MATH / MTHE 326: Functions of a Complex Variable

__ Final Exam:__

Monday, December 16th from 9 AM -12 PM in Bartlett Gym - Main Floor

Bring Your Queen's ID

__December Office Hours:__

MTW, 9th, 10th and 11th, 10:30 AM - 12:00 PM and 1:00 PM - 4:30 PM

**Book:** Fundamentals of Complex Analysis with Applications to Engineering and Science, 3rd Edition by E. B. Saff and A. D. Snider (ISBN-10: 0139078746). This is available in the bookstore.

**Credit Where It's Due: ** The lecture notes for the class were based off of those of Professor Mike Roth, who has taught a version of this course in years past. The homework assignments and solutions, aside from occasional modifications, were originally his.

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

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

**Exams:
**Midterm: October 30th, 6:30-8:30 PM, Stirling Hall A, The Exam, The Solutions

Final: December 16th, 9-12 AM, Bartlett Gym - Main Floor

**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 signicantly 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 for students of mathematics, physics, and engineering. We will focus on a careful development of the theory as well as some applications to physical problems, often concerning harmonic functions. Proofs will be written, problems will be solved, memories will be made.

**Exam Prep:**

The Review Packet, The Solutions

**The Course So Far:**

Day | Description | Notes | Practice Problems |

Week One |
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M 9/9 | Course Introduction: Why we care and operations with complex numbers | Notes 1 | §1.1: 8,10 |

T 9/10 | Geometric Interpretation of Complex Numbers | Notes 2 | §1.2: 10 §1.3: 8, 11,12 §1.5: 6, 12,18 |

Th 9/12 | Visualizing Complex Mappings | Notes 3 | §2.1: 6 |

Week Two |
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M 9/16 | Loci and Mobius Maps | Notes 4 | §1.2: 7 |

T 9/17 | Finish Mobius Maps, Exponentials of Complex Numbers | Notes 5 | §7.3: 7 §7.4: 1,21 §1.4: 5, 7, 11 |

Th 9/19 | Complex Logarithms, Complex Exponentiation | Notes 6 | §3.3: 1, 5; §3.5: 1, 4 |

Week Three |
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M 9/23 | Limits and Continuity | Notes 7 | §2.2: 6,7,17 |

T 9/24 | Limits, Continuity, and Complex Differentiation | Notes 8 | §2.1: 1 §2.3: 1,4,6,10 |

Th 9/26 | Precursor to the Cauchy Riemann Theorem | Notes 9 | |

Week Four |
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M 9/30 | The Cauchy-Riemann Theorem | Notes 10 | §2.4: 3,5,15 |

T 10/1 | Proof of Cauchy-Riemann Theorem, Conformality | Notes 11 | §7.2: 6 |

Th 10/3 | Harmonic Functions | Notes 12 | §2.5: 1,2,3,5 |

Week Five |
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M 10/7 | Finish Harmonic Functions, Begin Differentiation of Elementary Functions | Notes 13 | §3.3: 6 |

T 10/8 | Finish Differentiation of Elementary Functions | Notes 14 | §3.2: 8, 9, 10 |

Th 10/10 | Contour Integration, Properties and Bounds | Notes 15 | §4.1: 8; §4.2: 6, 12 |

Week Six |
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M 10/14 | Thanksgiving, No Class | ||

T 10/15 | Bounds of Complex Integrals, Fundamental Theorem of Complex Calculus | Notes 16 and 17 | § 4.3: 2, 4, 5, 7 |

Th 10/16 | Cauchy's Theorem | Notes 16 and 17 | §4.1: 5, §4.4:10, 11, 18 |

Week Seven |
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M 10/21 | Cauchy's Theorem: Version 2 | Notes 18 | §4.4: 1, 3, 4, |

T 10/22 | Cauchy's Theorem: Version 3 | Notes 19 | §4.5: 1 |

Th 10/24 | Consequences of Cauchy's Theorem | Notes 20 | |

Week Eight |
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M 10/28 | Cauchy's Integral Formula, Integrals of Cauchy Type | Notes 21 | §4.5: 2, 8, 13 |

T 10/29 | Cauchy's Differentiation Formula, Liouville's Theorem | Notes 22 | §4.6: 1, 3,5,6 |

Th 10/31 | Maximum modulus theorem | Notes 23 | |

Week Nine |
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M 11/4 | Global Max Modulus and Max/Min Principle for Harmonic Functions | Notes 24 | §4.6: 8, 14; §4.7: 4, 6 |

T 11/5 | Dirichlet Problem and Poisson's Formula | Notes 25 | §4.7: 8, 9, 10, 11 |

Th 11/7 | Proof of Poisson's Formula, Different Types of Convergence | Notes 26 | |

Week Ten |
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M 11/11 | Uniform Convergence and Analytic Functions, Power Series | Notes 27 | §5.1: 10, 11; §5.4: 5, 8 |

T 11/12 | Taylor Series | Notes 28 | §5.2: 1, 2, 11 §5.3: 1, 4, 5, 10 |

Th 11/14 | Laurent Series | Notes 29 | §5.5: 1, 2, 6, 7 |

Week Eleven |
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M 11/18 | Classification of Singularities and Cauchy's Residue Theorem | Notes 30 | §5.6: 1, 2, 5, 6 |

T 11/19 | Residue theorem and calculation of residues | Notes 31 | §6.1: 1,2,5,6,7 |

Th 11/21 | Finishing Residues and Evaluating Trigonometric Integrals | Notes 32 | §6.2: 1, 4, 6 |

Week Twelve |
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M 11/25 | "Big D" Method of Evaluating Improper Integrals | Notes 33 | §6.3: 1, 3, 4, 5 §6.4: 1, 3, 4, 6 |

T 11/26 | "Big D" Method With Short Hops | Notes 34 | §6.5: 2, 3, 4, 5, |

Th 11/28 | Course Review | Notes 35 |

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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