NITH SYLLABUS

MA-321REAL & COMPLEX ANALYSIS
CreditsLTP
4310

UNIT-1

RIEMANN INTEGRATION
  • Riemann Integration
  • Partition and Riemann sums
  • Necessary and sufficient conditions for Riemann integrability of a function
  • First and second mean value theorems of integral calculus
  • Fundamental theorem of integral calculus

UNIT-2

IMPROPER INTEGRALS
  • Improper integrals of first and second type
  • Beta function
  • Gamma function
  • Their properties
  • Relation between Beta and Gamma function
  • Convergence of improper integrals
  • Comparison test
  • Μ-test
  • Abel’s test
  • Dirichlet’s test

UNIT-3

METRIC SPACES
  • Definition and examples
  • Open
  • Closed and bounded sets
  • Interior
  • Closure and boundary
  • Convergence and completeness
  • Continuity and uniform continuity
  • Connectedness
  • Compactness and Seperability
  • Heine-Borel theorem

UNIT-4

PRELIMINARIES TO COMPLEX ANALYSIS
  • Basic properties: convergence
  • Compactness
  • Connectedness
  • Power series of complex valued function
  • Radius of convergence

UNIT-5

COMPLEX FUNCTIONS
  • Poisson’s integral formula for a circle
  • Cauchy’s inequality
  • Fundamental theorem of integral calculus for complex valued function
  • Fundamental theorem of algebra
  • Argument principle
  • Rouche’s theorem
BOOKS
  1. Advanced Engineering Mathematics: by Erwin Kreyszig, John Wiley and Sons.
  2. Advanced Engineering Mathematics, R. K. Jain & S. R. K Iyengar, Narosa Pub. House.
  3. Complex Analysis for Mathematics and Engineering by J.H. Mathews and R.W. Howell, Narosa Publishing House.
  4. Complex Variables and Applications by J.W. Brown and R.V. Churchill, McGraw Hill.
  5. Mathematical Analysis by T. M. Apostol,, Addison-Wesley Publishing Company.
  6. Mathematical Analysis by S.C. Malik & Savita Arora,, New Age International (P) Ltd
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