# MA243: Complex Analysis

Unit 6: Singularities and Power Series   *In this unit we will consider functions that are analytic in a disk with the center removed. (The center of the disk is called an isolated singularity.) We will be focusing on the behavior of the function close to the singularity and identify the three kinds of singularities: removable singularities, poles, and essential singularities. In the case where the singularities of the function are poles, the function is called a meromorphic function.

Isolated singularities can be classified by using the Laurent Series Expansion about the singularity, which will be introduced at the end of the unit, and understanding Laurent Series is fundamental for using Residue Theory, which will be the focus of Unit 7.*

Unit 6 Time Advisory
This unit will take you 19.5 hours to complete

☐    Subunit 6.1: 1 hour

☐    Subunit 6.2: 2 hours

☐    Subunit 6.3: 1.5 hours

☐    Subunit 6.4: 1 hour

☐    Subunit 6.5: 1 hour

☐    Subunit 6.6: 8 hours ☐    Subunit 6.6.1: 1.5 hours

☐    Subunit 6.6.2: 1 hour

☐    Subunit6.6.3: 4.5 hours

☐    Subunit 6.6.4: 1 hour

☐    Subunit 6.7: 5 hours

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define analytic functions and find power series representations of holomorphic functions. - Find Laurent series for meromorphic functions. - Define and calculate residues. - State fundamental results about the zeros of analytic functions and use them to establish results about the uniqueness of functions and series representations of functions. - Classify singularities as removable singularities, poles, or essential singularities. - State the Maximum Modulus Theorem and use it to establish uniqueness results. - State and use Riemann’s Theorem on Removable Singularities. - State results about the boundedness of a function near a singularity. - State and use the Casorati-Weierstrass Theorem. - Classify isolated singularities at infinity.

6.1 Power Series Representation of Analytic Functions   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “8.1: Taylor and Laurent Series: Power Series and Holomorphic Functions” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “8.1: Taylor and Laurent Series: Power Series and Holomorphic Functions” (PDF)

Instructions: Scroll down to page 90 (marked page 86) of the document and read the indicated section.

Terms of Use: The material above has been reposted with permission by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka.  It can be viewed in its original form here (PDF).  It may not be altered in any way.

6.2 Laurent Series and Residues Introduced   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “8.3: Taylor and Laurent Series: Laurent Series” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “8.3: Taylor and Laurent Series: Laurent Series” (PDF)

Instructions: Scroll down to page 95 (marked page 91) of the document and read the indicated section.

Terms of Use: The material above has been reposted with permission by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka.  It can be viewed in its original form here (PDF).  It may not be altered in any way.

• Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 22: Laurent Series Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 22: Laurent Series (Flash Video)

Instructions: Click on the link above, then scroll down to the indicated lecture.  Click on “click to play” to play the video (Time: 1 hour, 8 minutes).

6.3 Zeros of an Analytic Function   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “8.2: Taylor and Laurent Series: Classification of Zeros and the Identity Principle” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “8.2: Taylor and Laurent Series: Classification of Zeros and the Identity Principle” (PDF)

Instructions: Scroll down to page 93 (marked page 89) of the document and read the indicated section.

Terms of Use: The material above has been reposted with permission by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka.  It can be viewed in its original form here (PDF).  It may not be altered in any way.

6.4 Uniqueness of Series Representations   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “7.3: Taylor Series, Uniqueness, and the Maximum Principle: Uniqueness” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “7.3: Taylor Series, Uniqueness, and the Maximum Principle: Uniqueness” (PDF)

Instructions: Click on the link above, then click on “Chapter 7: Taylor Series, Uniqueness, and the Maximum Principle.”  The reference will open in PDF.  Scroll down to page 5 of the document and read the indicated section.

6.5 The Maximum Modulus Theorem   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “7.4: Taylor Series, Uniqueness, and the Maximum Principle: The Maximum Principle” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “7.3: Taylor Series, Uniqueness, and the Maximum Principle: The Maximum Principle” (PDF)

Instructions: Click on the link above, then click on “Chapter 7: Taylor Series, Uniqueness, and the Maximum Principle.” The reference will open in PDF.  Scroll down to page 8 of the document and read the indicated section.

6.6 Classification of Singularities   6.6.1 Removable Singularities   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.1: Isolated Singularities and Laurent Series: Removable Singularities” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.1: Isolated Singularities and Laurent Series: Removable Singularities” (PDF)

Instructions: Click on the link above, then click on “Chapter 8: Isolated Singularities and Laurent Series.” The reference will open in PDF.  Please read the indicated section.

• Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 23: Zeros and Singularities Link: Suffolk University: Professor Adam Glesser’s Lecture 23: Zeros and Singularities(Flash Video)

Instructions: Click on the link above, then scroll down to the indicated lecture.  Click on “click to play” to play the video (Time: 42 minutes).

Note that this lecture covers subunits 6.3 and 6.6.1 and introduces material for subunits 6.6.2 and 6.6.3.

6.6.2 Poles   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.2: Isolated Singularities and Laurent Series: Poles” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.2: Isolated Singularities and Laurent Series: Poles” (PDF)

Instructions: Click on the link above, then click on “Chapter 8: Isolated Singularities and Laurent Series.”  The reference will open in PDF.  Scroll down to page 4 of the document and read the indicated section.

6.6.3 Essential Singularities   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.3: Isolated Singularities and Laurent Series: Essential Singularities” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.3: Isolated Singularities and Laurent Series: Essential Singularities” (PDF)

Instructions: Click on the link above, then click on “Chapter 8: Isolated Singularities and Laurent Series.”  The reference will open in PDF.  Scroll down to page 4 of the document and read the indicated section.

• Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 24: Poles Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 24: Poles (Flash Video)

Instructions: Click on the link above, then scroll down to the indicated lecture.  Click on “click to play” to play the video (Time: 1 hour, 9 minutes).

• Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 5” Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 5” (PDF)

Instructions: Click on the link and scroll down to the link to HW#5, which will open in PDF.  Work through all problems. When finished, return to the first page and click on the “solutions” link.

• Assessment: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.2, Problem 5; Chapter 2.4, Problem 1; and Chapter 4.1, Problems 10 and 13” Link: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.2, Problem 5; Chapter 2.4,  Problem 1; and Chapter 4.1, Problems 10 and 13” (PDF)

Instructions: Click on the link and select “Chapter 2” which will open in PDF.  Problems for section 2.2 begin on page 17 and those for sections 2.4 begin on page 26.

Next, return to the main page and click on “Chapter 4.”  Problems for section 4.1 begin on page 6.

When finished, return to the main page and click on the “Solutions” link.

6.6.4 Isolated Singularities at Infinity   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.4: Isolated Singularities and Laurent Series: Isolated Singularities at Infinity” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.4: Isolated Singularities and Laurent Series: Isolated Singularities at Infinity” (PDF)

Instructions: Click on the link above, then click on “Chapter 8: Isolated Singularities and Laurent Series: Removable Singularities.”  The reference will open in PDF.  Scroll down to page 5 of the document and read the indicated section.

6.7 Laurent Series Revisited   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.6: Isolated Singularities and Laurent Series: Removable Singularities: Taylor Series, Uniqueness, and the Maximum Principle: Laurent Series” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.6: Isolated Singularities and Laurent Series: Laurent Series” (PDF)

Instructions: Click on the link above, then click on “Chapter 8: Isolated Singularities and Laurent Series.”  The reference will open in PDF.  Scroll down to page 7 of the document and read the indicated section.