# MA243: Complex Analysis

Unit 7: Residue Theory   *The unit opens with the Residue Theorem, one of the most useful results in complex analysis, which allows one to calculate many integrals which the usual techniques of real variables do not allow us to evaluate.  Recall that the “residue” of the function at the singularity will be defined by one of the coefficients of the Laurent expansion around the singularity.

We will then introduce the Argument Principle and Rouche’s Theorem, which pertain to meromorphic functions.  We will close the unit by using these results to evaluate a number of different types of definite integrals on the real line which can only be handled using techniques from Complex Analysis.*

This unit will take you 20 hours to complete

☐    Subunit 7.1: 2 hours

☐    Subunit 7.2: 4.5 hours

☐    Subunit 7.3: 13.5 hours ☐    Subunit 7.3.1: 1.5 hours

☐    Subunit 7.3.2: 1.5 hours

☐    Subunit 7.3.3: 2.5 hours

☐    Subunit 7.3.4: 1.5 hours

☐    Subunit 7.3.5: 6.5 hours

Unit7 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- State the Residue Theorem. - Use the Residue Theorem to calculate definite integrals on the real line. - Calculate the winding number of a curve with respect to a point. - State and use the Argument Principle. - State and use Rouche’s Theorem.

7.1 The Residue Theorem   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.1-10.2: Residue Theory: Cauchy’s Residue Theorem and Finding the Residue” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.1-10.2: Residue Theory: Cauchy’s Residue Theorem and Finding the Residue” (PDF)

Instructions: Click on the link above, then click on “Chapter 10: Residue Theory.”  The reference will open in PDF.  Please read the indicated sections.

• Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 18: Residue Theorem” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 18: Residue Theorem” (Windows Media Video)

Instructions: Click on the link above and scroll down to the indicated video.  Click on “Video” to download the lecture in WMV format.  Once it has downloaded, watch it in its entirety (Time: 36:05 minutes).

7.2 The Argument Principle and Rouche’s Theorem   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.3: Residue Theory: Principle of the Argument” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.3: Residue Theory: Principle of the Argument” (PDF)

Instructions: Click on the link above, then click on “Chapter 10: Residue Theory.”  The reference will open in PDF.  Scroll down to page 6 of the document and read the indicated section.

• Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 21: More on Residues” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 21: More on Residues” (Windows Media Video)

Instructions: Click on the link above and scroll down to the indicated video.  Click on “Video1” to download the lecture in WMV format.  Once it has downloaded, watch it in its entirety (Time: 29:19 minutes).  (The second video associated with this lecture, video 2, focuses on Laplace Transforms; for the purposes of this course, it is not necessary for you to watch this video.)

• Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 7, Problems 1,2, 4” Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 7, Problems 1, 2 and 4” (PDF)

Instructions: Click on the link and scroll down to the link to HW#7, which will open in PDF.  Work through the indicated 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 4.2, Problems 1-4, 10, 16, 20, 22-24, 26” Link: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.2, Problems 1-4, 10, 16, 20, 22-24, 26” (PDF)

Instructions: Click on the link and select “Chapter 4” which will open in PDF.  Scroll down to page 11 and work through the indicated problems.  When finished, return to the main page and click on the “Solutions” link.

7.3 Evaluation of Definite Integrals   7.3.1 Rational Functions on the Unit Circle   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.1-11.2: Evaluation of Definite Integrals: Introduction and Rational Functions on the Unit Circle” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.1-11.2: Evaluation of Definite Integrals: Introduction and Rational Functions on the Unit Circle” (PDF)

Instructions: Click on the link above, then click on “Chapter 11: Evaluation of Definite Integrals.”  The reference will open in PDF.  Please read the indicated sections.

7.3.2 Rational Functions on the Real Line   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.3: Evaluation of Definite Integrals: Rational Functions on the Real Line” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.3: Evaluation of Definite Integrals: Rational Functions on the Real Line” (PDF)

Instructions: Click on the link above, then click on “Chapter 11: Evaluation of Definite Integrals.”  The reference will open in PDF.  Scroll down to page 3 of the document and read the indicated section.

7.3.3 Rational and Trigonometric Functions on the Real Line   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.4: Evaluation of Definite Integrals: Rational and Trigonometric Functions on the Real Line” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.4: Evaluation of Definite Integrals: Rational and Trigonometric Functions on the Real Line” (PDF)

Instructions: Click on the link above, then click on “Chapter 11: Evaluation of Definite Integrals.”  The reference will open in PDF.  Scroll down to page 6 of the document and read the indicated section.

• Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 20: Calculus of Residues” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 20: Calculus of Residues” (Windows Media Video)

Instructions: Click on the link above and scroll down to the indicated video.  Click on “Video1” to download the lecture in WMV format.  Once it has downloaded, watch it in its entirety.  This is the first half of the lecture; we will continue with the second half below (Time: 48:53 minutes).

7.3.4 Avoiding a Singularity   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.5: Evaluation of Definite Integrals: Bending Round a Singularity” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.5: Evaluation of Definite Integrals: Bending Round a Singularity” (PDF)

Instructions: Click on the link above, then click on “Chapter 11: Evaluation of Definite Integrals.”  The reference will open in PDF.  Scroll down to page 13 of the document and read the indicated section.

7.3.5 Integrands with Branch Points   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.6: Evaluation of Definite Integrals: Integrands with Branch Points” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.6: Evaluation of Definite Integrals: Integrands with Branch Points” (PDF)

Instructions: Click on the link above, then click on “Chapter 11: Evaluation of Definite Integrals.”  The reference will open in PDF.  Scroll down to page 17 of the document and read the indicated section.

• Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 20: Calculus of Residues” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 20: Calculus of Residues” (Windows Media Video)

Instructions: Click on the link above and scroll down to the indicated video.  Click on “Video2” to download the lecture in WMV format.  Once it has downloaded, watch it in its entirety.  This is the second half of the lecture (Time: 36:10 minutes).

Note that if you would like to see more examples of residue computations, Professor Glesser also has three lectures on the topic (lectures 26-29).

• Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 6” Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 6” (PDF)

Instructions: Click on the link and scroll down to the link to HW#6, which will open in PDF.  Work through all problems.  When finished, return to the first page and click on the “solutions” link.