# MA243: Complex Analysis

Unit 8: Harmonic Functions and Conformal Mappings   In Unit 2, we talked about the Cauchy Riemann equations.  Because of these equations, we noticed that the real and imaginary parts of an analytic function satisfy the Laplace equation.  In other words, we learned that these parts are harmonic functions.  In this last unit, we will study some of the basic properties of harmonic functions, especially the Mean and Minimum Value Properties.

We will revisit transformations (especially the Möbius transformation) and discuss conformal mappings--mappings which preserve angles between arcs.  We will also explore two of the most important results in Complex Analysis: the Open Mapping Theorem and the Riemann Mapping Theorem.

Unit 8 Time Advisory
This unit will take you 12 hours to complete

☐    Subunit 8.1: 1.5 hours

☐    Subunit 8.2: 2.5 hours

☐    Subunit 8.3: 1.5 hours

☐    Subunit 8.4: 1.5 hours

☐    Subunit 8.5: 5 hours

Unit8 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define harmonic functions and state their properties. - State and use the Mean Value and Minimum Value Theorems. - Define conformal mapping. - Find the image of a region under a conformal mapping. - State, prove, and use the Open Mapping Theorem. - State the Riemann Mapping Theorem.

8.1 Properties of Harmonic Functions   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “6.1: Harmonic Functions: Definitions and Basic Properties” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “6.1: Harmonic Functions: Definitions and Basic Properties” (PDF)

Instructions: Scroll down to page 69 (marked page 65) 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: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 25: Harmonic Functions” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 25: Harmonic Functions” (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: 35:47 minutes).

8.2 Mean Value and Minimum Value Theorems   - Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “6.2: Harmonic Functions: Mean-Value and Maximum/Minimum Principle” Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “6.2: Harmonic Functions: Mean-Value and Maximum/Minimum Principle” (PDF)

Instructions: Scroll down to page 71 (marked page 67) 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.

• Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 11” Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 11” (PDF)

Instructions: Click on the link and scroll down to the link to HW#11, 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 2.4, Problems 5-7, 9, 10, 21, 23” Link: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.4, Problems 5-7, 9, 10,  21, 23” (PDF)

Instructions: Click on the link and select “Chapter 2” which will open in PDF.  Scroll down to page 26 and work through the indicated problems.  When finished, return to the main page and click on the “Solutions” link.

8.3 Transformations Revisited (Möbius and Otherwise)   - Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 22: Moebius Transforms” and “Lecture 23: More Transforms” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 22: Moebius Transforms” (Windows Media Video) and “Lecture 23: More Transforms” (Windows Media Video)

Instructions: Click on the link above and scroll down to the indicated videos.  Click on “Video” to download the lecture in WMV format.  Once each video has downloaded, watch it in its entirety.  (Time: 52:50 minutes and 30 minutes.)

8.4 Confomal Mappings   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “12.1: Harmonic Functions and Conformal Mappings: A Local Property of Analytic Functions” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “12.1: Harmonic Functions and Conformal Mappings: A Local Property of Analytic Functions” (PDF)

Instructions: Click on the link above, then click on “Chapter 12: Harmonic Functions and Conformal Mappings.”  The reference will open in PDF.  Scroll down to page 1 of the document and read the indicated section.

• Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 24: Conformal Maps” Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 24: Conformal Maps” (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: 18:55 minutes).

8.5 Global Properties of Analytic Functions and the Open Mapping Theorem   - Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “12.3: Harmonic Functions and Conformal Mappings: Global Properties of Analytic Functions” Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “12.3: Harmonic Functions and Conformal Mappings: Global Properties of Analytic Functions” (PDF)

Instructions: Click on the link above, then click on “Chapter 12: Harmonic Functions and Conformal Mappings.”  The reference will open in PDF.  Scroll down to page 5 of the document and read the indicated section.

• Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 12, Problems 4 and 5” Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 12, Problems 4 and 5” (PDF)

Instructions: Click on the link and scroll down to the link to HW#12, 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 4.3, Problems 1-5; and 4.5, Problems 1-5” Link: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.3, Problems 1-5; and 4.5, Problems 1-5” (PDF)

Instructions: Click on the link and select “Chapter 4” which will open in PDF.  Scroll down to page 17 for the problems from section 4.3 and page 20 for the problems from section 4.5.  Work through the indicated problems.  When finished, return to the main page and click on the “Solutions” link.

• Assessment: University of California at Berkeley: Professor Michael Christ’s “Fall 2005 Semester Final Exam” Link: University of California at Berkeley: Professor Michael Christ’s “Fall 2005 Semester Final Exam” (PDF)

Instructions: Click on the link and then select the “Teaching” tab at the top. Click on “Mathematics 185 – Complex Analysis 2009, Course Homepage” and scroll down to the “Exams” section.  Select the hyperlink labeled “Math 185, Fall 2005 semester final exam” which will display the exam in PDF.  Take an hour for the exam.  When finished, return to the main page and click on the hyperlink labeled “solutions” to download the solutions in PDF.