## Final exam

Final exam is scheduled on Monday December 17, in Currie Gym Studio 1, starting at 9:00.

The exam will involve questions similar to the ones given in the homework assignments from §4.1–4.6, §5.1–5.6, and §6.1–6.7, which basically means that the study topics are

• Contour integration, Cauchy's formula
• Taylor series, Laurent series, their convergence
• Residue calculus, poles
• Using residues to compute real integrals

If you study the above subjects well, there is 99% chance you will succeed in the exam, and of course, 100% chance you will succeed in life.

Practice these problems. I will not post answers to these problems.

Please do not hesitate to drop by my office (Burnside 1123) during the exam period.

## Class schedule

 Date Topics Mo 9/10 Complex numbers [§1.1-§1.3] We 9/12 Fractional powers [§1.4]. Limits and continuity [§2.2]. Mo 9/17 Complex derivative. Cauchy-Riemann equations [§2.3] We 9/19 Analyticity [§2.4]. Exponential [§3.1] Mo 9/24 Trigonometric and hyperbolic trigonometric functions [§3.2-3.3]. We 9/26 Logarithm [§3.4-3.5] Mo 10/1 Complex powers [§3.6]. Inverse trigonometric functions [§3.7]. Harmonic functions [§2.5] We 10/3 Line integration [§4.1]. Complex line integration [§4.2] Mo 10/8 Thanksgiving We 10/10 Cauchy-Goursat theorem [§4.3]. Deformation of paths [§4.4] Mo 10/15 Midterm We 10/17 Fundamental theorem of calculus [§4.4]. Cauchy's formula [§4.5] Mo 10/22 Mean value property. Maximum and minimum modulus principles [§4.6] We 10/24 Liouville's theorem. Fundamental theorem of algebra [§4.6] Mo 10/29 Series and sequences [§5.1-5.2]. Uniform convergence [§5.3] We 10/31 Taylor series [§5.4] Mo 11/05 Taylor series [§5.5]. We 11/07 Laurent series [§5.6] Mo 11/12 Isolation of zeroes [§5.7]. Residues [§6.1] We 11/14 Isolated singularities [§6.2] Mo 11/19 Residue calculus [§6.3]. Integration over a circle [§6.4] We 11/21 Integrals over the real line [§6.5] Mo 11/26 Jordan's lemma [§6.6] We 11/28 Examples [§6.5-§6.6] Mo 12/03 Indented contours [§6.7] We 12/05 Branch cuts [§6.8]. Review of selected topics Mo 12/17 Final exam

## Past assignments

Assignment 1 due We, Sep 26 [solutions]
§1.2: 14, 17
§1.3: 11
§1.4: 14, 27, 32
§1.5: 8
§2.1: 11, 17
§2.2: 8, 9, 15
§2.3: 7, 13
§2.4: 3, 4, 15

Assignment 2 due We, Oct 10
§3.1: 17, 22
§3.2: 14, 28
§3.3: 11, 15
§3.4: 21, 22, 25
§3.5: 2, 3
§3.6: 11, 15, 19, 20
§3.7: 4

Assignment 3 due We, Oct 24
§2.5: 9
§4.1: 5, 6
§4.2: 4, 6, 14
§4.3: 15, 17, 21
§4.4: 4, 16, 19
§4.5: 2, 3, 15

Assignment 4 due We, Nov 7
§4.6: 5, 12
§5.1: 5, 11, 14
§5.2: 5, 7, 10, 15
§5.3: 3, 5
§5.4: 8, 15, 20

Assignment 5 due We, Nov 21
§5.5: 1, 7, 10a, 19
§5.6: 14, 21, 26
§5.7: 7
§6.1: 3, 6, 7
§6.2: 5, 6, 9, 25

Assignment 6 not to be graded
§6.3: 4, 5, 26, 31
§6.4: 3, 14
§6.5: 24, 25, 30
§6.6: 8, 9, 13
§6.7: 5, 8, 13
§6.8: 2, 14

Midterm: problems with solutions

## Course outline

Lectures: MW 14:35–15:55, Burnside 1B45

Tutorials: R 10:35–11:25 Trottier 2100, R 11:35–12:25 Burnside 1B23

Office hours: W 13:30–14:30

Instructor: Dr. Gantumur Tsogtgerel

Teaching assistants: Ibrahim Al Balushi, Padina Suky, and Mashbat Suzuki

Calendar description: Analytic functions, Cauchy-Riemann equations, simple mappings, Cauchy's theorem, Cauchy's integral formula, Taylor and Laurent expansions, residue calculus. Properties of one and two-sided Fourier and Laplace transforms, the complex inversion integral, relation between the Fourier and Laplace transforms, application of transform techniques to the solution of differential equations. The Z-transform and applications to difference equations.

Textbook: David A. Wunsch, Complex Variables with Applications, 3rd edition, Addison Wesley 2004

Topics: We will cover essentially all of Chapters 1-5, half of Chapter 6, and the beginning of Chapter 7. These topics can be described as follows.

• Complex differentiation, Cauchy-Riemann equations, holomorphy, harmonicity
• Basic transcendental functions
• Contour integration, Cauchy's formula
• Taylor series, Laurent series, analyticity
• Residue calculus
• Laplace transform