Instructor: Antoine Mellet
Office:Math Building 4104
Classes: TuTh 11:00am-12:15pm (MTH 0105)
Office Hours: Tu. 1:30-3:30pm
First day Hand-out
Textbook:
Complex Variables and Applications, Eighth Edition. James W. Brown and Ruel V. Churchill. Published by McGraw-Hill. ISBN 978-0-07-305194-9
Dates:
| Lecture dates | Sections | Assignment | |
|---|---|---|---|
| 1. Tu. Sept. 1 | 1-4 | Basic definitions; The complex plane Algebra of complex numbers; Modulus |
Homework #1 Due Tu. Sept. 15th |
| 2. Thu Sept. 3 | 4-6 | Triangle inequality; Complex conjugates; Exponential form of complex numbers |
|
| 3. Tu. Sept. 8 | 7-10 | Using the exponential form to compute products, powers and fractions n-th roots of a complex number |
|
| 4. Thu. Sept. 10 | 10-1texier2 | Application of complex numbers to AC circuits Neighborhood of a point, open sets, closed sets, boundary of a set Functions of a complex variable |
Homework #2 Due Th. Sept. 17th |
| 5. Tu. Sept. 15 | 13-15 | Complex functions as transformations of the complex plane. Mappings by the exponential function Limits: Definition and first properties |
|
| 6. Thu. Sept. 17 | 16-19 | Further properties of limits; Point at infinity; Stereographic projection. Continuity: Definition and properties Derivatives: Definition. First examples |
Homework #3 Due Th. Sept. 24th |
| 7. Tu. Sept. 22 | 19-23 | Derivatives: Further examples, formulas and properties. Cauchy-Riemann equations in rectangular and polar coordinates. |
|
| 8. Thu. Sept. 24 | 24-26 | Analytic functions, singular points, entire functions. Connected set. Harmonic functions, harmonic conjugates |
Homework #4 Due Th. Oct. 1st |
| 9. Tu. Sept. 29 | Review | ||
| 10. Thu. Oct. 1 | Midterm #1 | ||
| 11. Tu. Oct. 6 | 29-33 | Elementary functions: The exponential function, the logarithmic function Branches of a multiple valued function Complex exponents |
|
| 12. Thu. Oct 8 | 34-36 | Trigonometric functions, Inverse trigonometric functions Hyperbolic functions Application of harmonic conjugate function to electrostatic |
Homework #5 Due Th. Oct. 15th |
| 13. Tu. Oct. 13 | 37-39 | Calculus for complex valued functions of a real variable. Arc, Simple Closed Curve (Jordan Curve), smooth arc, contour. |
|
| 14. Thu. Oct. 15 | 40-45 | Contour integrals: Definition, properties and examples Contour integrals and antiderivatives |
Homework #6 Due Th. Oct. 22th |
| 15. Tu. Oct. 20 | 46-49 | Cauchy Theorem, Cauchy-Goursat Theorem Simply connected domain; Multiply connected domain: Principle of deformation |
|
| 16. Thu. Oct. 22 | 50 | First Cauchy integral formula and applications | Homework #7 Due Th. Oct. 29th |
| 17. Tu. Oct. 27 | 51 | Cauchy integral formula for the derivatives. Applications to the computation of complex integrals |
|
| 18. Thu. Oct.29 | 52-55 | Consequences of Cauchy's formula: Cauchy's inequality, Liouville theorem, Fundamental theorem of Algebra. Computing a complex integral: Review. Sequences of complex numbers. |
Homework #8 Due Th. Nov. 5th |
| 19. Tu. Nov. 3 | 56 | Infinite series - Geometric series Application of complex integrals to fluid flows. |
|
| 20. Thu. Nov. 5 | Review | ||
| 21. Tue. Nov. 10 | Midterm #2 | ||
| 22. Thu. Nov. 12 | 57-59 (64-66) | Power series. Radius and circle of convergence. Properties of power series Taylor's Theorem |
Homework #9 Due Th. Nov. 19th |
| 23. Tue. Nov. 17 | 60,62 | Laurent series. Laurent's Theorem. Examples. | |
| 24. Thu. Nov. 19 | 68-70, 72 | Residues. Cauchy's residues theorem. Removable singularity/pole/essential singularities. |
Homework #10 Due Tue. Dec. 1st |
| 25. Tue. Nov 24 | 73-76 | Poles and zeroes of an analytic function | |
| 26. Tue. Dec. 1 | 78-82 | Application of residue: Evaluating improper integrals | Homework #11 |
| 27. Thu. Dec. 3 | Review | ||
| 28. Tue. Dec. 8 | Midterm #3 Solution | ||
| 29. Thu. Dec. 10 | Review | ||