MTHE 228: Complex Analysis

Final Exam

The final exam is from 7-10 PM on Thursday, April 23rd in Bartlett Gym (Enter PEC).

Queen's Approved, Non-Programmable Calculators are permitted, but no other resources.

Here is an updated Review Guide

Here's a list of Conceptual Questions, and Conceptual Solutions

Practice Final, Practice Final Solutions

Office Hours in Jeff 512 (or nearby) from 9 AM-12 PM and 1-5 PM on April 20th-23rd, unless otherwise stated.

Here is a hastily made video with the lost content from the last lecture. There will be a bonus question on the final concerning this material, but you are not otherwise responsible for it.

Midterm Exam

Exam, Exam Solutions

Practice Exam, Practice Exam Solutions

**Lecturer:** Dr. Thomas A. Hulse (you can call me "Tom")

**Location:**

Monday: Jeffery Hall 128, 10:30-11:20 AM

Tuesday: Jeffery Hall 126, 8:30-9:20 AM

Wednesday: Jeffery Hall 126, 10:30-11:20 AM

**Office:** Jeffery Hall 512 ph.: x3-6299

**e-mail:** hulse@mast.queensu.ca

**TA**: Andrew Staal, Meghan Beattie
**Tutorial:** Wednesdays 11:30 AM - 12:20 PM

Ahmed through Laurenzi will be in Jeff 225 with Meghan Beattie

Lewis through Xu will be in Jeff 234 with Andrew Staal

**Book:** The textbook for this course is *Fundamentals of Complex Analysis with Applications to Engineering and Science Third Edition*, by E. B. Saff and A. D. Snider (ISBN-10: 0139078746). This is available in the bookstore.

**Course Information: ** It is required that you read and understand the course syllabus.

**Homework: **Watch this page for homework assignments and solutions.

**Introduction: ** Calculus is a powerful mathematical tool which has proven to be of profound practical use in the sciences and one of the great achievements of human thought.

But you know that already.

The theory of calculus with complex numbers, also known as the theory of functions of a complex variable, is significantly richer than just the extension of calculus to the real plane. The beautiful underlying algebraic and geometric structure of the complex numbers allows us to describe a large class of extremely well-behaved and physically-relevant functions with a very small number of conditions and allows us to better understand functions of a real variable. It's an exciting twist on a well-known story, and will quite possibly answer many questions you never knew you had.

This course is an introduction to complex analysis, intended primarily for engineering students. Some attention will be paid to theory but mainly this course will be focused on developing your ability to work in the realm of calculus with complex numbers.

**The Course So Far:**

Day | Description | Notes | Practice Problems |

Week One |
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M 1/5 | Course Introduction. Complex Numbers, Operations | Notes 1 | Section 1.1: 6,7,10,15,16 |

T 1/6 | Complex Plane, Geometric Objects, Conjugation | Notes 2 | Section 1.2: 4, 5, 7 |

W1/7 | Vector Representation and Polar Form | Notes 3 | Section 1.3: 5,6,7,15,16 |

Week Two |
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M 1/12 | The Exponential Function | Notes 4 | Section 1.4: 1,4,10,12,14 |

T 1/13 | Powers and Roots | Notes 5 | Section 1.5: 4,5,11 |

W1/14 | Planar Sets | Notes 6 Tutorial Problems |
Section 1.6: 2,3,4,8
Assignment 1 Due |

Week Three |
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M 1/19 | Properties of Domains, Boundedness, and Beginning Functions | Notes 7 | Section 1.6: 5,19,21 Section 2.1: 2 |

T 1/20 | Functions of a Complex Variable, Visualizing the Range | Notes 8 | Section 2.1: 1,3,5, 6b |

W1/21 | Limits and Continuity | Notes 9 Tutorial Problems |
Section 2.2: 7,11,12 Assignment 2 Due |

Week Four |
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M 1/26 | Infinity, Admissibility, Differentiability | Notes 10 | Section 2.2: 11, 15, 25 |

T 1/27 | Rules of Differentiation, Analyticity | Notes 11 | Section 2.3: 7,9,14 [L'Hopital's Rule] |

W1/28 | Cauchy-Riemann Equations | Notes 12 Tutorial Problems |
Section 2.4: 2,5,10,11,13 Assignment 3 Due |

Week Five |
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M 2/2 | Zero Derivative and Harmonic Functions | Notes 13 | Section 2.4: 12 Section 2.5: 1,4,5,6 |

T 2/3 | More Harmonic Functions and The Fundamental Theorem of Algebra | Notes 14 | Section 2.5: 3 Section 3.1: 3 |

W2/4 | Factoring, Polynomial Long Division and Taylor Expansions | Notes 15 Tutorial Problems [See also: Example 2 on Page 101-102] |
Section 3.1: 5,8,11 Assignment 4 Due |

Week Six |
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M 2/9 | Partial Fractions Decomposition, Returning to the Exponential Function | Notes 16 | Section 3.1: 13, 15 Section 3.2: 1,2,4 |

T 2/10 | Complex Trigonometric and Hyperbolic Functions, Logarithms Begin | Notes 17 | Section 3.2: 5,9, 11,13,15,17,19 |

W2/11 | Complex Logarithms | Notes 18 Tutorial Problems |
Section 3.3: 1,3,5, 8, 9, 12, 13, Assignment 5 Due |

Week Off, 2/16-2/20 | |||

Week Seven |
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M 2/23 | Boundary Value Problems for Harmonic Functions | Notes 19 | Section 3.4: All of them |

T 2/24 | Complex Powers | Notes 20 | Section 3.5: 1,3,4,6,15a |

W2/25 | Contours | Notes 21 Tutorial Problems |
Section 4.1: 1,6, 7, 8, 9 Assignment 6 Due |

Week Eight |
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M 3/2 | Intro to Contour Integrals | Notes 22 | Section 4.1: 5,10 Section 4.2: 3 Exam Tonight! |

T 3/3 | Evaluating Contour Integrals | Notes 23 | Section 4.2: 5,6,7,8,9,10,11 |

W3/4 | Fundamental Theorem of Calculus on Contours | Notes 24 Tutorial Problems |
Section 4.2: 14 Section 4.3: 1,2,4,6,7 |

F3/6 | Assignment 7 Due by 10:20 AM in Jeff 512 mail slot | ||

Week Nine |
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M 3/9 | Continuous Deformation and Cauchy's Integral Theorem | Notes 25 | Section 4.4: 1,3,9,10 |

T 3/10 | Cauchy's Integral Formula | Notes 26 | Section 4.5: 1,2,3a,3b,3c |

W3/11 | Cauchy's Generalized Integral Formula | Notes 27 Tutorial Problems |
Section 4.5: 3d, 3e, 3f, 4,5,6,7 Assignment 8 Due |

Week Ten |
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M 3/16 | Liouville's Theorem and The Maximum Modulus Principle | Notes 28 | Section 4.6: 1,2,5,6,7,8,14 |

T 3/17 | Using Max Modulus and Max/Min Principle for Harmonic Functions | Note 29 | Section 4.6: 10,16,17 Section 4.7: 1,2,4 |

W3/18 | Dirichlet's Problem, Poisson's Integral Formula, and (Re-)Introduction to Series | Notes 30 Tutorial Problems |
Section 4.7: 6,8, 11 Section 5.1: 1,2,5,7 Assignment 9 Due |

Week Eleven |
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M 3/23 | Evaluations, More Series Examples | Notes 31 | Section 5.1: 13,14 |

T 3/24 | Taylor Series | Notes 32 | Section 5.2: 1,3,4,5,6,8 |

W3/25 | Power Series and Laurent Series | Notes 33 Tutorial Problems |
Assignment 10 Due Section 5.3: 3, 5a, 5b, 10 Section 5.5: 1,2,3,5 |

Week Twelve |
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M 3/30 | Finishing Laurent Series, Classifying Singularities | Notes 34 | Section 5.5: 6,9,11 |

T 3/31 | Classifying Singularities, Cauchy's Residue Theorem | Notes 35 | Section 5.6: 1,2 Section 6.1: 1,3 |

W4/1 | Using Cauchy's Residue Theorem to Evaluate Real Integrals | Notes 36 Tutorial Problems |
Assignment 11 Due Assignment 12. Section 6.2: 1,2,3,4,8 Section 6.3: (Optional) 1,2,3,4,5,6,7 |

Here's a picture of me wearing only one shoe and standing in front of the Oscar Mayer Weinermobile:

And here's a link to a video of my Linear Algebra Class being disrupted.

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