### MA 205 – COMPLEX ANALYSIS

**Course offered in:**

Autumn 2020

**Instructors:**

Prof. Sudarshan R. Gurjar

**Course Content:**

- Fundamental theorem of Algebra
- Some basic notions of topology
- Definition and properties of analytic functions
- Cauchy-Riemann equations, harmonic functions
- Power series and their properties
- Elementary functions
- Cauchy’s theorem and its applications
- Taylor series and Laurent expansions
- Residues and the Cauchy residue formula
- Evaluation of improper integrals
- Conformal mappings

**Prerequisites:**

There are no strict pre-requisites. Concepts like limits, continuity and differentiation using epsilon-delta from MA 105 part 1 (MA 109) will help so it’s advisable to brush up on them for the course.

**Feedback on Lectures:**

The lectures were live, much unlike most other courses online. They were smooth and there was rarely ever spill-over from one class to the next. The Professor kept a light-hearted informal tone and genuinely encouraged doubts which made this one of the more interesting ‘in-class’ experiences.

Sometimes, though rarely discussions would go on tangents which could be frustrating for a few but eye-opening for others. Since the course content is arguably on the more interesting side in comparison to other math courses, there should be little trouble maintaining interest during the course. Special attention and time was devoted to the more ‘beautiful’ concepts, an additional bonus.

All in all you can expect to have your minds blown by cool theorems and results derived and proved in this course.

**Feedback on Tutorials, Assignments and Exams:**

Tutorials were held weekly, they were well organized in spite of it being the first online tutorial offering. Tutorials would help much more than classes, so make sure not to miss them even if you miss a lecture or two.

There were no assignments.

There was a quiz halfway through the course, accounting for 35% of the grade. It was MCQ on SAFE with single correct answers and +2.5, -1 grading scheme. All questions were direct applications of in-class concepts.

The final exam had both integer type, multiple correct and single correct questions on SAFE. But the questions were direct application of concepts from the course barring a few ones requiring some thought. I’d say the Exams were easy to ace but also easy to slip up and lose a giant chunk of weightage as a result of some blunder or calculation error.

**Difficulty:**

Moderately difficult

**Grading Statistics:**

**Study Material and References:**

- R. V. Churchill and J. W. Brown, Complex variables and applications (7th Edition), McGraw-Hill (2003)
- E. Kreyszig, Advanced engineering mathematics (8th Edition), John Wiley (1999)
- J. M. Howie, Complex analysis, Springer-Verlag (2004)
- M. J. Ablowitz and A. S. Fokas, Complex Variables- Introduction and Applications, Cambridge University Press, 1998 (Indian Edition)

More advanced references :

- Lars Ahlfors – Complex Analysis
- John Conway – Functions of a Complex Variable
- Serge Lang – Complex Analysis

Solving the Tutorials yourself should be enough to ensure a good grade.

**Comments:**

This course was rigorous but the evaluation was not. An offline offering with subjective papers may have suited the mathematically inclined.

**Advanced courses that can be taken after this:**

For those pursuing a minor in Math, a lot of the content will be repeated in MA412M.

**Takeaways from the course:**

Concepts from this course will help in EE229 – Signal Processing I and EE302 – Control Systems.

The concepts are also a foundation for math related research and as electrical engineers you will also deal with a lot of complex numbers, so learning how to handle them wouldn’t hurt.

Although people may argue that the concepts in this course are hardly ‘useful’, I’d disagree because it helps develop a mathematic temperament, which goes a long way in many placement positions offered.

Review by – Prasann Viswanathan (iamprasannn@gmail.com)

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