Utkal University, Odisha, Mathematics Semester 3, Theory of Real Functions Syllabus

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Unit - 1 L’ Hospital’s Rules

Unit 1

L’ Hospital’s Rules

1.1 L’ Hospital’s Rules other Intermediate forms

1.2 Cauchys mean value theorem

1.3 Taylors theorem with Lagranges form of remainder and Cauchys form of remainder

1.4 Application of Taylors theorem to convex functions Relative extremes

1.5 Taylors series and Maclaurins series

1.6 Expansions of exponential and trigonometric functions

Unit - 2 Riemann Integration

Unit 2

Riemann integration

2.1 Riemann integration inequalities of upper and lower sums

2.2 Riemann conditions of integrability

2.3 Riemann sum and definition of Riemann integral through Riemann sums equivalence of two definitions

2.4 Riemann integrability of monotone and continuous functions

2.5 Properties of the Riemann integral

2.6 Definition and integrability of piecewise continuous and monotone functions

2.7 Intermediate value theorem for Integrals

2.8 Fundamental theorems of Calculus

Unit - 3 Improper Integrals

Unit 3

Improper integrals

3.1 Improper integrals Convergence of Beta and Gamma functions

3.2 Pointwise and uniform convergence of sequence of functions

3.3 Theorems on continuity derivability and integrability of the limit function of a sequence of functions

Unit - 4 Series Of Functions

Unit 4

Series of functions

4.1 Series of functions

4.2 Theorems on the continuity and derivability of the sum function of a series of functions

4.3 Cauchy criterion for uniform convergence and Weierstrass MTest Limit superior and Limit inferior

4.4. Power series radius of convergence Cauchy Hadamard Theorem

4.5. Differentiation and integration of power series

4.6. Abels Theorem

4.7. Weierstrass Approximation Theorem

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Other Subjects of Semester-3
Group theory-i
Partial differential equations and system of odes
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