Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles The Residue Theorem Bernd Schroder¨ logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Singularities Residues Residue Theorem Residue at Infinity Types of Isolated Singularities Residues at Poles Introduction 1. The Cauchy-Goursat Theorem says that if a function is analytic on and in a closed contour C, then the integral over the closed contour is zero. 2. But what if the function is not analytic? 3. We will avoid situations where the function “blows up” (goes to infinity) on the contour. So we will not need to generalize contour integrals to “improper contour integrals”. 4. But the situation in which the function is not analytic inside the contour turns out to be quite interesting. 5. We will prove the requisite theorem (the Residue Theorem) in this presentation and we will also lay the abstract groundwork. 6. We will then spend an extensive amount of time with examples that show how widely applicable the Residue Theorem is. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1;i;−i.