z)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
ℑ(z)
6
D
ℜ(z)
C
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
ℑ(z)
6
D
ℜ(z)
C
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
ℑ(z)
6
D
ℜ(z)
C
Bernd Schröder
Zeros and Poles of Analytic Functions
g(z) = g (z)
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane. The integral in the upper
half plane is zero by analyticity of f .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane. The integral in the upper
half plane is zero by analyticity of f . The integral in the lower half
plane is zero because it is the complex conjugate of the integral of f
over the reflected curve.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane. The integral in the upper
half plane is zero by analyticity of f . The integral in the lower half
plane is zero because it is the complex conjugate of the integral of f
over the reflected curve. Thus, by Morera’s Theorem, g is analytic.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane. The integral in the upper
half plane is zero by analyticity of f . The integral in the lower half
plane is zero because it is the complex conjugate of the integral of f
over the reflected curve. Thus, by Morera’s Theorem, g is analytic.
Because g is equal to f in the upper half of the domain, it must be
equal to f on D.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane. The integral in the upper
half plane is zero by analyticity of f . The integral in the lower half
plane is zero because it is the complex conjugate of the integral of f
over the reflected curve. Thus, by Morera’s Theorem, g is analytic.
Because g is equal to f in the upper half of the domain, it must be
equal to f on D. But that means f (z) = f (z) for all z in the domain.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Clearly, if f (z) = f (z) for all z, then for all z in S we have
f (z) = f (z) = f (z). That is, f is real valued in S.
Conversely, suppose that f is real valued in S and consider the
restriction g of f to the upper half plane. Then defining g(z) = g (z)
for all z in the lower half plane that are symmetric to elements in the
(original) domain of g defines a continuous function on D. Moreover,
Integrals of g over closed curves are zero: This is because the integral
over every closed curve can be split into an integral in the upper half
plane and an integral in the lower half plane. The integral in the upper
half plane is zero by analyticity of f . The integral in the lower half
plane is zero because it is the complex conjugate of the integral of f
over the reflected curve. Thus, by Morera’s Theorem, g is analytic.
Because g is equal to f in the upper half of the domain, it must be
equal to f on D. But that means f (z) = f (z) for all z in the domain.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n
n=0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n and
n=0
∞
q(z) = (z − z0 )m
∑ bk (z − z0 )k−m
k=m
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n and
n=0
∞
q(z) = (z − z0 )m
∑ bk (z − z0 )k−m we have a0 6= 0 and bm 6= 0. Thus
k=m
n
∑∞
n=0 an (z − z0 )
is analytic in a neighborhood of z0 .
k−m
∑∞
k=m bk (z − z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n and
n=0
∞
q(z) = (z − z0 )m
∑ bk (z − z0 )k−m we have a0 6= 0 and bm 6= 0. Thus
k=m
n
∑∞
n=0 an (z − z0 )
is analytic in a neighborhood of z0 . Hence
k−m
∑∞
k=m bk (z − z0 )
p(z)
q(z)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n and
n=0
∞
q(z) = (z − z0 )m
∑ bk (z − z0 )k−m we have a0 6= 0 and bm 6= 0. Thus
k=m
n
∑∞
n=0 an (z − z0 )
is analytic in a neighborhood of z0 . Hence
k−m
∑∞
k=m bk (z − z0 )
n
1
p(z)
∑∞
n=0 an (z − z0 )
=
k−m
q(z) (z − z0 )m ∑∞
k=m bk (z − z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n and
n=0
∞
q(z) = (z − z0 )m
∑ bk (z − z0 )k−m we have a0 6= 0 and bm 6= 0. Thus
k=m
n
∑∞
n=0 an (z − z0 )
is analytic in a neighborhood of z0 . Hence
k−m
∑∞
k=m bk (z − z0 )
n
1
p(z)
∑∞
n=0 an (z − z0 )
=
has a pole of order m at z0 .
k−m
q(z) (z − z0 )m ∑∞
k=m bk (z − z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let p and q be analytic in a neighborhood of z0 and so that
p(z0 ) 6= 0 while q has a zero of order m at z0 . Then the quotient pq has
a pole of order m at z0 .
∞
Proof. For the expansions p(z) =
∑ an (z − z0 )n and
n=0
∞
q(z) = (z − z0 )m
∑ bk (z − z0 )k−m we have a0 6= 0 and bm 6= 0. Thus
k=m
n
∑∞
n=0 an (z − z0 )
is analytic in a neighborhood of z0 . Hence
k−m
∑∞
k=m bk (z − z0 )
n
1
p(z)
∑∞
n=0 an (z − z0 )
=
has a pole of order m at z0 .
k−m
q(z) (z − z0 )m ∑∞
k=m bk (z − z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Example. The function f (z) =
Reflection Principle
Poles
Behavior Near Singularities
1
has a pole of order 2 at the
1 − cos(z)
origin.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Example. The function f (z) =
Reflection Principle
Poles
Behavior Near Singularities
1
has a pole of order 2 at the
1 − cos(z)
origin.
That’s because 1 − cos(z) has a zero of order 2 at 0
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Example. The function f (z) =
Reflection Principle
Poles
Behavior Near Singularities
1
has a pole of order 2 at the
1 − cos(z)
origin.
That’s because 1 − cos(z) has a zero of order 2 at 0 (shown earlier).
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
z0
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof. The fact that the pole is simple follows from the preceding
theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof. The fact that the pole is simple follows from the preceding
theorem. For the residue, we recall that
p(z)
q(z)
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof. The fact that the pole is simple follows from the preceding
theorem. For the residue, we recall that
p(z)
∑∞ an (z − z0 )n
= (z − z0 )−1 ∞n=0
q(z)
∑k=1 bk (z − z0 )k−1
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof. The fact that the pole is simple follows from the preceding
theorem. For the residue, we recall that
p(z)
∑∞ an (z − z0 )n
= (z − z0 )−1 ∞n=0
, which means that the residue at
q(z)
∑k=1 bk (z − z0 )k−1
a0
z = z0 is
b1
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof. The fact that the pole is simple follows from the preceding
theorem. For the residue, we recall that
p(z)
∑∞ an (z − z0 )n
= (z − z0 )−1 ∞n=0
, which means that the residue at
q(z)
∑k=1 bk (z − z0 )k−1
p(z0 )
a0
=
.
z = z0 is
b1 q0 (z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Corollary. Let p and q be analytic in a neighborhood of z0 and so
that p(z0 ) 6= 0, q(z0 ) = 0 and q0 (z0 ) 6= 0. Then pq has a simple pole at
p(z)
p(z0 )
= 0
.
z0 and the residue is Resz=z0
q(z) q (z0 )
Proof. The fact that the pole is simple follows from the preceding
theorem. For the residue, we recall that
p(z)
∑∞ an (z − z0 )n
= (z − z0 )−1 ∞n=0
, which means that the residue at
q(z)
∑k=1 bk (z − z0 )k−1
p(z0 )
a0
=
.
z = z0 is
b1 q0 (z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The cotangent function has a simple pole at the origin.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The cotangent function has a simple pole at the origin.
This follows from cot(z) =
Bernd Schröder
Zeros and Poles of Analytic Functions
cos(z)
and the preceding theorem.
sin(z)
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example. The cotangent function has a simple pole at the origin.
cos(z)
and the preceding theorem. The
sin(z)
cos(0)
residue of the cotangent at z = 0 is Resz=0 cot(z) =
= 1.
cos(0)
This follows from cot(z) =
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Example.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
Bernd Schröder
Zeros and Poles of Analytic Functions
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
Because 1 + i =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ iπ
2e 4 , there really is a singularity at 1 + i.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ π
Because 1 + i = 2ei 4 , there really is a singularity at 1 + i. Because
z4 + 4 has four distinct zeros, we can apply the theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ π
Because 1 + i = 2ei 4 , there really is a singularity at 1 + i. Because
z4 + 4 has four distinct zeros, we can apply the theorem.
z
Resz=1+i 4
z +4
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ π
Because 1 + i = 2ei 4 , there really is a singularity at 1 + i. Because
z4 + 4 has four distinct zeros, we can apply the theorem.
z
z Resz=1+i 4
= 3 √ π
z +4
4z z= 2ei 4
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ π
Because 1 + i = 2ei 4 , there really is a singularity at 1 + i. Because
z4 + 4 has four distinct zeros, we can apply the theorem.
√ iπ
z
z 2e 4
Resz=1+i 4
= 3 √ π = √
π 3
z +4
4z z= 2ei 4
4
2ei 4
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ π
Because 1 + i = 2ei 4 , there really is a singularity at 1 + i. Because
z4 + 4 has four distinct zeros, we can apply the theorem.
√ iπ
z
z 2e 4
1
Resz=1+i 4
= 3 √ π = √
= iπ
3
π
z +4
4z z= 2ei 4
8e 2
4
2ei 4
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Example. Find the residue of f (z) =
z
z4 + 4
Poles
Behavior Near Singularities
at z0 = 1 + i.
√ π
Because 1 + i = 2ei 4 , there really is a singularity at 1 + i. Because
z4 + 4 has four distinct zeros, we can apply the theorem.
√ iπ
z
z 2e 4
1
i
Resz=1+i 4
= 3 √ π = √
= iπ = −
3
π
z +4
4z z= 2ei 4
8
8e 2
4
2ei 4
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Proof. Let m be the order of the pole at z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Proof. Let m be the order of the pole at z0 . For |z − z0 | < r for some
∞
r > 0 we have f (z) = (z − z0 )−m ∑ an (z − z0 )n .
n=0
Bernd Schröder
Zeros and Poles of Analytic Functions
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Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Proof. Let m be the order of the pole at z0 . For |z − z0 | < r for some
∞
r > 0 we have f (z) = (z − z0 )−m ∑ an (z − z0 )n . The power series
n=0
converges to a0 6= 0 as z → z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Proof. Let m be the order of the pole at z0 . For |z − z0 | < r for some
∞
r > 0 we have f (z) = (z − z0 )−m ∑ an (z − z0 )n . The power series
n=0
converges to a0 6= 0 as z → z0 and |(z − z0 )−m | → ∞ as z → z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted
neighborhood of z0 . If z0 is a
pole of the function f , then lim f (z) = ∞.
z→z0
Proof. Let m be the order of the pole at z0 . For |z − z0 | < r for some
∞
r > 0 we have f (z) = (z − z0 )−m ∑ an (z − z0 )n . The power series
n=0
converges to a0 6= 0 as z → z0 and |(z − z0 )−m | → ∞ as z → z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . Then z0
is a removable singularity of the function f if and only if f is bounded
in a deleted neighborhood of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . Then z0
is a removable singularity of the function f if and only if f is bounded
in a deleted neighborhood of z0 .
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . Then z0
is a removable singularity of the function f if and only if f is bounded
in a deleted neighborhood of z0 .
Proof. Clearly, if z0 is a removable singularity of f , then for
∞
0 < |z − z0 | < r for some r > 0 we have f (z) =
∑ an (z − z0 )n
n=0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . Then z0
is a removable singularity of the function f if and only if f is bounded
in a deleted neighborhood of z0 .
Proof. Clearly, if z0 is a removable singularity of f , then for
∞
0 < |z − z0 | < r for some r > 0 we have f (z) =
means that f converges to a0 as z → z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
∑ an (z − z0 )n , which
n=0
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . Then z0
is a removable singularity of the function f if and only if f is bounded
in a deleted neighborhood of z0 .
Proof. Clearly, if z0 is a removable singularity of f , then for
∞
0 < |z − z0 | < r for some r > 0 we have f (z) =
∑ an (z − z0 )n , which
n=0
means that f converges to a0 as z → z0 . In particular, f is bounded in a
deleted neighborhood of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
f (z) =
∞
∞
n=0
n=1
∑ an (z − z0 )n + ∑ bn (z − z0 )−n .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
f
(z)
|bn | = dz
−n+1
2πi C (z − z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz|
≤
2π C
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz| ≤ ρMρ n−1
≤
2π C
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz| ≤ ρMρ n−1
≤
2π C
= Mρ n
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz| ≤ ρMρ n−1
≤
2π C
= Mρ n → 0
(ρ → 0)
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz| ≤ ρMρ n−1
≤
2π C
= Mρ n → 0
(ρ → 0)
which means that all bn are zero
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz| ≤ ρMρ n−1
≤
2π C
= Mρ n → 0
(ρ → 0)
which means that all bn are zero and hence f has a removable
singularity at z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Proof. Conversely, suppose that f is bounded on a deleted
neighborhood of z0 and consider the Laurent expansion
∞
f (z) =
∞
n
a
(z
−
z
)
+
0
∑ n
∑ bn (z − z0 )−n . Let M be so that f (z) ≤ M
n=0
n=1
for 0 < |z − z0 | < r for some r > 0. Then for the bn we obtain, with C
being a circle of radius ρ between 0 and r around z0 :
1 Z
1 Z
f
(z)
n−1
|bn | = f (z)(z − z0 )
dz
dz = −n+1
2πi C
2πi C (z − z0 )
Z 1
f (z) (z − z0 )n−1 |dz| ≤ ρMρ n−1
≤
2π C
= Mρ n → 0
(ρ → 0)
which means that all bn are zero and hence f has a removable
singularity at z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
aδ >0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
f (z) − w0 > ε.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0, then
1
f (z) − w0 = g(z)
is analytic in a deleted neighborhood of z0 and it is
bounded there.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0, then
1
f (z) − w0 = g(z)
is analytic in a deleted neighborhood of z0 and it is
bounded there. This implies that f (z) − w0 has a removable singularity
at z0 .
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0, then
1
f (z) − w0 = g(z)
is analytic in a deleted neighborhood of z0 and it is
bounded there. This implies that f (z) − w0 has a removable singularity
at z0 . Hence f has a removable singularity at z0
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0, then
1
f (z) − w0 = g(z)
is analytic in a deleted neighborhood of z0 and it is
bounded there. This implies that f (z) − w0 has a removable singularity
at z0 . Hence f has a removable singularity at z0 , a contradiction.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0, then
1
f (z) − w0 = g(z)
is analytic in a deleted neighborhood of z0 and it is
bounded there. This implies that f (z) − w0 has a removable singularity
at z0 . Hence f has a removable singularity at z0 , a contradiction.
If g(z0 ) = 0, then, with m being the order of the zero of g at z0 ,
1
f (z) = w0 + g(z)
has a pole of order m at z = z0 , a contradiction.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science
Zeros
Uniqueness of Analytic Functions
Reflection Principle
Poles
Behavior Near Singularities
Theorem. Let f be analytic in a deleted neighborhood of z0 . If z0 is
an essential singularity of the function f then for every complex
number w0 , for every ε >
every δ > 0 there is a z with
0 and for
0 < |z − z0 | < δ so that f (z) − w0 < ε.
Proof. Suppose for a contradiction that there is a w0 and an ε > 0 and
a δ > 0 so that for every z with 0 < |z − z0 | < δ we have
1
f (z) − w0 > ε. Then g(z) =
is analytic in a neighborhood
f (z) − w0
of z0 and it is bounded there. Thus g has a removable singularity at z0
If the value that makes g analytic at z0 satisfies g(z0 ) 6= 0, then
1
f (z) − w0 = g(z)
is analytic in a deleted neighborhood of z0 and it is
bounded there. This implies that f (z) − w0 has a removable singularity
at z0 . Hence f has a removable singularity at z0 , a contradiction.
If g(z0 ) = 0, then, with m being the order of the zero of g at z0 ,
1
f (z) = w0 + g(z)
has a pole of order m at z = z0 , a contradiction.
Bernd Schröder
Zeros and Poles of Analytic Functions
logo1
Louisiana Tech University, College of Engineering and Science