Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
The Cauchy Integral Formula
Bernd Schröder
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
1. One of the most important consequences of the Cauchy-Goursat
Integral Theorem is that the value of an analytic function at a
point can be obtained from the values of the analytic function on
a contour surrounding the point
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
1. One of the most important consequences of the Cauchy-Goursat
Integral Theorem is that the value of an analytic function at a
point can be obtained from the values of the analytic function on
a contour surrounding the point (as long as the function is
defined on a neighborhood of the contour and its inside).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
1. One of the most important consequences of the Cauchy-Goursat
Integral Theorem is that the value of an analytic function at a
point can be obtained from the values of the analytic function on
a contour surrounding the point (as long as the function is
defined on a neighborhood of the contour and its inside).
2. The result itself is known as Cauchy’s Integral Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
1. One of the most important consequences of the Cauchy-Goursat
Integral Theorem is that the value of an analytic function at a
point can be obtained from the values of the analytic function on
a contour surrounding the point (as long as the function is
defined on a neighborhood of the contour and its inside).
2. The result itself is known as Cauchy’s Integral Theorem.
3. Among its consequences is, for example, the Fundamental
Theorem of Algebra, which says that every nonconstant complex
polynomial has at least one complex zero.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
4. Once we have introduced series, another consequence is the fact
that every analytic function is locally equal to a power series.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
4. Once we have introduced series, another consequence is the fact
that every analytic function is locally equal to a power series.
This very powerful result is a cornerstone of complex analysis.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
4. Once we have introduced series, another consequence is the fact
that every analytic function is locally equal to a power series.
This very powerful result is a cornerstone of complex analysis.
5. In this presentation we will at least be able to prove that analytic
functions have derivatives of any order.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Introduction
4. Once we have introduced series, another consequence is the fact
that every analytic function is locally equal to a power series.
This very powerful result is a cornerstone of complex analysis.
5. In this presentation we will at least be able to prove that analytic
functions have derivatives of any order.
6. ... and the above are only highlights of the consequences of
Cauchy’s Integral Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula. Let C be a simple closed
positively oriented piecewise smooth curve, and let the function f be
analytic in a neighborhood of C and its interior.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula. Let C be a simple closed
positively oriented piecewise smooth curve, and let the function f be
analytic in a neighborhood of C and its interior. Then for every z0 in
the interior of C we have that
f (z0 ) =
Bernd Schröder
The Cauchy Integral Formula
1
2πi
f (z)
dz.
C z − z0
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
1
2πi
1
dz
C(r,z0 ) z − z0
Z
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
1
2πi
1
dz =
C(r,z0 ) z − z0
Z
Bernd Schröder
The Cauchy Integral Formula
1
2πi
Z 2π
0
1
ireit dt
(z0 + reit ) − z0
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
1
2πi
1
dz =
C(r,z0 ) z − z0
Z
=
Bernd Schröder
The Cauchy Integral Formula
2π
1
1
ireit dt
2πi 0 (z0 + reit ) − z0
Z 2π
1
1
ireit dt
2πi 0 reit
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
1
2πi
1
dz =
C(r,z0 ) z − z0
Z
=
=
Bernd Schröder
The Cauchy Integral Formula
2π
1
1
ireit dt
2πi 0 (z0 + reit ) − z0
Z 2π
1
1
ireit dt
2πi 0 reit
Z
1 2π
dt
2π 0
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
1
2πi
1
dz =
C(r,z0 ) z − z0
Z
=
=
Bernd Schröder
The Cauchy Integral Formula
2π
1
1
ireit dt
2πi 0 (z0 + reit ) − z0
Z 2π
1
1
ireit dt
2πi 0 reit
Z
1 2π
dt = 1
2π 0
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Example. Let’s first check out if the theorem works for f (z) = 1 and
the circle C(r, z0 ) of radius r around z0 .
1
2πi
1
dz =
C(r,z0 ) z − z0
Z
=
=
Bernd Schröder
The Cauchy Integral Formula
2π
1
1
ireit dt
2πi 0 (z0 + reit ) − z0
Z 2π
1
1
ireit dt
2πi 0 reit
Z
1 2π
dt = 1 = f (z0 )
2π 0
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
-
C(r, z0 )
r
]
?
O
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
-
C(r, z0 )
r
]
?
O
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
r
-
C(r, z0 )
r
]
?
O
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
r
r-
C(r, z0 )
r
]
?
O
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
r
rr C(r, z )
r 0?
]
O
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
r
I r rC(r, z0 )
r
?
O
]
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
r
R
I r rC(r, z0 )
r
?
O
]
C
-
Bernd Schröder
The Cauchy Integral Formula
1
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
D
r
r
R
I r rC(r, z0 )
r
?
O
]
C
-
1
f (z)
dz =
C z − z0
Z
Bernd Schröder
The Cauchy Integral Formula
f (z)
dz.
C(r,z0 ) z − z0
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) ≤
z − z0 d|z|
C(r,z )
0
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) Z
1
d|z| ≤ max f (z) − f (z0 )
≤
d|z|
z − z0
C(r,z0 )
C(r,z0 )
C(r,z0 ) r
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) Z
1
d|z| ≤ max f (z) − f (z0 )
≤
d|z|
z − z0
C(r,z0 )
C(r,z0 )
C(r,z0 ) r
1
= max f (z) − f (z0 )2πr
r
C(r,z0 )
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) Z
1
d|z| ≤ max f (z) − f (z0 )
≤
d|z|
z − z0
C(r,z0 )
C(r,z0 )
C(r,z0 ) r
1
= max f (z) − f (z0 )2πr = 2π max f (z) − f (z0 )
r
C(r,z0 )
C(r,z0 )
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) Z
1
d|z| ≤ max f (z) − f (z0 )
≤
d|z|
z − z0
C(r,z0 )
C(r,z0 )
C(r,z0 ) r
1
= max f (z) − f (z0 )2πr = 2π max f (z) − f (z0 )
r
C(r,z0 )
C(r,z0 )
and if we choose r small enough, we can make the last term arbitrarily
small.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) Z
1
d|z| ≤ max f (z) − f (z0 )
≤
d|z|
z − z0
C(r,z0 )
C(r,z0 )
C(r,z0 ) r
1
= max f (z) − f (z0 )2πr = 2π max f (z) − f (z0 )
r
C(r,z0 )
C(r,z0 )
and if we choose r small enough, we can make the last term arbitrarily
small. But that means that the original difference must be zero.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof of Cauchy’s Integral Formula.
Z
f
(z)
C z − z0 dz − 2πif (z0 )
Z
Z
f (z)
f (z0 ) = dz −
dz
C(r,z0 ) z − z0
C(r,z0 ) z − z0
Z
f (z) − f (z0 ) dz
= z − z0
C(r,z0 )
Z
f (z) − f (z0 ) Z
1
d|z| ≤ max f (z) − f (z0 )
≤
d|z|
z − z0
C(r,z0 )
C(r,z0 )
C(r,z0 ) r
1
= max f (z) − f (z0 )2πr = 2π max f (z) − f (z0 )
r
C(r,z0 )
C(r,z0 )
and if we choose r small enough, we can make the last term arbitrarily
small. But that means that the original difference must be zero.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
1
Analyzing f (z0 ) =
2πi
Bernd Schröder
The Cauchy Integral Formula
Fundamental Theorem of Algebra
Maximum Modulus Principle
f (z)
dz.
γ z − z0
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
1
Analyzing f (z0 ) =
2πi
Fundamental Theorem of Algebra
Maximum Modulus Principle
f (z)
dz.
γ z − z0
Z
1. Note that the right side is a function of z0 .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
1
Analyzing f (z0 ) =
2πi
Fundamental Theorem of Algebra
Maximum Modulus Principle
f (z)
dz.
γ z − z0
Z
1. Note that the right side is a function of z0 .
1
2. For z fixed,
is differentiable in z0 .
z − z0
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
1
Analyzing f (z0 ) =
2πi
Fundamental Theorem of Algebra
Maximum Modulus Principle
f (z)
dz.
γ z − z0
Z
1. Note that the right side is a function of z0 .
1
2. For z fixed,
is differentiable in z0 .
z − z0
3. So if we could move the derivative into the integral, we could get
a formula for f 0 .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
1
Analyzing f (z0 ) =
2πi
Fundamental Theorem of Algebra
Maximum Modulus Principle
f (z)
dz.
γ z − z0
Z
1. Note that the right side is a function of z0 .
1
2. For z fixed,
is differentiable in z0 .
z − z0
3. So if we could move the derivative into the integral, we could get
a formula for f 0 .
1
4. And, anticipating that the new integrand will involve
,
(z − z0 )2
there is no reason to think that the process should stop there.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
1
Analyzing f (z0 ) =
2πi
Fundamental Theorem of Algebra
Maximum Modulus Principle
f (z)
dz.
γ z − z0
Z
1. Note that the right side is a function of z0 .
1
2. For z fixed,
is differentiable in z0 .
z − z0
3. So if we could move the derivative into the integral, we could get
a formula for f 0 .
1
4. And, anticipating that the new integrand will involve
,
(z − z0 )2
there is no reason to think that the process should stop there.
5. So for analytic functions, being once differentiable should imply
that we have derivatives of any order.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Warning.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Warning. The next result (as motivated on the preceding panel) does
not work for functions of a real variable.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Warning. The next result (as motivated on the preceding panel) does
not work
of a real variable. Consider
for functions
x2 ; for x > 0,
f (x) =
−x2 ; for x ≤ 0.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Warning. The next result (as motivated on the preceding panel) does
not work
of a real variable. Consider
for functions
x2 ; for x > 0,
Its derivative is 2|x|
f (x) =
−x2 ; for x ≤ 0.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Warning. The next result (as motivated on the preceding panel) does
not work
of a real variable. Consider
for functions
x2 ; for x > 0,
Its derivative is 2|x|, which is not
f (x) =
−x2 ; for x ≤ 0.
differentiable at 0.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula (extended).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula (extended). Let C be a simple
closed positively oriented piecewise smooth curve, and let the
function f be analytic in a neighborhood of C and its interior.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Integral Formula (extended). Let C be a simple
closed positively oriented piecewise smooth curve, and let the
function f be analytic in a neighborhood of C and its interior. Then
for every z0 in the interior of C and every natural number n we have
that f is n-times differentiable at z0 and its derivative is
f
Bernd Schröder
The Cauchy Integral Formula
(n)
n!
(z0 ) =
2πi
f (z)
dz.
n+1
C (z − z0 )
Z
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Induction step, n → (n + 1).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Induction step, n → (n + 1).
f (n) (w) − f (n) (z0 )
w→z0
w − z0
lim
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Induction step, n → (n + 1).
f (n) (w) − f (n) (z0 )
w→z0
w − z0
Z
Z
f (z)
f (z)
1
n!
n!
= lim
dz −
dz
w→z0 w − z0
2πi C (z − w)n+1
2πi C (z − z0 )n+1
lim
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Induction step, n → (n + 1).
f (n) (w) − f (n) (z0 )
w→z0
w − z0
Z
Z
f (z)
f (z)
1
n!
n!
= lim
dz −
dz
w→z0 w − z0
2πi C (z − w)n+1
2πi C (z − z0 )n+1
Z
n!
1
f (z)
f (z)
= lim
−
dz
n+1
w→z0 2πi w − z0 C (z − w)
(z − z0 )n+1
lim
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Induction step, n → (n + 1).
f (n) (w) − f (n) (z0 )
w→z0
w − z0
Z
Z
f (z)
f (z)
1
n!
n!
= lim
dz −
dz
w→z0 w − z0
2πi C (z − w)n+1
2πi C (z − z0 )n+1
Z
n!
1
f (z)
f (z)
= lim
−
dz
n+1
w→z0 2πi w − z0 C (z − w)
(z − z0 )n+1
lim
n!
= lim
w→z0 2πi
Bernd Schröder
The Cauchy Integral Formula
Z
f (z)
C(z0 ,r)
1
(z−w)n+1
− (z−z1 )n+1
0
w − z0
dz
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. Induction on n.
Base step, n = 0. This is Cauchy’s Integral Formula.
Induction step, n → (n + 1).
f (n) (w) − f (n) (z0 )
w→z0
w − z0
Z
Z
f (z)
f (z)
1
n!
n!
= lim
dz −
dz
w→z0 w − z0
2πi C (z − w)n+1
2πi C (z − z0 )n+1
Z
n!
1
f (z)
f (z)
= lim
−
dz
n+1
w→z0 2πi w − z0 C (z − w)
(z − z0 )n+1
lim
1
Z
− (z−z1 )n+1
n!
(z−w)n+1
0
= lim
f (z)
dz
w→z0 2πi C(z0 ,r)
w − z0
Z
n!
1
=
f (z)(−(n + 1))
(−1) dz
2πi C(z0 ,r)
(z − z0 )n+2
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
n!
2πi
Bernd Schröder
The Cauchy Integral Formula
Fundamental Theorem of Algebra
Z
f (z)(−(n + 1))
C(z0 ,r)
Maximum Modulus Principle
1
(−1) dz
(z − z0 )n+2
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
1
(−1) dz
n+2
(z
−
z
C(z0 ,r)
0)
Z
f (z)
(n + 1)!
=
dz
n+2
2πi
C(z0 ,r) (z − z0 )
n!
2πi
Bernd Schröder
The Cauchy Integral Formula
Z
f (z)(−(n + 1))
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
1
(−1) dz
n+2
(z
−
z
C(z0 ,r)
0)
Z
f (z)
(n + 1)!
=
dz
n+2
2πi
C(z0 ,r) (z − z0 )
Z
(n + 1)!
f (z)
=
dz
2πi
(z
−
z0 )n+2
C
n!
2πi
Bernd Schröder
The Cauchy Integral Formula
Z
f (z)(−(n + 1))
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
1
(−1) dz
n+2
(z
−
z
C(z0 ,r)
0)
Z
f (z)
(n + 1)!
=
dz
n+2
2πi
C(z0 ,r) (z − z0 )
Z
(n + 1)!
f (z)
=
dz
2πi
(z
−
z0 )n+2
C
n!
2πi
Bernd Schröder
The Cauchy Integral Formula
Z
f (z)(−(n + 1))
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders. That means all derivatives are differentiable
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders. That means all derivatives are differentiable and thus all
derivatives are analytic on the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders. That means all derivatives are differentiable and thus all
derivatives are analytic on the domain. Therefore real and imaginary
part have
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders. That means all derivatives are differentiable and thus all
derivatives are analytic on the domain. Therefore real and imaginary
part have (by repeated application of the Cauchy-Riemann equations)
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders. That means all derivatives are differentiable and thus all
derivatives are analytic on the domain. Therefore real and imaginary
part have (by repeated application of the Cauchy-Riemann equations)
continuous partial derivatives of all orders throughout the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be an analytic function on an open domain. Then all
derivatives of f are analytic on this domain, too. Moreover, the
component functions (the real and imaginary parts) have continuous
partial derivatives of all orders throughout the domain.
Proof. By the preceding result, we see that f has derivatives of all
orders. That means all derivatives are differentiable and thus all
derivatives are analytic on the domain. Therefore real and imaginary
part have (by repeated application of the Cauchy-Riemann equations)
continuous partial derivatives of all orders throughout the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0.
contained in the open set we have
C
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0. Then f is analytic in
contained in the open set we have
the open set.
Bernd Schröder
The Cauchy Integral Formula
C
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0. Then f is analytic in
contained in the open set we have
the open set.
C
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0. Then f is analytic in
contained in the open set we have
the open set.
C
Proof. By theorem from an earlier presentation, f has an
antiderivative F.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0. Then f is analytic in
contained in the open set we have
the open set.
C
Proof. By theorem from an earlier presentation, f has an
antiderivative F. By the preceding theorem, all derivatives of F
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0. Then f is analytic in
contained in the open set we have
the open set.
C
Proof. By theorem from an earlier presentation, f has an
antiderivative F. By the preceding theorem, all derivatives of F
(including f ) are analytic.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Morera’s Theorem. Let f be a continuous complex
function on an open set so that for Zevery simple closed curve C
f (z) dz = 0. Then f is analytic in
contained in the open set we have
the open set.
C
Proof. By theorem from an earlier presentation, f has an
antiderivative F. By the preceding theorem, all derivatives of F
(including f ) are analytic.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
n! Z
f
(z)
(n)
dz
f (z0 ) = 2πi CR (z0 ) (z − z0 )n+1 Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
n! Z
f
(z)
(n)
dz
f (z0 ) = 2πi CR (z0 ) (z − z0 )n+1 Z
f (z) n!
d|z|
≤
2π CR (z0 ) (z − z0 )n+1 Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
n! Z
f
(z)
(n)
dz
f (z0 ) = 2πi CR (z0 ) (z − z0 )n+1 Z
f (z) n!
d|z|
≤
2π CR (z0 ) (z − z0 )n+1 Z
n!
MR
≤
d|z|
2π CR (z0 ) Rn+1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
n! Z
f
(z)
(n)
dz
f (z0 ) = 2πi CR (z0 ) (z − z0 )n+1 Z
f (z) n!
d|z|
≤
2π CR (z0 ) (z − z0 )n+1 Z
n!
MR
≤
d|z|
2π CR (z0 ) Rn+1
n!
MR
≤
2πR n+1
2π
R
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
n! Z
f
(z)
(n)
dz
f (z0 ) = 2πi CR (z0 ) (z − z0 )n+1 Z
f (z) n!
d|z|
≤
2π CR (z0 ) (z − z0 )n+1 Z
n!
MR
≤
d|z|
2π CR (z0 ) Rn+1
n!MR
n!
MR
≤
2πR n+1 =
Rn
2π
R
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Cauchy’s Inequality. Let f be analytic on a circle CR (z0 )
of radius R centered at z0 and in the circle’s interior. If |f | is bounded
n!M
R
by MR on the circle, then for all n we have f (n) (z0 ) ≤
.
Rn
Proof.
n! Z
f
(z)
(n)
dz
f (z0 ) = 2πi CR (z0 ) (z − z0 )n+1 Z
f (z) n!
d|z|
≤
2π CR (z0 ) (z − z0 )n+1 Z
n!
MR
≤
d|z|
2π CR (z0 ) Rn+1
n!MR
n!
MR
≤
2πR n+1 =
Rn
2π
R
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.)
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 )
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 ) ≤ B
R
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 ) ≤ B → 0 (R → ∞).
R
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 ) ≤ B → 0 (R → ∞).
R
Because R was arbitrary, we infer that f 0 (z0 ) = 0.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 ) ≤ B → 0 (R → ∞).
R
Because R was arbitrary, we infer that f 0 (z0 ) = 0. Because z0 was
arbitrary, we have f 0 = 0.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 ) ≤ B → 0 (R → ∞).
R
Because R was arbitrary, we infer that f 0 (z0 ) = 0. Because z0 was
arbitrary, we have f 0 = 0. But f 0 = 0 implies that f is constant.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Liouville’s Theorem. Let f be an entire function. (That is,
f is analytic in C.) If f is bounded, then f is constant.
Proof. Let |f | be bounded by B. Let R > 0 be arbitrary. By Cauchy’s
Inequality with n = 1, for every z0 in C we have
0
f (z0 ) ≤ B → 0 (R → ∞).
R
Because R was arbitrary, we infer that f 0 (z0 ) = 0. Because z0 was
arbitrary, we have f 0 = 0. But f 0 = 0 implies that f is constant.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then
analytic in the complex plane.
Bernd Schröder
The Cauchy Integral Formula
1
p
is
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
=0
z→∞ p(z)
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
But the only way 1p can be unbounded inside C(R, 0) is for the
denominator p(z) to go to zero somewhere
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
But the only way 1p can be unbounded inside C(R, 0) is for the
denominator p(z) to go to zero somewhere, which was excluded by
hypothesis.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
But the only way 1p can be unbounded inside C(R, 0) is for the
denominator p(z) to go to zero somewhere, which was excluded by
hypothesis. Thus 1p is bounded on C.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
But the only way 1p can be unbounded inside C(R, 0) is for the
denominator p(z) to go to zero somewhere, which was excluded by
hypothesis. Thus 1p is bounded on C.
Now by Liouville’s Theorem we infer that 1p is constant.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
But the only way 1p can be unbounded inside C(R, 0) is for the
denominator p(z) to go to zero somewhere, which was excluded by
hypothesis. Thus 1p is bounded on C.
Now by Liouville’s Theorem we infer that 1p is constant. Hence p is
constant.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Fundamental Theorem of Algebra. Every nonconstant
complex polynomial has at least one complex zero.
Proof. Let p be a complex polynomial without zeros. Then 1p is
analytic in the complex plane. We claim that 1p is bounded on C. To
1
see this claim, first note that lim
= 0, because lim p(z) = ∞.
z→∞ p(z)
z→∞
1
Thus p is bounded outside a circle C(R, 0) for sufficiently large R.
But the only way 1p can be unbounded inside C(R, 0) is for the
denominator p(z) to go to zero somewhere, which was excluded by
hypothesis. Thus 1p is bounded on C.
Now by Liouville’s Theorem we infer that 1p is constant. Hence p is
constant.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Lemma.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Lemma. If f is analytic in some neighborhood |z − z0 | < ε and
|f (z)| ≤ |f (z0 )| for all z in that neighborhood
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Lemma. If f is analytic in some neighborhood |z − z0 | < ε and
|f (z)| ≤ |f (z0 )| for all z in that neighborhood, then in fact f (z) = f (z0 )
for all z in that neighborhood.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Z
f (z0 )
1
d|z|
≤
2π C(z0 ,r) r
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Z
f (z0 )
1
d|z| = f (z0 )
≤
2π C(z0 ,r) r
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Z
f (z0 )
1
d|z| = f (z0 )
≤
2π C(z0 ,r) r
and the latter integral will be strictly smaller
than
f (z0 ) if there is
even a single point z on C(z0 , r) where f (z) < f (z0 ).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Z
f (z0 )
1
d|z| = f (z0 )
≤
2π C(z0 ,r) r
and the latter integral will be strictly smaller
than
f (z0 ) if there is
even a single point z on C(z0 , r) where f (z) < f (z0 ). Thus for all
r < ε and all |z − z0 | = r we must have f (z) = f (z0 ).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Z
f (z0 )
1
d|z| = f (z0 )
≤
2π C(z0 ,r) r
and the latter integral will be strictly smaller
than
f (z0 ) if there is
even a single point z on C(z0 , r) where f (z) < f (z0 ). Thus for all
r < ε and all |z − z0 | = r we must have f (z) = f (z0 ). But if |f | is
constant for |z − z0 | < ε, then so is f .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Proof. By Cauchy’s Formula, for all circles C(z0 , r) with r < ε we
have that
Z
f (z)
f (z0 ) = 1
2πi C(z ,r) z − z0 dz
0
Z
f (z) 1
d|z|
≤
2π C(z0 ,r) z − z0 Z
f (z0 )
1
d|z| = f (z0 )
≤
2π C(z0 ,r) r
and the latter integral will be strictly smaller
than
f (z0 ) if there is
even a single point z on C(z0 , r) where f (z) < f (z0 ). Thus for all
r < ε and all |z − z0 | = r we must have f (z) = f (z0 ). But if |f | is
constant for |z − z0 | < ε, then so is f .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain. Now there is a disk of maximum
radius around z0 on which f is equal to f (z0 ).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain. Now there is a disk of maximum
radius around z0 on which f is equal to f (z0 ). Thus the point on C that
is in this disk and closest to z1 (on the arc C) has the same properties
as z0 .
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain. Now there is a disk of maximum
radius around z0 on which f is equal to f (z0 ). Thus the point on C that
is in this disk and closest to z1 (on the arc C) has the same properties
as z0 . Continue with a disk of maximum radius around this new point.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain. Now there is a disk of maximum
radius around z0 on which f is equal to f (z0 ). Thus the point on C that
is in this disk and closest to z1 (on the arc C) has the same properties
as z0 . Continue with a disk of maximum radius around this new point.
Repeat until we have that f (z1 ) = f (z0 ).
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain. Now there is a disk of maximum
radius around z0 on which f is equal to f (z0 ). Thus the point on C that
is in this disk and closest to z1 (on the arc C) has the same properties
as z0 . Continue with a disk of maximum radius around this new point.
Repeat until we have that f (z1 ) = f (z0 ). This would prove that f is
constant, a contradiction.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Theorem. Maximum Modulus Principle. Let f be analytic and not
constant on an open domain. Then f does not assume a maximum
value on the domain. That is, there is no z0 in the domain so that
|f (z)| ≤ |f (z0 )| for all z in the domain.
Proof. Suppose for a contradiction that such a z0 does exist. By the
preceding lemma, f would be constant on a disk around z0 . Moreover,
we could choose the radius of the disk arbitrarily large, as long as it
stays inside the domain. Now let z1 be in the domain. Then there is an
arc C from z0 to z1 and the arc has a positive (minimum) distance
from the boundary of the domain. Now there is a disk of maximum
radius around z0 on which f is equal to f (z0 ). Thus the point on C that
is in this disk and closest to z1 (on the arc C) has the same properties
as z0 . Continue with a disk of maximum radius around this new point.
Repeat until we have that f (z1 ) = f (z0 ). This would prove that f is
constant, a contradiction.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
z0
u
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
u
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
&%
"!
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
&%
"!
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
h
z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
hi z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
hi
k z1
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
n z1
hi
k
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
n z1
hi
k
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Visualization
'$
z0
# u
'$
&%
"!
&%
C
u
i
h
n z1
hi
k
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be continuous on a closed and bounded region and
let it be analytic and nonconstant in the interior of the region.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be continuous on a closed and bounded region and
let it be analytic and nonconstant in the interior of the region. Then
the largest value of |f | will be assumed at some point on the boundary
of the region and it will not be reached in the interior.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be continuous on a closed and bounded region and
let it be analytic and nonconstant in the interior of the region. Then
the largest value of |f | will be assumed at some point on the boundary
of the region and it will not be reached in the interior.
Proof.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be continuous on a closed and bounded region and
let it be analytic and nonconstant in the interior of the region. Then
the largest value of |f | will be assumed at some point on the boundary
of the region and it will not be reached in the interior.
Proof. Direct consequence of the fact that continuous functions will
assume a maximum on closed and bounded regions and the Maximum
Modulus Principle.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science
Cauchy Integral Formula
Infinite Differentiability
Fundamental Theorem of Algebra
Maximum Modulus Principle
Corollary. Let f be continuous on a closed and bounded region and
let it be analytic and nonconstant in the interior of the region. Then
the largest value of |f | will be assumed at some point on the boundary
of the region and it will not be reached in the interior.
Proof. Direct consequence of the fact that continuous functions will
assume a maximum on closed and bounded regions and the Maximum
Modulus Principle.
Bernd Schröder
The Cauchy Integral Formula
logo1
Louisiana Tech University, College of Engineering and Science