This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis & especially conformal mappings. Students should have a background in real analysis (as in the course Real Analysis I), including the ability to write a simple proof in an analysis context. Complex Analysis is a topic that is extremely useful in many applied topics such as numerical analysis, electrical engineering, physics, chaos theory, & much more, & you will see some of these applications throughout the course. In addition, complex analysis is a subject that is, in a sense, very complete. The concept of complex differentiation is much more restrictive than that of real differentiation & as a result the corresponding theory of complex differentiable functions is a particularly nice one.

Pre-Requisites:

 Calculus-II

Intended Learning Outcomes

On successful completion of this course students will be able to:

1.  Explain the fundamental concepts of complex analysis and their role in modern mathematics and applied contexts
2.  Demonstrate accurate and efficient use of complex analysis techniques.
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from complex analysis.
4. Apply problem-solving using complex analysis techniques applied to diverse situations in physics, engineering and other mathematical contexts.

Contents

1. Introduction: The algebra of complex numbers
2. Geometric representation of complex numbers
3. Polar form of complex numbers
4. Powers & roots of complex numbers
5. Functions of Complex Variables
6. Limit
7. Continuity
8. Differentiable functions, the Cauchy-Riemann equations
9. Analytic functions, entire functions, harmonic functions
10. Elementary functions: The exponential, Trigonometric functions
11. Hyperbolic, Logarithmic & Inverse elementary functions
12. Complex Integrals: Contours & contour integrals, antiderivatives, independence of path
13.  Cauchy-Goursat theorem, Cauchy integral formula, Lioville’s theorem, Morerea’s theorem
14. Maximum Modulus Principle
15. Series: Power series, Radius of convergence & analyticity
16. Taylor’s & Laurent’s series
17.  Integration & differentiation of power series, isolated singular points
18. Cauchy’s residue theorem with applications
19. Types of singularities & calculus of residues, Zeros & Poles, Mobius transforms
20. Conformal mappings & transformations

Recommended Books

1. Mathews J. H., & Howell, R.W. (2006). Complex analysis for mathematics & engineering (5th ed.). Burlington: Jones & Bartlett Publication.
2. Churchill, R.V., & Brown, J.W. (2013). Complex variables & applications (9th ed.). New York: McGraw-Hill.

Suggusted Books:

1. Remmert, R. (1998). Theory of complex functions (1st ed.). New York: Springer-Verlag.
2. Rudin, W. (1987). Real & complex analysis (3rd ed.). New York: McGraw-Hill.

Description of system of evaluation (homework, midterms, final,  assignments etc.): 

1. Homework, Exercises, Attendance, Assignments: 20%
2. Midterm: 30%, Final Term: 50 %

Key Dates and Time of Class Meeting

Wednesday: 9:30am-11:00am (Regular)  & 12:30pm-2:00pm (Self Support)

Friday: 11:00am-12:30am (Regular)  & 03:30pm-5:00pm (Self Support)

Commencement of Classes                                                   Febraury 24, 2021

Mid Term Examination                                                            April 19-23, 2021

Final Term Examination                                                          June 21-25, 2021

Declaration of Result                                                              July 02, 2021