National University of Sciences and Technology
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MATH-802 Analysis
Campus SNS
Programs PG
Session Fall Semester 2016
Course Title Analysis
Course Code MATH-802
Credit Hours 3-0
Course Objectives This course is designed to provide in-depth knowledge of Real and Complex Analysis
Detail Content Real Analysis
  1. The sandwiching theorem, the nested intervals theorem, the Bolzano-Wierstrass theorem.
  2. Uniform continuity, Cauchy criterion for convergence.
  3. The Heine-Borel Theorem.
  4. Review of Rolle’s, mean-value theorems and L’Hopital rule.
  5. Inverse functions in R1, Inverse function theorem, inverse differentiation theorem.
  6. The Darboux integral for functions in R1, Upper and lower Darboux integrals, Fundamental theorems of calculus – 1st and 2nd forms, The Riemann integral.
  7. Continuity in RN, Darboux integral in RN, The Riemann integral in RN.
  8. Sequences of functions and uniform convergence of sequences.
  9. Uniform convergence of series
  10. Fourier series; Fourier sine and cosine series; Half-range series; Convergence theorems.
  11. The derivative of a function defined by an integral; Leibnitz rule; Convergence and divergence of improper integrals; The Gamma function.
  12. Functions of bounded variation; the Riemann-Stieltjes integral.
  13. The implicit function theorems; Change of variables in multiple integrals; The Lagrange multiplier rule
Complex Analysis
  1. Analyticity of complex functions.
  2. Transcendental functions in the complex plane
  3. Complex integration.
  4. Cauchy’s integral theorem and Cauchy’s integral formula.
  5. Sequence and series and their convergence.
  6. Power series, Taylor series, and Laurent series.
  7. Singularities and zeros.
  8. Residue integration
  9. Evaluation of real integrals
  10. Conformal mapping
Text/Ref Books
  1. M. H. Protter, and C. B. Morrey, Jr.: A First Course in Real Analysis, Springer (1991)
  2. E. B. Saff, and A. D. Snider: Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Prentice Hall (1993)
Time Schedule Fall Semester 2014
Faculty/Resource Person Dr. Matloob Anwar
PhD GCU Lahore
Discipline: Mathematical Analysis