Course Prerequisite: Calculus II

Description and Objectives

When the real numbers are replaced by the complex numbers in the definition of the derivative of a function, the resulting (complex-) differentiable functions turn out to have many remarkable properties. These functions have numerous applications in areas such as engineering, physics, differential equations and number theory. The focus of this course is on the study of complex valued functions and their most important basic properties 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.

Intended Learning Outcomes

The student should be able to: 

• Represent complex numbers algebraically and geometrically,
• Define and analyze limits and continuity for complex functions as well as consequences of continuity,
• Apply the concept and consequences of analyticity and the Cauchy-Riemann equations and of results on harmonic and entire functions including the fundamental theorem of algebra,
•  Analyze sequences and series of analytic functions and types of convergence,
•  Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula, and
• Represent functions as Taylor, power and Laurent series, classify singularities and poles, find residues and evaluate complex integrals using the residue theorem.

Course Contents

1. Introduction: The algebra of complex numbers
2. Geometric representation of complex numbers, Powers and roots of complex numbers
3. Functions of Complex Variables: Definition, limit and continuity
4. Branches of functions, Differentiable and analytic functions
5. The Cauchy-Riemann equations, Entire functions, Harmonic functions
6. Elementary functions:The exponential,Logarithmic Trigonometric, Hyperbolic and Inverse trionometric and hyperbolic functions
7. Open mapping theorem. Maximum modulus theorem.
8. Complex Integrals:Contours and contour integrals, Cauchy-Goursat theorem, Cauchy integral formula
9. Lioville’s theorem, Morerea’s theorem
10. Series:Power series, Radius of convergence and analyticity, Integration and differentiation of power series
11. Taylor’s and Laurent’s series
12.  Singularities, Poles and residues
13. Types of singular points, Calculus of residues, contour integration,
14.  Cauchy’s residue theorem with applications.
15. Mobius transforms, Conformal mappings and transformations.

Recommended Books

1. Churchill R. V., J. W. Brown. Complex Variables and Applications ,9th ed. (McGraw  Hill, New York, 1989)
2. Mathews J. H. and R. W. Howell. Complex Analysis for Mathematics and Engineering, (2006)
3. Lang S. Complex Analysis, (Springer-Verlag, 1999)

System of Evaluation

Sessional: 20 (Presentation  15, Attendance 05, )

Mid-Term Exam:  30 (Detailed assignment (of min 15 pages) 20, viva voce 10)

Final-Term Exam: 50 

Key Dates and Time of Class Meeting

Monday& Tuesday              12:30 PM - 02:00 PM (BS- VI New Ex PPP)

Wednesday & Thursday      09:30 AM - 11:00 AM (BS - VI Regular)

Wednesday & Thursday      02:00 PM - 03:30 PM (BS - VI Self Support)

Commencement of Classes     January 13, 2020

Mid Term Examination              March 09 - 13, 2020

Final Term Examination            May 04 - 08, 2020

Declaration of Result                May 19, 2020