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.
Unit 2 Time Advisory
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.
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
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).
Terms of Use: Please respect the copyright and terms of use
displayed on the webpages above.
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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.
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 “complex-differentiable.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.Assessment: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 4, 5, 6, 7, 14-18” Link: University of Illinois, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 1, Problems 4, 5, 6, 7, 14-18” (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 “complex-differentiable.”
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.