B.Sc. Mathematics
6th Semester Syllabus
Select Paper in this Semester
MJC-10 : Complex Analysis
Double-tap topic to mark complete • Long-press for YouTube & AI
Unit 1: Complex Numbers and Functions
◦Introduction to complex number and its geometrical interpretation
◦Algebra of complex numbers
◦Functions of complex variables
◦Limit of a complex function
◦Continuity and uniform continuity
◦Differentiability
◦Analytic and regular functions
◦Cauchy-Riemann equation and it's applications
Unit 2: Elementary Functions and Contour Integrals
◦Exponential function
◦Logarithmic function
◦Branches and derivatives of logarithms
◦Trigonometric and hyperbolic functions
◦Derivatives of functions
◦Definite integrals of functions
◦Contours
◦Contour integrals with examples
◦Upper bounds for moduli of contour integrals
Unit 3: Complex Integration
◦Complex integration
◦Cauchy's theorem
◦Cauchy's Goursat theorem
◦Primitives
◦Cauchy's integral formula
◦Cauchy's integral formula for the derivative of an analytic function
◦Morera's theorem
◦Poisson's integral formula for a circle
◦Cauchy's inequality
◦Liouville's Theorem and Fundamental theorem of Algebra
Unit 4: Series Expansions
◦Convergence of sequences and series
◦Taylor series with examples
◦Laurent series and its examples
◦Absolute and uniform convergence of power series
◦Uniqueness of series representations of power series
Unit 5: Transformations
◦Linear Transformation
◦Jacobian of a transformation
◦Elementary transformations: translation
◦Rotation
◦Magnification
◦Inversion
◦Mobius transformation (bilinear transformation)
◦Cross ratio
◦Preservation of cross ratio under bilinear transformation
◦Fixed point of a bilinear transformation
Reference Books:
- Brown, James Ward, & Churchill, R. V. (2014). Complex Variables and Applications (9th ed.). McGraw-Hill Education. New York
- S. Ponnusamy, (2011) Foundation of complex Analysis, Alpha Science International Ltd. UK
- Bak, Joseph & Newman, Donald J. (2010). Complex analysis (3rd ed.). Undergraduate Texts in Mathematics, Springer. New York
- Zills, Dennis G., & Shanahan, Patrick D. (2003). A First Course in Complex Analysis with Applications. Jones & Bartlett Publishers, Inc
- Mathews, John H., & Howell, Rusell W. (2012). Complex Analysis for Mathematics and Engineering (6th ed.). Jones & Bartlett Learning. Narosa, Delhi. Indian Edition