# MA243: Complex Analysis

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.

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.*

This unit will take you 6 hours to complete

☐    Subunit 3.1: 1.5 hours

☐    Subunit 3.2: 1.5 hours

☐    Subunit 3.3: 3 hours

Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define convergent and Cauchy sequences. - State and use the Monotone Sequence Property. - State the Archimedean Property. - Define convergence of series and state some properties necessary for the convergence of a series. - Define absolute convergence. - Define and use the integral test. - Define uniform convergence of a sequence of functions. - State a sufficient condition for the limit of a sequence of functions on a domain to be continuous on that domain. - State several sufficient conditions for the convergence of the integrals of a sequence of functions. - Define power series and define region and radius of convergence. - State the region of convergence for the geometric series and use it to solve problems involving convergence of power series. - Use the power series expansion for ez to calculate the power series expansion of the trigonometric functions.

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.

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.

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).

• Assessment: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: HW # 2, Problems 4-6, HW #4, Problem 1 Link: Washington University in St. Louis: Professor M. Victor Wickerhauser’s Complex Variables: HW # 2, Problems 4-6 (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.