# MA243: Complex Analysis

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.

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 real-valued 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.

This unit will take you 13.5 hours to complete

☐    Subunit 2.1: 1.5 hour

☐    Subunit 2.2: 1.5 hour

☐    Subunit 2.3: 1.5 hour

☐    Subunit 2.4: 1.5 hour

☐    Subunit 2.5: 2.5 hours

☐    Subunit 2.6: 5 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define limits and continuity in the context of the complex plane and functions of a complex variable. - Define the derivative of a complex function, determine whether a function of a complex variable is holomorphic, and find the derivative if it exists. - State the Cauchy-Riemann equations and use them to determine whether a function is holomorphic. - State Laplace’s equation and define and find harmonic conjugates.

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 (YouTube)

Instructions: Click on the link above 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.

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](http://math.sfsu.edu/beck/complex.html) (PDF).  It may not be
altered in any way.
``````

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.

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 complex-differentiable 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).

2.5 The Cauchy-Riemann 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 Cauchy-Riemann 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 Cauchy-Riemann 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 Cauchy-Riemann Equations Link: Suffolk University: Professor Adam Glesser’s F09 Suffolk Math 481, Lecture 6: Limits, Analyticity, and the Cauchy-Riemann Equations (YouTube)

Instructions: Click on the link above to play the video (Time: 1 hour, 12 minutes).

Note that this lecture covers subunits 2.2-2.5.

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.

• 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 “complex-differentiable.”