Prerequisite(s)

Calculus -II  Credit Hours :3+0

 Description and Objectives of the course

Complex Analysis is the study of complex numbers together with their derivatives, manipulation, and other properties .Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of the physical problems. 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.The focus of this course is on the study of complex valued functions and their most important basic properties especially Conformal mappings. Aalso have a properties of complex integration  and having the ability to compute such integrals. 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 learing outcomes

After successful completion of this course,

1. Students will be able to demonstrate the basic concepts underlying complex analysis and describe the basic properties of complex integration.
2. They will decide the analyticity of the complex valued function and find its series development.
3. 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,
4. Evaluate complex contour integrals directly and by the fundamental theorem, apply the Cauchy integral theorem in its various versions, and the Cauchy integral formula,
5. 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 ,10th ed. (McGraw  Hill, New York, 1989)
2. Mathews J. H. and R. W. Howell. Complex Analysis for Mathematics and Engineering, (2006).

​Suggested Books

1. Lang S. Complex Analysis, (Springer-Verlag, 1999)
2. Remmert R. Theory of Complex  Functions, (Springer-Verlag, 1991)
3. Rudin W. Real and Complex Analysis, (McGraw-Hill, 1987).

    

Assessment  Criteria

       Sessional: 20 (Presentation / Assignment 10, Attendance 05, Quiz 05)

       Mid-Term Exam:   30

       Final-Term Exam: 50

Key Dates and Time of  Class Meeting

Thursay                                                  11 AM-12:30 PM (Reg)                          02:00 PM-03:30 PM) (SS)

Friday                                                      8::00 AM-9::30 AM (Reg)                       11 AM-12:30 PM (SS)

Commencement of Classes                                                   March 02, 2020

Mid Term Examination                                                            April 27 to May 04, 2020

Final Term Examination                                                          June 22-26, 2020

Declaration of Result                                                              July 03, 2020