### MA 205 – COMPLEX ANALYSIS

**Session:** 2012-13

**Professor:** Prof. Anant.R.Shastri

**Course Content:**

- Sequences and Series of Complex Numbers
- Topological and geometrical aspects of Complex Numbers
- Complex Differentiation (Analytic Functions, Power Series)
- Cauchy Riemann Equations
- Holomorphic Functions
- Contour Integration (Cauchy’s theorem/formulae)
- Zeroes and Poles (Laurent Series, Winding Number)
- Fractional Linear Transformations
- Some celebrated integrals

**Pre requisites:** MA 105(Not in strict sense though)

**Feedback on lectures:**

Atmosphere was somber and disappointing with low voice of the instructor in a large class room. Only the front benchers were able to concentrate well. It was somewhat a wastage of an experienced Professor. Those who concentrated scored well.

**Feedback on tutorials, assignments and exams:**

There were 5 tutorials. All were somewhat difficult and needed efforts/ideas. Tutorials were discussed in tutorial sessions but many students were unhappy with incomprehensive explanations by the TAs. There was a quiz and an endsem. The quiz was mostly on previously known topics and was relatively easy. The endsem was unprecedented in terms of the theory aspects. Several statements of theorems/corollaries were asked and the problems asked were decent to difficult.

**Difficulty level:** Difficult.

**Grading:** Decent grading with 25 FRs out of 282. BB, CD and DD were high in number while remaining were between 15-25.

**Study Material and References:**

Lecture slides were put up on the MOODLE. Notes covered theory and good number of examples. Text: Basic Complex Analysis of One Variable by Anant.R.Shastri(Instructor Himself) The text was an elaboration of what is in slides with no much use. Slides are sufficient.

**Miscellaneous:**

Endsem was inordinately difficult to many owing to the theory asked in it. Endsem was for 80 marks. The instructor was very particular about statements of theorems and was furious at students who tried to defend their statements and steps written in exams. For some, marks were deducted right away for not being MATHEMATICAL in their solutions!!

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

- MA 412-Complex Analysis
- MA 521-Theory of Analytic Functions
- MA 532-Analytic Number Theory

**Application of course in other courses and practice:**

- Main application of complex analysis is to evaluate real integrals that cannot be evaluated by normal real number techniques, via a complex contour integration using Cauchy’s theorems,Jordan’s Inequality and Pole Bypassing technique.
- Winding Number concept finds applications in determining stability of systems through Nyquist plots in EE 324(Control Systems).
- Concept of Analytic functions is useful in Signal Processing via the concept Kramers-Kronig Relations which relates real and imaginary parts of analytic complex functions exploiting the causality of systems. Analogies are developed between several mechanical/electrical systems and signals and this concept is used to find some variables treated as real part using other variables treated as complex part.
- Residue Theory, winding numbers etc are used in other Math courses mentioned above.
- Complex Numbers are widely used in advanced studies with relatively less applications in B.tech courses.

Reviewed by - **Tharun Kumar Reddy** (tharunkumarreddy24@gmail.com)