The Residue Theorem

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

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

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

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

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

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

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

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. ℑ(z) 6

i r

- ℜ(z) −1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ℑ(z) 6

i r

- ℜ(z) −1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−1 1r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−1 r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z) 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z) −1 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem r

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z) −1 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z) −1 1

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

- ℜ(z) −1 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

i r

- ℜ(z) −1 1r

−i

1 Example. The function f (z) = is not analytic at (z − 1)(1 + z2) z = 1,i,−i. ℑ(z) 6

i r

- ℜ(z) −1 1r

−i r

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· ··· - ℜ(z) −r1 r r r r r 1r

−i

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 and at z = −1,− ,− ,− ,.... 2 3 4 ℑ(z) 6

··· ··· - ℜ(z) −r1 r r r r r 1r

−i

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 z = 1, , , ,... and at 0 2 3 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ℑ(z) 6

··· ··· - ℜ(z) −r1 r r r r r 1r

−i

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· ··· r r r r r r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

- ℜ(z) −1 1

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· ··· r r r r r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

- ℜ(z) −1 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· ··· r r r r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

- ℜ(z) −1 r 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· ··· r r r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

- ℜ(z) −1 r r 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· r r r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· - ℜ(z) −1 r r 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· r r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· - ℜ(z) −1 r r r 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· - ℜ(z) −r1 r r r 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ··· r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· - ℜ(z) −r1 r r r r 1r

−i

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

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· - ℜ(z) −r1 r r r r r 1r

−i

1 Example. The function f (z) = π is not analytic at sin z 1 1 1 1 1 1 z = 1, , , ,... and at 0 and at z = −1,− ,− ,− ,.... 2 3 4 2 3 4 ℑ(z) 6

··· ··· - ℜ(z) −r1 r r r r r 1r

−i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

Notes. 1 1. z = 1,i,−i are isolated singularities of f (z) = . (z − 1)(1 + z2) 1 1 1 1 2. z = 1, , , ,... are isolated singularities of f (z) = π . 2 3 4 sin z 1 But 0 is not an isolated singularity of f (z) = π . sin z 3. 0 is not an isolated singularity of f (z) = Log(z) or of any root function. (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , then z0 is called an isolated singularity of f .

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Notes. 1 1. z = 1,i,−i are isolated singularities of f (z) = . (z − 1)(1 + z2) 1 1 1 1 2. z = 1, , , ,... are isolated singularities of f (z) = π . 2 3 4 sin z 1 But 0 is not an isolated singularity of f (z) = π . sin z 3. 0 is not an isolated singularity of f (z) = Log(z) or of any root function. (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1. z = 1,i,−i are isolated singularities of f (z) = . (z − 1)(1 + z2) 1 1 1 1 2. z = 1, , , ,... are isolated singularities of f (z) = π . 2 3 4 sin z 1 But 0 is not an isolated singularity of f (z) = π . sin z 3. 0 is not an isolated singularity of f (z) = Log(z) or of any root function. (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

Notes.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 1 2. z = 1, , , ,... are isolated singularities of f (z) = π . 2 3 4 sin z 1 But 0 is not an isolated singularity of f (z) = π . sin z 3. 0 is not an isolated singularity of f (z) = Log(z) or of any root function. (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

Notes. 1 1. z = 1,i,−i are isolated singularities of f (z) = . (z − 1)(1 + z2)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 But 0 is not an isolated singularity of f (z) = π . sin z 3. 0 is not an isolated singularity of f (z) = Log(z) or of any root function. (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

Notes. 1 1. z = 1,i,−i are isolated singularities of f (z) = . (z − 1)(1 + z2) 1 1 1 1 2. z = 1, , , ,... are isolated singularities of f (z) = π . 2 3 4 sin z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 3. 0 is not an isolated singularity of f (z) = Log(z) or of any root function. (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem (Remember that every branch cut must contain zero, so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , so these functions will not be analytic on a set 0 < |z| < R.)

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then z0 is called an isolated singularity of f .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem If the function f is analytic for 0 < |z − z0| < R, then it has ∞ n a Laurent expansion ∑ cn(z − z0) about z0. The coefﬁcient c−1 is n=−∞

called the residue of f at z0. It is also denoted Resz=z0 (f ) := c−1.

The next result will show the relevance of residues.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , then it has ∞ n a Laurent expansion ∑ cn(z − z0) about z0. The coefﬁcient c−1 is n=−∞

called the residue of f at z0. It is also denoted Resz=z0 (f ) := c−1.

The next result will show the relevance of residues.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem The coefﬁcient c−1 is

called the residue of f at z0. It is also denoted Resz=z0 (f ) := c−1.

The next result will show the relevance of residues.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then it has ∞ n a Laurent expansion ∑ cn(z − z0) about z0. n=−∞

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem It is also denoted Resz=z0 (f ) := c−1.

The next result will show the relevance of residues.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then it has ∞ n a Laurent expansion ∑ cn(z − z0) about z0. The coefﬁcient c−1 is n=−∞ called the residue of f at z0.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem The next result will show the relevance of residues.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then it has ∞ n a Laurent expansion ∑ cn(z − z0) about z0. The coefﬁcient c−1 is n=−∞

called the residue of f at z0. It is also denoted Resz=z0 (f ) := c−1.

Deﬁnition. If the function f is analytic for 0 < |z − z0| < R, then it has ∞ n a Laurent expansion ∑ cn(z − z0) about z0. The coefﬁcient c−1 is n=−∞

called the residue of f at z0. It is also denoted Resz=z0 (f ) := c−1.

The next result will show the relevance of residues.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C Proof. From the theorem on Laurent expansions, we have that 1 I f (ξ) 1 I Res (f ) = a = d = f ( ) d , z=z0 −1 (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem and C is a positively oriented simple closed contour around z that is contained Z 0

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z

, then f (z) dz = 2πiResz=z0 (f ). C Proof. From the theorem on Laurent expansions, we have that 1 I f (ξ) 1 I Res (f ) = a = d = f ( ) d , z=z0 −1 (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z0 that is contained

in 0 < |z − z0| < R

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Proof. From the theorem on Laurent expansions, we have that 1 I f (ξ) 1 I Res (f ) = a = d = f ( ) d , z=z0 −1 (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem From the theorem on Laurent expansions, we have that 1 I f (ξ) 1 I Res (f ) = a = d = f ( ) d , z=z0 −1 (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C Proof.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 I f (ξ) 1 I = a = d = f ( ) d , −1 (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C Proof. From the theorem on Laurent expansions, we have that

Resz=z0 (f )

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 I f (ξ) 1 I = d = f ( ) d , (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C Proof. From the theorem on Laurent expansions, we have that

Resz=z0 (f ) = a−1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 I = f (ξ) dξ, 2πi C

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C Proof. From the theorem on Laurent expansions, we have that 1 I f (ξ) Res (f ) = a = d z=z0 −1 (−1)+1 ξ 2πi C (ξ − z0)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ,

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

in 0 < |z − z0| < R, then f (z) dz = 2πiResz=z0 (f ). C Proof. From the theorem on Laurent expansions, we have that 1 I f (ξ) 1 I Res (f ) = a = d = f ( ) d z=z0 −1 (−1)+1 ξ ξ ξ 2πi C (ξ − z0) 2πi C

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

where C is any circle around z0 with radius < R.

Theorem. If the function f is analytic for 0 < |z − z0| < R and C is a positively oriented simple closed contour around z that is contained Z 0

where C is any circle around z0 with radius < R. Replacement of the circle with any contour around the origin requires an argument similar to the one that shows that we can use circles of any radius.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem

rR r I - 0 ] r ? O r r C

- - 1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem rR r I r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

- 0 ] ? O r C

- - 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem R r I r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

r - 0 ] ? O r C

- - 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem R I r r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

r r - 0 ] ? O r C

- - 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem R I

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

r r - 0 ] r ? O r C

- - 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem R I

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

r r - 0 ] r ? O r r C

- - 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem R

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof (concl.)

r r I - 0 ] r ? O r r C

- - 1

Proof (concl.)

rR r I - 0 ] r ? O r r C

- - 1

Proof (concl.)

rR r I - 0 ] r ? O r r C

- - 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Let C be the unit circle, traversed in the positive Z ez orientation. Then dz = 2πi C z ez The function has only the singularity at 0 inside the contour. The z ez 1 ∞ zn residue of = + ∑ at z = 0 is 1. Now apply the z z n=0 (n + 1)! preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z ez Then dz = 2πi C z ez The function has only the singularity at 0 inside the contour. The z ez 1 ∞ zn residue of = + ∑ at z = 0 is 1. Now apply the z z n=0 (n + 1)! preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive orientation.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ez The function has only the singularity at 0 inside the contour. The z ez 1 ∞ zn residue of = + ∑ at z = 0 is 1. Now apply the z z n=0 (n + 1)! preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez orientation. Then dz = 2πi C z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem The ez 1 ∞ zn residue of = + ∑ at z = 0 is 1. Now apply the z z n=0 (n + 1)! preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez orientation. Then dz = 2πi C z ez The function has only the singularity at 0 inside the contour. z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 ∞ zn = + ∑ at z = 0 is 1. Now apply the z n=0 (n + 1)! preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez orientation. Then dz = 2πi C z ez The function has only the singularity at 0 inside the contour. The z ez residue of z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem at z = 0 is 1. Now apply the preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Now apply the preceding theorem. Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Note that direct computation gives the same result, because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , because the ∞ zn integral of ∑ over any closed contour is 0 and n=0 (n + 1)! Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez orientation. Then dz = 2πi C z ez The function has only the singularity at 0 inside the contour. The z ez 1 ∞ zn residue of = + ∑ at z = 0 is 1. Now apply the z z n=0 (n + 1)! preceding theorem. Note that direct computation gives the same result

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem and

Z 1 Z 2π 1 dz = ieiθ d = i. iθ θ 2π C z 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z 2π 1 = ieiθ d = i. iθ θ 2π 0 e

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem = 2πi.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Let C be the unit circle, traversed in the positive Z ez2 orientation. Then 2 dz = 0 C z ez2 The function has only the singularity at 0 inside the contour. The z2 ez2 1 ∞ z2n residue of 2 = 2 + ∑ at z = 0 is 0. Now apply the z z n=0 (n + 1)! preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z ez2 Then 2 dz = 0 C z ez2 The function has only the singularity at 0 inside the contour. The z2 ez2 1 ∞ z2n residue of 2 = 2 + ∑ at z = 0 is 0. Now apply the z z n=0 (n + 1)! preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive orientation.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ez2 The function has only the singularity at 0 inside the contour. The z2 ez2 1 ∞ z2n residue of 2 = 2 + ∑ at z = 0 is 0. Now apply the z z n=0 (n + 1)! preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez2 orientation. Then 2 dz = 0 C z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem The ez2 1 ∞ z2n residue of 2 = 2 + ∑ at z = 0 is 0. Now apply the z z n=0 (n + 1)! preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez2 orientation. Then 2 dz = 0 C z ez2 The function has only the singularity at 0 inside the contour. z2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 ∞ z2n = 2 + ∑ at z = 0 is 0. Now apply the z n=0 (n + 1)! preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez2 orientation. Then 2 dz = 0 C z ez2 The function has only the singularity at 0 inside the contour. The z2 ez2 residue of z2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem at z = 0 is 0. Now apply the preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Let C be the unit circle, traversed in the positive Z ez2 orientation. Then 2 dz = 0 C z ez2 The function has only the singularity at 0 inside the contour. The z2 ez2 1 ∞ z2n residue of 2 = 2 + ∑ z z n=0 (n + 1)!

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Now apply the preceding theorem.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Residue Theorem. Let C be a simple closed positively oriented contour, let z1,...,zn be points in the interior of C, and let the function f be analytic on C and in its interior, except possibly at the zj. Then 1 Z n f (ξ) dξ = ∑ Resz=zj (f ). 2πi C j=1

Proof. Let Cj be positively oriented circle around zj so that no two of C1,...,Cn intersect and so that all are contained in the interior of C. Then by extension of Cauchy-Goursat theorem Z n Z n f (z) dz = ∑ f (z) dz = 2πi ∑ Resz=zj (f ). C j=1 Cj j=1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Proof. Let Cj be positively oriented circle around zj so that no two of C1,...,Cn intersect and so that all are contained in the interior of C. Then by extension of Cauchy-Goursat theorem Z n Z n f (z) dz = ∑ f (z) dz = 2πi ∑ Resz=zj (f ). C j=1 Cj j=1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. Residue Theorem. Let C be a simple closed positively oriented contour, let z1,...,zn be points in the interior of C, and let the function f be analytic on C and in its interior, except possibly at the zj. Then 1 Z n f (ξ) dξ = ∑ Resz=zj (f ). 2πi C j=1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Let Cj be positively oriented circle around zj so that no two of C1,...,Cn intersect and so that all are contained in the interior of C. Then by extension of Cauchy-Goursat theorem Z n Z n f (z) dz = ∑ f (z) dz = 2πi ∑ Resz=zj (f ). C j=1 Cj j=1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem so that no two of C1,...,Cn intersect and so that all are contained in the interior of C. Then by extension of Cauchy-Goursat theorem Z n Z n f (z) dz = ∑ f (z) dz = 2πi ∑ Resz=zj (f ). C j=1 Cj j=1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof. Let Cj be positively oriented circle around zj

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Then by extension of Cauchy-Goursat theorem Z n Z n f (z) dz = ∑ f (z) dz = 2πi ∑ Resz=zj (f ). C j=1 Cj j=1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof. Let Cj be positively oriented circle around zj so that no two of C1,...,Cn intersect and so that all are contained in the interior of C.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem n

= 2πi ∑ Resz=zj (f ). j=1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

i ? r - 63 ℜ(z) −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ℑ(z) 6

i ? r - 63 ℜ(z) −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem

i ? r 63 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem

i ? r 3 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

- 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i ? r 3 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

- 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r 3 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

? - 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r 3 −i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

? - 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r −i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

? - 63 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

? r - 63 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem −i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

i ? r - 63 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem −i

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

i ? r - 63 ℜ(z)

r -

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

ℑ(z) 6

i ? r - 63 ℜ(z) −i r -

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i:

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 = 1 − = 1 − z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 = z2 + 1 z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = 1 − z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 = = 1 − z2 + 1 z2 + 1 z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 i = − = i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 Res z=i z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i = 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 Res = − z=i z2 + 1 i + i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i:

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 = 1 − = 1 − z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 = z2 + 1 z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = 1 − z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 = = 1 − z2 + 1 z2 + 1 z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 i = − = − −i − i 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 Res z=−i z2 + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i = − 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 Res = − z=−i z2 + 1 −i − i

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Residue at i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z + i z − i z2 1 i Res = − = z=i z2 + 1 i + i 2 Residue at −i: z2 z2 + 1 − 1 1 1 1 = = 1 − = 1 − z2 + 1 z2 + 1 z2 + 1 z − i z + i z2 1 i Res = − = − z=−i z2 + 1 −i − i 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z z2 z2 z2 2 dz = 2πi Resz=i 2 + Resz=−i 2 C z + 1 z + 1 z + 1 i i = 2πi + − = 0 2 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 z2 = 2πi Res + Res z=i z2 + 1 z=−i z2 + 1 i i = 2πi + − = 0 2 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Z z2 2 dz C z + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i i = 2πi + − = 0 2 2

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Z z2 z2 z2 2 dz = 2πi Resz=i 2 + Resz=−i 2 C z + 1 z + 1 z + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem = 0

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Z z2 z2 z2 2 dz = 2πi Resz=i 2 + Resz=−i 2 C z + 1 z + 1 z + 1 i i = 2πi + − 2 2

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

Z z2 z2 z2 2 dz = 2πi Resz=i 2 + Resz=−i 2 C z + 1 z + 1 z + 1 i i = 2πi + − = 0 2 2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Let f be analytic on C except for a finite number of singular points z1,...,zn. Assume that R1 is so that all |zj| ≤ R1. For − R0 > R1 let C0 be the circle around the origin of radius R0, traversed in the clockwise, that is, the mathematically negative direction. Then we define the residue at infinity of f as 1 Z Resz=∞(f ) = f (z) dz. − 2πi C0

Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞ ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Assume that R1 is so that all |zj| ≤ R1. For − R0 > R1 let C0 be the circle around the origin of radius R0, traversed in the clockwise, that is, the mathematically negative direction. Then we deﬁne the residue at inﬁnity of f as 1 Z Resz=∞(f ) = f (z) dz. − 2πi C0

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. Let f be analytic on C except for a ﬁnite number of singular points z1,...,zn.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem For − R0 > R1 let C0 be the circle around the origin of radius R0, traversed in the clockwise, that is, the mathematically negative direction. Then we deﬁne the residue at inﬁnity of f as 1 Z Resz=∞(f ) = f (z) dz. − 2πi C0

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. Let f be analytic on C except for a ﬁnite number of singular points z1,...,zn. Assume that R1 is so that all |zj| ≤ R1.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , that is, the mathematically negative direction. Then we deﬁne the residue at inﬁnity of f as 1 Z Resz=∞(f ) = f (z) dz. − 2πi C0

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem direction. Then we deﬁne the residue at inﬁnity of f as 1 Z Resz=∞(f ) = f (z) dz. − 2πi C0

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. Let f be analytic on C except for a ﬁnite number of singular points z1,...,zn. Assume that R1 is so that all |zj| ≤ R1. For − R0 > R1 let C0 be the circle around the origin of radius R0, traversed in the clockwise, that is, the mathematically negative

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Then we deﬁne the residue at inﬁnity of f as 1 Z Resz=∞(f ) = f (z) dz. − 2πi C0

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞ ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z Z ∞ 1 1 n = − f (z) dz = − cnz dz + + ∑ 2πi C0 2πi C0 n=−∞ ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Z Resz=∞(f ) = f (z) dz − 2πi C0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z ∞ 1 n = − cnz dz + ∑ 2πi C0 n=−∞ ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Z 1 Z Resz=∞(f ) = f (z) dz = − f (z) dz − + 2πi C0 2πi C0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 Z ∞ c 1 Z 1 ∞ c = − n−2 dz = − n−2 dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞

= −c−1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 Z 1 ∞ c = − n−2 dz + 2 ∑ n−2 2πi C0 z n=−∞ z 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞ ∞ 1 Z cn−2 = −c−1 = − dz + ∑ n 2πi C0 n=−∞ z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 Z 1 1 1 1 = − f dz = −Resz=0 f + 2 2 2πi C0 z z z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞ ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = −Res f z=0 z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Z Z Z ∞ 1 1 1 n Resz=∞(f ) = f (z) dz = − f (z) dz = − cnz dz − + + ∑ 2πi C0 2πi C0 2πi C0 n=−∞ ∞ ∞ 1 Z cn−2 1 Z 1 cn−2 = −c−1 = − dz = − dz + ∑ n + 2 ∑ n−2 2πi C0 n=−∞ z 2πi C0 z n=−∞ z 1 Z 1 1 = − f dz + 2 2πi C0 z z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem If f is analytic on C, except for a ﬁnite number of singular points that lie in the interior of a positively oriented simple closed contour C, then Z 1 1 f (z) dz = 2πiResz=0 2 f . C z z

Proof. See previous panel.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Proof. See previous panel.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If f is analytic on C, except for a ﬁnite number of singular points that lie in the interior of a positively oriented simple closed contour C, then Z 1 1 f (z) dz = 2πiResz=0 2 f . C z z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem See previous panel.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. If f is analytic on C, except for a ﬁnite number of singular points that lie in the interior of a positively oriented simple closed contour C, then Z 1 1 f (z) dz = 2πiResz=0 2 f . C z z

Proof.

Theorem. If f is analytic on C, except for a ﬁnite number of singular points that lie in the interior of a positively oriented simple closed contour C, then Z 1 1 f (z) dz = 2πiResz=0 2 f . C z z

Proof. See previous panel.

Theorem. If f is analytic on C, except for a ﬁnite number of singular points that lie in the interior of a positively oriented simple closed contour C, then Z 1 1 f (z) dz = 2πiResz=0 2 f . C z z

Proof. See previous panel.

i ? r - 63 ℜ(z) −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ℑ(z) 6

i ? r - 63 ℜ(z) −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem

i ? r 63 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

- ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem

i ? r 3 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

- 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i ? r 3 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

- 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r 3 −i r -

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

? - 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r 3 −i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

? - 6 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i r −i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

? - 63 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i

−i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

? r - 63 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem −i r

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

i ? r - 63 ℜ(z)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem −i

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

i ? r - 63 ℜ(z)

r -

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3]. ℑ(z) 6

i ? r - 63 ℜ(z) −i r -

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 2 1 1 1 = z = z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 1 f z2 z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = z2 1 + z2 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 f = z z2 z z2 1 2 z + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 n 2n = 2 ∑ (−1) z z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 1 1 = z2 1 − (−z2)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = 2πiRes f = 0. z=0 z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z and hence f (z) dz C

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem = 0.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f C z z

z2 Example. Integrate the function f (z) = over the positively z2 + 1 oriented boundary of the square of numbers z = x + iy with (x,y) in [−3,3] × [−3,3].

1 2 1 1 1 1 1 f = z = z2 z z2 1 2 z2 1 + z2 z + 1 ∞ 1 1 1 n 2n = 2 2 = 2 ∑ (−1) z z 1 − (−z ) z n=0 ∞ 1 n 2n−2 = 2 + ∑ (−1) z z n=1 Z 1 1 and hence f (z) dz = 2πiResz=0 2 f = 0. C z z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Let z0 be a complex number, let r > 0 and let f be analytic in a deleted neighborhood of z0. In this case z0 is called an isolated singularity of f . Consider the Laurent expansion of f around z0 ∞ n b1 b2 b3 f (z) = ∑ an(z − z0) + + 2 + 3 + ··· n=0 z − z0 (z − z0) (z − z0)

The sum of the negative powers of z − z0 is also called the principal part of f .

1. If bn = 0 for all n, then z0 is called a removable singularity. 2. If there is a positive number m so that bm 6= 0 and bn = 0 for all n > m, then z0 is called a pole of order m. 3. If the number m in part 2 equals 1, then z0 is also called a simple pole.

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem In this case z0 is called an isolated singularity of f . Consider the Laurent expansion of f around z0 ∞ n b1 b2 b3 f (z) = ∑ an(z − z0) + + 2 + 3 + ··· n=0 z − z0 (z − z0) (z − z0)

The sum of the negative powers of z − z0 is also called the principal part of f .

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. Let z0 be a complex number, let r > 0 and let f be analytic in a deleted neighborhood of z0.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Consider the Laurent expansion of f around z0 ∞ n b1 b2 b3 f (z) = ∑ an(z − z0) + + 2 + 3 + ··· n=0 z − z0 (z − z0) (z − z0)

The sum of the negative powers of z − z0 is also called the principal part of f .

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. Let z0 be a complex number, let r > 0 and let f be analytic in a deleted neighborhood of z0. In this case z0 is called an isolated singularity of f .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem The sum of the negative powers of z − z0 is also called the principal part of f .

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Deﬁnition. Let z0 be a complex number, let r > 0 and let f be analytic in a deleted neighborhood of z0. In this case z0 is called an isolated singularity of f . Consider the Laurent expansion of f around z0 ∞ n b1 b2 b3 f (z) = ∑ an(z − z0) + + 2 + 3 + ··· n=0 z − z0 (z − z0) (z − z0)

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1. If bn = 0 for all n, then z0 is called a removable singularity. 2. If there is a positive number m so that bm 6= 0 and bn = 0 for all n > m, then z0 is called a pole of order m. 3. If the number m in part 2 equals 1, then z0 is also called a simple pole.

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

The sum of the negative powers of z − z0 is also called the principal part of f .

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 2. If there is a positive number m so that bm 6= 0 and bn = 0 for all n > m, then z0 is called a pole of order m. 3. If the number m in part 2 equals 1, then z0 is also called a simple pole.

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

The sum of the negative powers of z − z0 is also called the principal part of f .

1. If bn = 0 for all n, then z0 is called a removable singularity.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 3. If the number m in part 2 equals 1, then z0 is also called a simple pole.

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

The sum of the negative powers of z − z0 is also called the principal part of f .

1. If bn = 0 for all n, then z0 is called a removable singularity. 2. If there is a positive number m so that bm 6= 0 and bn = 0 for all n > m, then z0 is called a pole of order m.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

The sum of the negative powers of z − z0 is also called the principal part of f .

4. If z0 is not removable and there is no number as in part 2, then z0 is called an essential singularity.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem sin(z) The function f (z) = has a removable singularity at z z = 0. sin(z) 1 = sin(z) z z 1 ∞ (−1)n = ∑ z2n+1 z n=0 (2n + 1)! ∞ (−1)n = ∑ z2n n=0 (2n + 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem sin(z) 1 = sin(z) z z 1 ∞ (−1)n = ∑ z2n+1 z n=0 (2n + 1)! ∞ (−1)n = ∑ z2n n=0 (2n + 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

sin(z) Example. The function f (z) = has a removable singularity at z z = 0.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 = sin(z) z 1 ∞ (−1)n = ∑ z2n+1 z n=0 (2n + 1)! ∞ (−1)n = ∑ z2n n=0 (2n + 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

sin(z) Example. The function f (z) = has a removable singularity at z z = 0. sin(z) z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 ∞ (−1)n = ∑ z2n+1 z n=0 (2n + 1)! ∞ (−1)n = ∑ z2n n=0 (2n + 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

sin(z) Example. The function f (z) = has a removable singularity at z z = 0. sin(z) 1 = sin(z) z z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ (−1)n = ∑ z2n n=0 (2n + 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

sin(z) Example. The function f (z) = has a removable singularity at z z = 0. sin(z) 1 = sin(z) z z 1 ∞ (−1)n = ∑ z2n+1 z n=0 (2n + 1)!

sin(z) Example. The function f (z) = has a removable singularity at z z = 0. sin(z) 1 = sin(z) z z 1 ∞ (−1)n = ∑ z2n+1 z n=0 (2n + 1)! ∞ (−1)n = ∑ z2n n=0 (2n + 1)!

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 The function f (z) = has simple poles at z = 1 and at 1 − z2 z = −1. 1 1 1 = 1 − z2 z + 1 z − 1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 = 1 − z2 z + 1 z − 1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = has simple poles at z = 1 and at 1 − z2 z = −1.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = z + 1 z − 1

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = has simple poles at z = 1 and at 1 − z2 z = −1. 1 1 − z2

1 Example. The function f (z) = has simple poles at z = 1 and at 1 − z2 z = −1. 1 1 1 = 1 − z2 z + 1 z − 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 + z2 The function f (z) = has a pole of order 3 at z = 0. z3 1 + z2 1 1 = + z3 z3 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 + z2 1 1 = + z3 z3 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 + z2 Example. The function f (z) = has a pole of order 3 at z = 0. z3

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 = + z3 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 + z2 Example. The function f (z) = has a pole of order 3 at z = 0. z3 1 + z2 z3

1 + z2 Example. The function f (z) = has a pole of order 3 at z = 0. z3 1 + z2 1 1 = + z3 z3 z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 The function f (z) = e z has an essential singularity at z = 0. ∞ 1 1 1 z e = ∑ n n=0 n! z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 1 1 z e = ∑ n n=0 n! z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = e z has an essential singularity at z = 0.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 1 = ∑ n n=0 n! z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. The function f (z) = e z has an essential singularity at z = 0.

1 e z

1 Example. The function f (z) = e z has an essential singularity at z = 0. ∞ 1 1 1 z e = ∑ n n=0 n! z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Let the function f be analytic in a deleted neighborhood of the point z0. Then z0 is a pole of order m if and only if there is function Φ that is analytic in a neighborhood of z0, Φ(z0) 6= 0 and so that Φ(z) f (z) = m (z − z0)

for all z in a deleted neighborhood of z0. Moreover, Φ(m−1)(z ) Res (f ) = 0 z=z0 (m − 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Then z0 is a pole of order m if and only if there is function Φ that is analytic in a neighborhood of z0, Φ(z0) 6= 0 and so that Φ(z) f (z) = m (z − z0)

for all z in a deleted neighborhood of z0. Moreover, Φ(m−1)(z ) Res (f ) = 0 z=z0 (m − 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. Let the function f be analytic in a deleted neighborhood of the point z0.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , Φ(z0) 6= 0 and so that Φ(z) f (z) = m (z − z0)

for all z in a deleted neighborhood of z0. Moreover, Φ(m−1)(z ) Res (f ) = 0 z=z0 (m − 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. Let the function f be analytic in a deleted neighborhood of the point z0. Then z0 is a pole of order m if and only if there is function Φ that is analytic in a neighborhood of z0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem and so that Φ(z) f (z) = m (z − z0)

for all z in a deleted neighborhood of z0. Moreover, Φ(m−1)(z ) Res (f ) = 0 z=z0 (m − 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Theorem. Let the function f be analytic in a deleted neighborhood of the point z0. Then z0 is a pole of order m if and only if there is function Φ that is analytic in a neighborhood of z0, Φ(z0) 6= 0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Moreover, Φ(m−1)(z ) Res (f ) = 0 z=z0 (m − 1)!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

for all z in a deleted neighborhood of z0.

for all z in a deleted neighborhood of z0. Moreover, Φ(m−1)(z ) Res (f ) = 0 z=z0 (m − 1)!

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem If f has a pole of order m around z0, then the Laurent expansion of z for 0 < |z − z0| < R (for some R) is (with c−m 6= 0) ∞ ∞ n 1 n+m f (z) = ∑ cn(z − z0) = m ∑ cn(z − z0) n=−m (z − z0) n=−m ∞ 1 k Φ(z) = m ∑ ck−m(z − z0) =: m . (z − z0) k=0 (z − z0) Φ(z) Conversely, if f (z) = m with Φ(z0) 6= 0, then the above (z − z0) computation in reverse shows that f has a pole of order m. Moreover,

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 n+m = m ∑ cn(z − z0) (z − z0) n=−m ∞ 1 k Φ(z) = m ∑ ck−m(z − z0) =: m . (z − z0) k=0 (z − z0) Φ(z) Conversely, if f (z) = m with Φ(z0) 6= 0, then the above (z − z0) computation in reverse shows that f has a pole of order m. Moreover,

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof. If f has a pole of order m around z0, then the Laurent expansion of z for 0 < |z − z0| < R (for some R) is (with c−m 6= 0) ∞ n f (z) = ∑ cn(z − z0) n=−m

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem ∞ 1 k Φ(z) = m ∑ ck−m(z − z0) =: m . (z − z0) k=0 (z − z0) Φ(z) Conversely, if f (z) = m with Φ(z0) 6= 0, then the above (z − z0) computation in reverse shows that f has a pole of order m. Moreover,

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof. If f has a pole of order m around z0, then the Laurent expansion of z for 0 < |z − z0| < R (for some R) is (with c−m 6= 0) ∞ ∞ n 1 n+m f (z) = ∑ cn(z − z0) = m ∑ cn(z − z0) n=−m (z − z0) n=−m

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ(z) =: m . (z − z0) Φ(z) Conversely, if f (z) = m with Φ(z0) 6= 0, then the above (z − z0) computation in reverse shows that f has a pole of order m. Moreover,

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof. If f has a pole of order m around z0, then the Laurent expansion of z for 0 < |z − z0| < R (for some R) is (with c−m 6= 0) ∞ ∞ n 1 n+m f (z) = ∑ cn(z − z0) = m ∑ cn(z − z0) n=−m (z − z0) n=−m ∞ 1 k = m ∑ ck−m(z − z0) (z − z0) k=0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ(z) Conversely, if f (z) = m with Φ(z0) 6= 0, then the above (z − z0) computation in reverse shows that f has a pole of order m. Moreover,

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Moreover,

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Proof. If f has a pole of order m around z0, then the Laurent expansion of z for 0 < |z − z0| < R (for some R) is (with c−m 6= 0) ∞ ∞ n 1 n+m f (z) = ∑ cn(z − z0) = m ∑ cn(z − z0) n=−m (z − z0) n=−m ∞ 1 k Φ(z) = m ∑ ck−m(z − z0) =: m . (z − z0) k=0 (z − z0) Φ(z) Conversely, if f (z) = m with Φ(z0) 6= 0, then the above (z − z0) computation in reverse shows that f has a pole of order m.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ(m−1)(z ) is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent expansion of f

in this situation, the coefﬁcient c−1 = Resz=z0 (f ) of the Laurent Φ(m−1)(z ) expansion of f is the coefﬁcient a = 0 of the power m−1 (m − 1)! series expansion of Φ.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1. z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1, m = 3 Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem The only singularity of f is z = 1. z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1, m = 3 Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1, m = 3 Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 = z3 + 3z2 + 3z + 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1, m = 3 Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1. z + 13 f (z) = z − 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ(z) = z3 + 3z2 + 3z + 1, m = 3 Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1. z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , m = 3 Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1. z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ0(z) = 3z2 + 6z + 3 Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1. z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1, m = 3

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ00(z) = 6z + 6 Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example. Find all poles, their order and their residues for z + 13 f (z) = . z − 1 The only singularity of f is z = 1. z + 13 1 f (z) = = z3 + 3z2 + 3z + 1 z − 1 (z − 1)3 Φ(z) = z3 + 3z2 + 3z + 1, m = 3 Φ0(z) = 3z2 + 6z + 3

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Φ00(1) = 12 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 12 Res (f ) = = 6 z=1 2!

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem = 6

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Log(z) π + 2i Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2 Log(z) Φ(z) = , m = 2 (z + i)2 1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2 Log(z) Φ(z) = , m = 2 (z + i)2 1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 Log(z) = (z − i)2 (z + i)2 Log(z) Φ(z) = , m = 2 (z + i)2 1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) = (z2 + 1)2 (z + i)(z − i)2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Log(z) Φ(z) = , m = 2 (z + i)2 1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , m = 2

1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2 Log(z) Φ(z) = (z + i)2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2 Log(z) Φ(z) = , m = 2 (z + i)2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 (z + i) − 2Log(z) = z (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2 Log(z) Φ(z) = , m = 2 (z + i)2 1 (z + i)2 − 2(z + i)Log(z) Φ0(z) = z (z + i)4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem i + i − 2Log(i)i 2i − 2i π i 2i + π Φ0(i) = = 2 = i(i + i)3 i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 2i − 2i π i 2i + π = 2 = i(2i)3 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Log(z) π + 2i Example. Show that Resz=i = . (z2 + 1)2 8 Log(z) Log(z) 1 Log(z) = = (z2 + 1)2 (z + i)(z − i)2 (z − i)2 (z + i)2 Log(z) Φ(z) = , m = 2 (z + i)2 1 (z + i)2 − 2(z + i)Log(z) 1 (z + i) − 2Log(z) Φ0(z) = z = z (z + i)4 (z + i)3 z + i − 2Log(z)z = z(z + i)3 i + i − 2Log(i)i Φ0(i) = i(i + i)3

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 2i + π = 8 Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem Log(z) 1 π + 2i π + 2i Resz=i = = (z2 + 1)2 1! 8 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem π + 2i = 8

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z 1 1 1 1 1 f = = z2 z z2 1 3 1 1 1+4z z z + 4 z z z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

Example.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z 1 1 1 1 1 f = = z2 z z2 1 3 1 1 1+4z z z + 4 z z z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem , or we need the residue 1 1 at 0 of f . z2 z 1 1 1 1 1 f = = z2 z z2 1 3 1 1 1+4z z z + 4 z z z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 1 1 f = = z2 z z2 1 3 1 1 1+4z z z + 4 z z z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 1 = = z2 1 3 1 1 1+4z z z + 4 z z z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z 1 1 f z2 z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 = 1 1+4z z z z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z 1 1 1 1 f = z2 z z2 1 3 1 z z + 4

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem z2 = 1 + 4z 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z 1 1 1 1 1 f = = z2 z z2 1 3 1 1 1+4z z z + 4 z z

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem 1 1 As f does not have a singularity at 0, the integral must be 0. z2 z

Singularities Residues Residue Theorem Residue at Inﬁnity Types of Isolated Singularities Residues at Poles

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

We will need the residues of f at 0 and at −4, or we need the residue 1 1 at 0 of f . z2 z 1 1 1 1 1 f = = z2 z z2 1 3 1 1 1+4z z z + 4 z z z2 = 1 + 4z

1 Example. Integrate f (z) = around the positively oriented z3(z + 4) circle of radius 5 around the origin.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Residue Theorem