Integral Theorems Notes | EduRev

: Integral Theorems Notes | EduRev

 Page 1


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Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Integral Theorems
Bernd Schr¨ oder
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
Page 2


logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Integral Theorems
Bernd Schr¨ oder
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
Page 3


logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Integral Theorems
Bernd Schr¨ oder
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero
, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
Page 4


logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Integral Theorems
Bernd Schr¨ oder
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero
, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
Page 5


logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Integral Theorems
Bernd Schr¨ oder
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero
, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
logo1
Antiderivatives Cauchy-Goursat Theorem Two Kinds of Domains
Introduction
1. Much of the strength of complex analysis derives from the fact
that the integral of an analytic function over a simple closed
contour is zero, as long as the function is analytic on the contour
andin the contour.
2. The above is called the Cauchy-Goursat Theorem.
3. We will start by analyzing integrals across closed contours a bit
more carefully.
4. Then we will prove the Cauchy-Goursat Theorem.
5. Then we will consider a few properties of domains that relate to
the Cauchy-Goursat Theorem.
6. The original motivation to investigate integrals over closed
contours probably comes from considerations of potentials in
physics. For potentials in physics, integrals over closed curves
must be zero.
Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science
Integral Theorems
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