# MA243: Complex Analysis

Unit 5: Integration   We earlier defined analytic or holomorphic functions as functions differentiable in an open set.  We will now see that the derivative of an analytic function is continuous and that in fact higher derivatives also exist, using complex integration to prove it.

We will also discuss curves in the complex plane—that is, functions from a closed bounded interval of real numbers into the complex plane—before defining the integral of a complex function along a curve in the complex plane (contour or line integrals).

We will then learn about the main theorem of complex analysis: Cauchy’s Theorem.  As a consequence of the theorem, we will see that an analytic function can be represented as a contour integral where the function’s argument acts as a parameter (Cauchy’s Integral Formula). Cauchy’s Theorem will also enable us to prove that an analytic function is infinitely differentiable and that the sum of a power series is analytic inside the domain of convergence.  Conversely, any analytic function can be represented by a power series.

We will finish by proving some of the most important consequences of Cauchy’s Theorem: the Fundamental Theorem of Algebra, Liouville’s Theorem, and Morera’s Theorem.

This unit will take you 20 hours to complete

☐    Subunit 5.1: 2 hours

☐    Subunit 5.2: 4.5 hours

☐    Subunit 5.2.1: 2 hours

☐    Subunit 5.2.2: 1.5 hours

☐    Subunit 5.2.3: 1 hour

☐    Subunit 5.3: 3 hours

☐    Subunit 5.4: 2.5 hours

☐    Subunit 5.5: 2 hours

☐    Subunit 5.5.1: 1 hour

☐    Subunit 5.5.2-5.5.3: 1 hour

☐    Subunit 5.6: 6 hours

Unit5 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define key terms relating the curves in the complex plane, including: simple, closed, smooth, pathwise-smooth, positively oriented, etc. - Find the length of a smooth curve. - Write a parameterization of a curve that traverses a given path. - Define homotopy. - State and prove Cauchy’s Theorem. Use its corollaries. - State and prove Cauchy’s Integral Formula. - State the Jordan Curve Theorem. - Calculate contour integrals. - Bound contour integrals. - Define equivalent curves. - State the Fundamental Theorem of Algebra and Liouville’s Theorem and derive them as consequences of Cauchy’s Theorem. - Define primitive and state the First and Second Fundamental Theorems of Calculus in the context of contour integrals. - State and prove Morera’s theorem.

5.1 Curves in the Complex Plane   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.1: Complex Integrals: Curves in the Complex Plane” Link: Imperial College, London: W.W.L.Chen’s  Introduction to Complex Analysis: “4.1: Complex Integrals: Curves in the Complex Plane” (PDF)

Instructions: Click on the link above, then click on “Chapter 4: Complex Integrals.”  The reference will open in PDF.  Please read the indicated (short) section, noting in particular the illustrations of paths and different types of curves.

• Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “4.1 Integration: Definition and Basic Properties” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “4.1 Integration: Definition and Basic Properties” (PDF)

Instructions: Scroll down to page 48 (marked page 44) of the document and read the indicated section. Note in particular the calculation of arc length, which should be familiar to you from Calculus.

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 11: Contours and Contour Integrals Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 11: Contours and Contour Integrals (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, 6 minutes).

Note that we have skipped Lecture 10 because it is merely a long preparation for this lecture, making most of the same definitions in the context of R2.

5.2 Contour Integrals   5.2.1 Definitions   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.2: Complex Integrals: Contour Integrals” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.2: Complex Integrals: Contour Integrals” (PDF)

Instructions: Click on the link above, then click on “Chapter 4: Complex Integrals.”  The reference will open in PDF.  Scroll down to page 3 of the document and read the indicated section.

• Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 12: Examples of Contour Integrals Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 12: Examples of Contour Integrals (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, 3 minutes).

5.2.2 Inequalities   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.3: Complex Integrals: Inequalities for Contour Integrals” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.3: Complex Integrals: Inequalities for Contour Integrals” (PDF)

Instructions: Click on the link above, then click on “Chapter 4: Complex 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 12: Upper Bounds for Integrals” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 12: Upper Bounds for Integrals” (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: 11 minutes).

5.2.3 Equivalent Curves   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.4: Complex Integrals: Equivalent Curves” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.4: Complex Integrals: Equivalent Curves” (PDF)

Instructions: Click on the link above, then click on “Chapter 4: Complex Integrals.”  The reference will open in PDF.  Scroll down to page 7 of the document and read the indicated section.

5.3 Cauchy’s Theorem   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “4.2: Integration: Cauchy’s Theorem” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “4.2: Integration: Cauchy’s Theorem” (PDF)

Instructions: Scroll down to page 51 (marked page 47) of the document and read the indicated section.

This is perhaps the most important theorem that you will learn in this course, and you should commit it to memory.  Note that a second proof can be found in Professor Chen’s notes, Chapter 5, Sections 1 and 2.

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: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 13: Cauchy-Goursat Theorem” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 13: Cauchy-Goursat Theorem” (Windows Media Video)

Instructions: Click on the link above and scroll down to the indicated video.  Click on “Video1” to download the first part of the lecture in WMV format.  Once it has downloaded, watch it in its entirety (Time: 39:03 minutes). Next, click on “Video2” to download the second part of the lecture, and watch it in its entirety (Time: 12:25 minutes).

Note that this lecture covers subunits 5.2.3 and 5.3.

5.4 Cauchy’s Integral Formula and the Jordan Curve Theorem   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “4.3: Integration: Cauchy’s Integral Formula” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “Integration: Cauchy’s Integral Formula” (PDF)

Instructions: Scroll down to page 53 (marked page 49) of the document and read the indicated section.

Cauchy’s Integral Formula should also be committed to memory.

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: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 14: Cauchy Integral Formula” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 14: Cauchy Integral Formula” (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: 35:17 minutes).

This video is intended to cover subunits 5.4 and 5.5.1.  Note, however, that it also covers some of the material for subunits 5.5.2-5.5.6.  This latter material will be covered again in the video lecture listed in subunit 5.5.6 at a slower pace.

5.5 Consequences of Cauchy’s Theorem   5.5.1 Derivatives   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “5.1: Consequences of Cauchy’s Theorem: Extensions of Cauchy’s Formula” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “5.1: Consequences of Cauchy’s Theorem: Extensions of Cauchy’s Formula” (PDF)

Instructions: Scroll down to page 59 (marked page 55) 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.

5.5.2 Fundamental Theorem of Algebra   The Reading for this course is covered by 5.5.3.

5.5.3 Liouville’s Theorem   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “5.2: Consequences of Cauchy’s Theorem: Taking Cauchy’s Formula to the Limit” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “5.2: Consequences of Cauchy’s Theorem: Taking Cauchy’s Formula to the Limit” (PDF)

Instructions: Scroll down to page 61 (marked page 57) of the document and read the indicated section.

Note that this reading covers subunits 5.5.2 and 5.5.3.

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.

5.6 Antiderivatives and Morera’s Theorem   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “5.3: Consequences of Cauchy’s Theorem: Antiderivatives” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “5.3: Consequences of Cauchy’s Theorem: Antiderivatives” (PDF)

Instructions: Scroll down to page 64 (marked page 60) 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 20: Some Applications Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 20: Some Applications (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, 13 minutes).

Note that video covers subunits 5.5.2-5.6.  Also note that much of the material in this video was presented at a faster pace in the video lecture for subunit 5.4.