Complex Analysis
Purpose of Course showclose
This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Simply put, you will be working with lines and sets and very specific functions on the complex plane—drawing pictures of them and teasing out all of their idiosyncrasies. You will again find yourself calculating line integrals, just as in multivariable calculus. However, the techniques you learn in this course will help you get past many of the seeming deadends you ran up against in calculus. Indeed, most of the definite integrals you will learn to evaluate in Unit 7 come directly from problems in physics and cannot be solved except through techniques from complex variables.
We will begin by studying the minimal algebraically closed extension of real numbers: the complex numbers. The Fundamental Theorem of Algebra states that any nonconstant polynomial with complex coefficients has a zero in the complex numbers. This makes life in the complex plane very interesting. We will also review a bit of the geometry of the complex plane and relevant topological concepts, such as connectedness.
In Unit 2, we will study differential calculus in the complex domain. The concept of analytic or holomorphic function will be introduced as complex differentiability in an open subset of the complex numbers. The CauchyRiemann equations will establish a connection between analytic functions and differentiable functions depending on two real variables. In Unit 3, we will review power series, which will be the link between holomorphic and analytic functions. In Unit 4, we will introduce certain special functions, including exponentials and trigonometric and logarithmic functions. We will consider the Möbius Transformation in some detail.
In Units 5, 6, and 7 we will study Cauchy Theory, as well as its most important applications, including the Residue Theorem. We will compute Laurent series, and we will use the Residue Theorem to evaluate certain integrals on the real line which cannot be dealt with through methods from real variables alone. Our final unit, Unit 8, will discuss harmonic functions of two real variables, which are functions with continuous second partial derivatives that satisfy the Laplace equation, conformal mappings, and the Open Mapping Theorem.
Course Information showclose
Course Designer: Clare Wickman
Primary Resources:
San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis
Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis
University of Illinois, UrbanaChampaign: Professor Robert Ash’s and W.P. Novinger’s Complex Variables
Requirements for Completion: Completion of all readings, assignments, and assessments.
Time Commitment: 113 hours
Tips/Suggestions: In many of the problems you will encounter in this course, you will gain a clearer understanding of the setup if you draw a picture (e.g. of the contour or the set).
Learning Outcomes showclose
 Fundamentals: Manipulate complex numbers in various representations, define fundamental topological concepts in the context of the complex plane, and define and calculate limits and derivatives of functions of a complex variable.
 Key Functions: Represent analytic functions as power series on their domains and verify that they are welldefined. Define a branch of the complex logarithm. Classify singularities and find Laurent series for meromorphic functions.
 Key Results: State and prove fundamental results, including: Cauchy’s Theorem and Cauchy’s Integral Formula, the Fundamental Theorem of Algebra, Morera’s Theorem and Liouville’s Theorem. Use them to prove related results.
 Key Application: Calculate contour integrals. Calculate definite integrals on the real line using the Residue Theorem.
 Mappings: Define linear fractional transformations and prove their essential characteristics. Find the image of a region under a conformal mapping. State, prove, and use the Open Mapping Theorem.
Course Requirements showclose
√ Have access to a computer.
√ Have continuous broadband Internet access.
√ Have the ability/permission to install plugins or software (e.g. Adobe Reader or Flash).
√ Have the ability to download and save files and documents to a computer.
√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).
√ Have competency in the English language.
√ Have read the Saylor Student Handbook.
√ Have completed MA101, MA102, MA103, MA211, and MA221 or their equivalents. Completion of MA241 or its equivalent is recommended, but concurrent enrollment is acceptable.
Unit Outline show close
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Unit 1: Complex Numbers
The set of real numbers is a field that is not algebraically closed. In other words, some polynomials with real coefficients do not have a root in the real numbers (e.g., x^{2}+1=0 has no solution in the real numbers). In order to extend the set of real numbers to a larger field that is algebraically closed, we will consider the plane equipped with an addition rule and a multiplication rule that extends the corresponding operations in the real numbers. This new field is known as “the complex numbers.” In the complex numbers, there is a number whose square is equal to negative one: the “imaginary number” i.
Unit 1 Time Advisory show close
In this unit, we will study the algebraic and geometric properties of the complex numbers. Complex numbers are akin to 2D vectors, and therefore addition and multiplication of complex numbers have a very nice geometric interpretation. We will also consider some basic topology of the complex plane: open sets, paths, etc. In order to discuss functions that become infinite as the variable approaches a given point, we will introduce the extended complex plane and its representation as the unit sphere in the three dimensional real space by means of the stereographic projection.
Unit 1 Learning Outcomes show close

1.1 The Field of Complex Numbers
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “1.01.1: Introduction and Definitions and Algebraic Properties”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “1.01.1: Introduction and Definitions and Algebraic Properties” (PDF)
Instructions: Scroll down to page 5 (marked page 1) of the document and read the indicated sections. Please note that you will be returning to this resource throughout the course, so you may prefer to save the PDF to your desktop for quick reference.
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 1: Introduction
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 1: Brief Outline (YouTube)
Instructions: In this 30 minute lecture, Professor Glesser will briefly define the complex numbers and then lay out the key concerns of Complex Analysis. Pay special attention to his explanation of the difficulties of establishing the existence of limits of functions on the complex plane.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “1.01.1: Introduction and Definitions and Algebraic Properties”
 1.2 The Complex Plane

1.2.1 Complex Arithmetic and Polar Representation
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s: A First Course in Complex Analysis: “1.2: From Algebra to Geometry and Back”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s: A First Course in Complex Analysis: “1.2: From Algebra to Geometry and Back” (PDF)
Instructions: Scroll down to page 7 (marked page 3) 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 2: The Field of Complex Numbers
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 2: The Field of Complex Numbers (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, 12 minutes)
About the Media: In this lecture, Professor Glesser will establish the fact that the complex numbers are a field.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s: A First Course in Complex Analysis: “1.2: From Algebra to Geometry and Back”

1.2.2 DeMoivre’s Theorem
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “1.3: Complex Numbers: Rational Powers”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “1.3: Complex Numbers: Rational Powers” (PDF)
Instructions: Click on the link above, then click on “Chapter 1: Complex Numbers.” The reference will open in PDF. Scroll down to page 2 of the document and read the indicated section.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “1.3: Complex Numbers: Rational Powers”

1.2.3 Geometric Properties
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “1.3: Geometric Properties”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “1.3: Geometric Properties” (PDF)
Instructions: Scroll down to page 10 (marked page 6) 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 3: Polar Form and Lecture 4: The Geometry of Complex Numbers
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 3: Polar Form and Lecture 4: The Geometry of Complex Numbers (Flash Video)
Instructions: Click on the link above, then scroll down to lecture 3. Click on “click to play” to play the video. (Time: 1 hour, 9 minutes)
Next, scroll down to lecture 4. Click on “click to play” to play the video. (Time: 1 hour, 50 minutes)
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “1.3: Geometric Properties”

1.2.4 Elementary Topology
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “2.12.2: Foundations of Complex Analysis: Three Approaches and Point Sets in the Complex Plane”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “2.12.2: Foundations of Complex Analysis: Three Approaches and Point Sets in the Complex Plane” (PDF)
Instructions: Click on the link above, then click on “Chapter 2: Foundations of Complex Analysis.” The reference will open in pdf. Scroll down to page 1 of the document and read the indicated sections.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “2.12.2: Foundations of Complex Analysis: Three Approaches and Point Sets in the Complex Plane”

1.3 The Extended Plane and the Stereographic Projection
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “2.4: Foundations of Complex Analysis: Extended Complex Plane”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “2.4: Foundations of Complex Analysis: Extended Complex Plane” (PDF)
Instructions: Click on the link above, then click on “Chapter 2: Foundations of Complex Analysis.” The reference will open in pdf. Scroll down to page 6 of the document and read the indicated sections.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 1”
Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 1” (PDF)
Instructions: Click on the first link and scroll down to the link to HW#1, which will open in PDF. Work through all problems. When finished, return to the first page and click on the “solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash’s and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 1, 2, 3, 10, 12, 13”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash’s and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 1, 2, 3, 10, 12, 13” (PDF)
Instructions: Click on the link and select “Chapter 1” which will open in PDF. Then scroll down to page 9 and work through the indicated problems. When finished, return to the main page and click on the “Solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “2.4: Foundations of Complex Analysis: Extended Complex Plane”

Unit 2: Complex Functions
In this unit, we will learn about functions that depend on a single complex variable that have values in the complex plane. We will begin by discussing complex differentiability, which will enable us to introduce the definition of a holomorphic function. A holomorphic function can be defined as a function differentiable in an open set, or as a function that is differentiable with a continuous derivative in an open set. These two possible definitions are equivalent according to Goursat’s Theorem.
Unit 2 Time Advisory show close
On the other hand, when interpreting a complex function as a function of two real variables (the real and imaginary parts of the complex argument), the separation of the function values into their real and imaginary parts will produce two realvalued functions each depending on two real variables. The Cauchy Riemann equations will provide us with the conditions that those two functions must satisfy in order for the original complex function to be holomorphic.
We will conclude this unit by discussing Laplace’s equation and harmonic functions, topics important both in Complex Analysis and in Partial Differential Equations.
Unit 2 Learning Outcomes show close

2.1 Introduction to Complex Functions
 Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 5: Powers and Graphs of Complex Functions
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 5: Powers and Graphs of Complex Functions (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 hours, 6 minutes).
About the Media: In this lecture, Professor Glesser will briefly define the complex numbers and then lay out the key concerns of Complex Analysis. Pay special attention to his explanation of the difficulties of establishing the existence of limits of functions on the complex plane.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Lecture: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 5: Powers and Graphs of Complex Functions

2.2 Limits and Continuity
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.1: Differentiation: First Steps”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.1: Differentiation: First Steps” (PDF)
Instructions: Scroll down to page 19 (marked page 15) 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.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.1: Differentiation: First Steps”

2.3 Derivative of a Complex Function
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.2: Differentiation: Differentiability and Holomorphicity”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.2: Differentiation: Differentiability and Holomorphicity” (PDF)
Instructions: Scroll down to page 21 (marked page 17) 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.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.2: Differentiation: Differentiability and Holomorphicity”

2.4 Holomorphic Functions
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “3.3: Complex Differentiation: Analytic Functions”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “3.3: Complex Differentiation: Analytic Functions” (PDF)
Instructions: Click on the link above, then click on “Chapter 3: Complex Differentiation.” The reference will open in PDF. Scroll down to page 4 of the document and read the indicated section.
Note that Professor Chen defines “analytic” to mean complexdifferentiable functions, while Professor Beck et al. use “holomorphic” instead. In fact, “holomorphic” is the more technically correct word; “analytic” more generally refers to functions which can be represented as power series. However, a key result of Complex Analysis (which will be proven later) is that a function of a complex variable is holomorphic (on a domain) if and only if it is analytic (on that domain).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “3.3: Complex Differentiation: Analytic Functions”

2.5 The CauchyRiemann Equations
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.4: Differentiation: The CauchyRiemann Equations”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.4: Differentiation: The CauchyRiemann Equations” (PDF)
Instructions: Scroll down to page 24 (marked page 20) 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 6: Limits, Analyticity, and the CauchyRiemann Equations
Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 6: Limits, Analyticity, and the CauchyRiemann Equations (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, 12 minutes).
Note that this lecture covers subunits 2.22.5.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “2.4: Differentiation: The CauchyRiemann Equations”

2.6 Laplace’s Equation and Harmonic Conjugates
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “3.6: Complex Differentiation: Laplace’s Equation and Harmonic Conjugates”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “3.6: Complex Differentiation: Laplace’s Equation and Harmonic Conjugates” (PDF)
Instructions: Click on the link above, then click on “Chapter 3: Complex Differentiation.” The reference will open in PDF. Scroll down to page 10 of the document and read the indicated section.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 2, Problems 2 and 3”
Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 2, Problems 2 and 3” (PDF)
Instructions: Click on the link and scroll down to the link to HW#2, which will open in PDF. Work through the indicated problems. When finished, return to the first page and click on the “solutions” link.
Recall that some authors use “analytic” and “holomorphic” interchangeably to mean “complexdifferentiable.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 4, 5, 6, 7, 1418”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 4, 5, 6, 7, 1418” (PDF)
Instructions: Click on the link and select “Chapter 1” which will open in PDF. Then scroll down to page 9 and work through the indicated problems. When finished, return to the main page and click on the “Solutions” link.
Recall that some authors use “analytic” an “holomorphic” interchangeably to mean “complexdifferentiable.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “3.6: Complex Differentiation: Laplace’s Equation and Harmonic Conjugates”

Unit 3: Power Series
A power series with complex coefficients can be considered a generalization of a polynomial function. Since the terms are polynomials, they are also analytic functions. Therefore, it seems reasonable to expect that the sum of the series will be an analytic function. In this unit, we will study the basic properties of power series in order to prepare us to use them to represent analytic functions in Unit 6.
Unit 3 Time Advisory show close
If you are already familiar with the material in this unit, i.e. through MA241, then feel free to move on to Unit 4. See learning outcomes below.
Unit 3 Learning Outcomes show close

3.1 Sequences and Series
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.1: Power Series: Sequences and Completeness” and “7.2 Power Series: Series”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.1: Power Series: Sequences and Completeness” and “7.2 Power Series: Series” (PDF)
Instructions: Scroll down to page 74 (marked page 70) of the document and read the indicated sections.
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.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.1: Power Series: Sequences and Completeness” and “7.2 Power Series: Series”

3.2 Uniform Convergence
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.3: Power Series: Sequences and Series of Functions”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.3: Power Series: Sequences and Series of Functions” (PDF)
Instructions: Scroll down to page 79 (marked page 75) 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.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.3: Power Series: Sequences and Series of Functions”

3.3 Power Series
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.4: Power Series: Region of Convergence”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.4: Power Series: Region of Convergence” (PDF)
Instructions: Scroll down to page 82 (marked page 78) 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 17: Power Series”
Link: Louisiana Tech University: Professor Bernd Schröder’sIntroduction to Complex Analysis: “Lecture 17: Power Series” (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: 56 minutes).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: HW # 2, Problems 46, HW #4, Problem 1
Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: HW # 2, Problems 46 (PDF) and Homework #4, Problem 1 (PDF)
Instructions: Click on the first link and scroll down to the links to Homework 2 and 4, which will open in PDF. Work through the indicated problems. When finished, return to the first page and click on the “solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “7.4: Power Series: Region of Convergence”

Unit 4: Examples of Complex Functions
Now that we have reviewed power series, we will use them to define the exponential function and the functions cosine and sine in the complex domain. We will see how these functions are related. In order to define an inverse for the exponential function, we will need to use the complex logarithm. Pay close attention to the construction of the different branches of the logarithm.
Unit 4 Time Advisory show close
We will also study Möbius Transformations, a type of a linear fractional transformation (the quotient of two polynomials of degree one or constant), learning about some of their geometric properties, including how they transform the Projective Plane.
Unit 4 Learning Outcomes show close

4.1 Möbius Transformations
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.1: Examples of Functions: Möbius Transformations”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.1: Examples of Functions: Möbius Transformations” (PDF)
Instructions: Scroll down to page 30 (marked page 26) of the document and read the indicated section.
Much more will be said about Möbius Transformations in Unit 8.
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.  Web Media: YouTube: Professor Douglas Arnold and Jonathan Rogness’s “Mobius Transformations Revealed”
Link: YouTube: Professor Douglas Arnold and Jonathan Rogness’s “Mobius Transformations Revealed” (YouTube)
Instructions: Click on the link above and watch this short and beautiful video on the relationship of the Mobius Transformation to the stereographic projection. (Time: 2:45 minutes)
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.1: Examples of Functions: Möbius Transformations”

4.2 The Cross Ratio
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.2: Examples of Functions: Infinity and the Cross Ratio”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.2: Examples of Functions: Infinity and the Cross Ratio” (PDF)
Instructions: Scroll down to page 33 (marked page 29) 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.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.2: Examples of Functions: Infinity and the Cross Ratio”

4.3 Exponential and Trigonometric Functions
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.4: Examples of Functions: Exponential and Trigonometric Functions”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.4: Examples of Functions: Exponential and Trigonometric Functions” (PDF)
Instructions: Scroll down to page 38 (marked page 34) 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.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.4: Examples of Functions: Exponential and Trigonometric Functions”

4.4 The Logarithm and Complex Exponentials
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.5: Examples of Functions: The Logarithm and Complex Exponentials”
Link: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.5: Examples of Functions: The Logarithm and Complex Exponentials” (PDF)
Instructions: Scroll down to page 41 (marked page 37) of the document and read the indicated section.
The process of constructing the logarithm is tricky because, simply put, it requires us to find a good definition of the “argument” of a complex number. However, as we know, if z=re^{i?}, then z=re^{i(?+2πk)} for any integer value of k. Here, ? is the argument of z, and for any z it is cannot be uniquely determined unless we choose a range to which argument will be restricted. This situation will lead us to define the different branches of the logarithm, and will necessitate careful bookkeeping of arguments whenever the logarithm is encountered, most especially in Unit 7 when we begin to calculate residues.
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 7: Complex Trig Functions and the Log Function and Lecture 8: Complex Powers of Complex Numbers
Link: Suffolk University: Professor Adam Glesser’s Lecture 7: Complex Trig Functions and the Log Function (Flash Video) and Lecture 8: Complex Powers of Complex Numbers (Flash Video)
Instructions: Click on the first link above, then scroll down to Lecture 7. Click on “click to play” to play the video. (Time: 1 hour, 6 minutes.)
Next, click on the second link above, and scroll down to Lecture 8. Click on “click to play” to play the video. (Time: 42 minutes.)
Note that these lectures cover the material in subunits 4.3 and 4.4.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.2, Problems 2, 5, 6”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.2, Problems 2, 5, 6” (PDF)
Instructions: Click on the link, scroll down to page 17, and work through the indicated problems. When finished, return to the main page and click on the “Solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of California at Berkeley: Professor Michael Christ’s “Math 185, Fall 2005 First Midterm Exam”
Link: University of California at Berkeley: Professor Michael Christ’s “Math 185, Fall 2005 First Midterm 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 first midterm 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 for Math 185 Fall 2005 midterm exam 1” to download the solutions in PDF.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: San Francisco State University: Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka’s A First Course in Complex Analysis: “3.5: Examples of Functions: The Logarithm and Complex Exponentials”

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.
Unit 5 Time Advisory show close
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.
Unit 5 Learning Outcomes show close

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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 R^{2}.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.1: Complex Integrals: Curves in the Complex Plane”
 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.2: Complex Integrals: Contour Integrals”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.3: Complex Integrals: Inequalities for Contour Integrals”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “4.4: Complex Integrals: Equivalent Curves”

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: CauchyGoursat Theorem”
Link: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 13: CauchyGoursat 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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 5.5.25.5.6. This latter material will be covered again in the video lecture listed in subunit 5.5.6 at a slower pace.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”
 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.
 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”

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.
 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”

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 5.5.25.6. Also note that much of the material in this video was presented at a faster pace in the video lecture for subunit 5.4.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 4, Problems 2 and 3” and “HW #9, Problem 4”
Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 4, Problems 2 and 3” (PDF) and “HW # 9, Problem 4” (PDF)
Instructions: Click on the link and scroll down to the links to HW#4 and HW #9, which will open in PDF. Work through the indicated problems. When finished, return to the first page and click on the “solutions” links.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.1, Problems 13, 2.2, Problems 5 and 13, and 2.4, Problems 13, 1619”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.1, Problems 13; 2.2, Problems 5 and 13; and 2.4, Problems 13, 1619” (PDF)
Instructions: Click on the link and select “Chapter 2” which will open in PDF. Problems for section 2.1 begin on page 9, those for sections 2.2 begin on page 17, and those for section 2.4 on page 26. When finished, return to the main page and click on the “Solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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.
Unit 6 Time Advisory show close
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 Learning Outcomes show close

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.
 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”

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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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.
 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”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “7.3: Taylor Series, Uniqueness, and the Maximum Principle: Uniqueness”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”
 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.1: Isolated Singularities and Laurent Series: Removable Singularities”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.2: Isolated Singularities and Laurent Series: Poles”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: 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, UrbanaChampaign: 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.3: Isolated Singularities and Laurent Series: Essential Singularities”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “8.4: Isolated Singularities and Laurent Series: Isolated Singularities at Infinity”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW #8”
Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: “HW # 8” (PDF)
Instructions: Click on the link and scroll down to the HW #8, which will open in PDF. Work through the indicated problems. When finished, return to the first page and click on the “solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.1, Problems 36”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.1, Problems 36” (PDF)
Instructions: Click on the link and select “Chapter 4” which will open in PDF. Scroll down to page 5 and work through the indicated problems. When finished, return to the main page and click on the “Solutions” link.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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.
Unit 7 Time Advisory show close
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.
Unit 7 Learning Outcomes show close

7.1 The Residue Theorem
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.110.2: Residue Theory: Cauchy’s Residue Theorem and Finding the Residue”
Link: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.110.2: 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.110.2: Residue Theory: Cauchy’s Residue Theorem and Finding the Residue”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.)
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.2, Problems 14, 10, 16, 20, 2224, 26”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.2, Problems 14, 10, 16, 20, 2224, 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “10.3: Residue Theory: Principle of the Argument”
 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: “11.111.2: 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: “11.111.2: 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.111.2: Evaluation of Definite Integrals: Introduction and Rational Functions on the Unit Circle”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.5: Evaluation of Definite Integrals: Bending Round a Singularity”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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 2629).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.2, Problems 79, 19”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.2, Problems 79, 19” (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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of California at Berkeley: Professor Michael Christ’s “Math 185, Fall 2005 Second Midterm Exam”
Link: University of California at Berkeley: Professor Michael Christ’s “Math 185, Fall 2005 Second Midterm 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 second midterm 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 for Math 185 Fall 2005 midterm exam 2” to download the solutions in PDF.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Reading: Imperial College, London: W.W.L.Chen’s Introduction to Complex Analysis: “11.6: Evaluation of Definite Integrals: Integrands with Branch Points”

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.
Unit 8 Time Advisory show close
We will revisit transformations (especially the Möbius transformation) and discuss conformal mappingsmappings 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 Learning Outcomes show close

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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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: MeanValue 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: MeanValue 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.4, Problems 57, 9, 10, 21, 23”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.4, Problems 57, 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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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: MeanValue and Maximum/Minimum Principle”

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.)
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 Lecture: Louisiana Tech University: Professor Bernd Schröder’s Introduction to Complex Analysis: “Lecture 22: Moebius Transforms” and “Lecture 23: More Transforms”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  Assessment: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.3, Problems 15; and 4.5, Problems 15”
Link: University of Illinois, UrbanaChampaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 4.3, Problems 15; and 4.5, Problems 15” (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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.  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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
 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”

Final Exam
 Final Exam: The Saylor Foundation's MA243 Final Exam
Link: The Saylor Foundation's MA243 Final Exam
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
 Final Exam: The Saylor Foundation's MA243 Final Exam