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.
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 Time Advisory
This unit will take you 13 hours to complete
☐ Subunit 4.1: 2 hours
☐ Subunit 4.2: 2 hours
☐ Subunit 4.3: 2 hours
☐ Subunit 4.4: 7 hours
Unit4 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define linear fractional and Möbius transformations.
- State and prove certain essential properties of Möbius
transformations.
- Define the cross ratio.
- Define what it means for a function of a complex variable to map a
point to infinity.
- State properties of the extended complex plane.
- Find the point mapped by a Möbius transformation to infinity and the
point to which infinity is mapped.
- Given the function values for three distinct points, calculate a
Möbius transformation which agrees with those three mappings.
- State and verify properties of the exponential, trigonometric, and
hyperbolic functions given the definition of the complex
exponential.
- Define the complex logarithm and define a branch of the logarithm.
- Define the principal argument and the argument of a complex number.
- Show that a well-defined branch of the logarithm is differentiable.
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.
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.
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.
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 9: Complex Trig Functions and the Log Function (YouTube) and Lecture 10: Complex Powers of Complex Numbers (YouTube)
Instructions: Click on the links above to play the videos. (Time: 1 hour, 6 minutes; 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, Urbana-Champaign: Professor Robert Ash and W.P. Novinger’s Complex Variables: “Chapter 2.2, Problems 2, 5, 6” Link: University of Illinois, Urbana-Champaign: 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.