Page 1 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 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 TheoremsRead More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!