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:

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

**Contents**

- Introduction: The algebra of complex numbers
- Geometric representation of complex numbers
- Polar form of complex numbers
- Powers & roots of complex numbers
- Functions of Complex Variables
- Limit
- Continuity
- Differentiable functions, the Cauchy-Riemann equations
- Analytic functions, entire functions, harmonic functions
- Elementary functions: The exponential, Trigonometric functions
- Hyperbolic, Logarithmic & Inverse elementary functions
- Complex Integrals: Contours & contour integrals, antiderivatives, independence of path
- Cauchy-Goursat theorem, Cauchy integral formula, Lioville’s theorem, Morerea’s theorem
- Maximum Modulus Principle
- Series: Power series, Radius of convergence & analyticity
- Taylor’s & Laurent’s series
- Integration & differentiation of power series, isolated singular points
- Cauchy’s residue theorem with applications
- Types of singularities & calculus of residues, Zeros & Poles, Mobius transforms
- Conformal mappings & transformations

**Recommended Books**

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

**Suggusted Books:**

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

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

- Homework, Exercises, Attendance, Assignments: 20%
- 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