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Annexure ‘AAB-CD-01’

Course Title: Advance Real Analysis

|L |T |P/S |SW/FW |TOTAL CREDIT UNITS |

|3 |1 | |0 |4 |

Course Code: to be decided later

Credit Units: 4

Level: PG

|# |Course Title |Weightage (%) |

|1 |Course Objectives: | |

| |to understand the basic concepts of real analysis and its physical properties | |

| |to develop fundamental knowledge and understanding of the many techniques in Real variable . | |

| |to make the students aware of General theory of differentiation and integration under the sign | |

| |of integration, question of convergence of series, Dirichlet’s integral, Laplace and Laplace | |

| |Steiltjes transform are employed in the theory of probability distributions. Similarly, | |

| |Bolzano-Weirstrass,Heine Borel theorems etc. are very much useful in Statistical Inference . | |

| |to apply statistical concepts to various fields of statistics to analyze and interpret data. | |

|2 |Prerequisites: | |

| | NIL | |

|3 |Student Learning Outcomes: | |

| |The students will be able to learn various continuity of functions. | |

| |The students will able to acquire knowledge on convergences. | |

| |The students will able to apply the properties of mgf and cf for distributions. | |

| |The students will able to define sequences of the functions. | |

| |The course enables the students to develop the skill set to solve the problems based on real | |

| |life situation. | |

|Course Contents / Syllabus: |

|4 |Module I: |20% Weightage |

| |Monotone functions and functions of bounded variation. Real valued functions, continuous | |

| |functions, Absolute continuity of functions, standard properties, uniform continuity, sequence | |

| |of functions, uniform convergence, power series and radius of convergence. | |

|5 |Module II: |20% Weightage |

| |Riemann-Stieltjes integration, standard properties, multiple integrals and their evaluation by | |

| |repeated integration, change of variable in multiple integration. Uniform convergence in | |

| |improper integrals, differentiation under the sign of integral - Leibnitz rule, Integration | |

| |under the sign of differentiation. Dirichlet integral. | |

|6 |Module III: |30% Weightage |

| |Introduction to n-dimensional Euclidean space, open and closed intervals (rectangles), compact | |

| |sets, Bolzano-Weierstrass theorem, Heine-Borel theorem. Maxima-minima of functions of several | |

| |variables, constrained maxima-minima of functions. | |

|7 |Module IV: Applications of mgf and cf for continuous distributions |30% Weightage |

| |Laplace and Laplace-Steiltjes transforms. Solutitions of linear differential.Properties of | |

| |Laplace transforms, Transforms of derivatives, Transforms of integrals, Evalualtion of integrals| |

| |using Laplace transform, convolution theorem, Applications to differential equations, | |

| |simultaneous linear equations with constant coefficient, unit step functions and Periodic | |

| |functions. | |

|8 |Pedagogy for Course Delivery: | |

| | | |

| |The class will be taught using theory and practical methods using software in a separate Lab | |

| |sessions. In addition to numerical applications, the real life problems and situations will be | |

| |assigned to the students and they are encouraged to get a feasible solution that could deliver | |

| |meaningful and acceptable solutions by the end users. The focus will be given to incorporate | |

| |probability and related measures to develop a risk model for various applications. | |

|9 | | |

| |Assessment/ Examination Scheme: | |

| | | |

| |Theory L/T (%) | |

| |Lab/Practical/Studio (%) | |

| |End Term Examination | |

| | | |

| | | |

| |30% | |

| | | |

| |NA | |

| | | |

| |70% | |

| | | |

| |Theory Assessment (L&T): | |

| |Continuous Assessment/Internal Assessment | |

| |End Term Examination | |

| | | |

| | | |

| | | |

| |Components (Drop down) | |

| | | |

| |Mid-Term Exam | |

| | | |

| |Project | |

| | | |

| |Viva | |

| | | |

| |Attendance | |

| | | |

| | | |

| |Weightage (%) | |

| | | |

| |10% | |

| | | |

| |10% | |

| | | |

| |5% | |

| | | |

| |5% | |

| | | |

| |70% | |

| | | |

| | | |

Text & References:

• Rudin, Walter (1976). Principles of Mathematical Analysis, McGraw Hill.

• Apostol, T. M. (1985). Mathematical Analysis, Narosa, Indian Ed.

• Narayan, S., (2010). Elements of Real Analysis, S. Chand and Sons.

• Miller, K. S. (1957). Advanced Real Calculus, Harper, New York

• Courant, R. and John, F. (1965). Introduction to Calculus and Analysis, Wiley

• Bartle, R.G. (1976): Elements of Real Analysis, John Wiley & Sons.

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