Detailed Syllabus:
Unit 1 [9 hours]
Sequence of real No, Bounded and Unbounded Sets, Supremum, Infimum, Limit points of a
set, Closed Set, Countable and uncountable sets. Sequences, Limit points of a Sequence. Limits
Inferior and Superior, Convergent sequence, Non convergent sequence, Cauchy General Principle
of Convergence, bounded and monotone sequence, Infinite Series, Positive Term Series,
Convergence of series of real numbers, Necessary condition, Absolute convergence and power
series, Convergence tests for series.
Unit 2 [9 hours]
Mean value theorems (Rolle’s Theorem, Cauchy Mean Value Theorem, Lagrange’s Mean
Value Theorem), Indeterminate forms, Taylors Series, Partial derivatives. Integration as a limit of a
sum, some integrable functions, Fundamental theorem of Calculus, Mean Value Theorems of
Integral calculus, Integration by parts, Change of variable in an integral, Second Mean value
theorem, Multiple integrals
Unit 3 [7.5 hours]
Vector, Vector operations, Products, Areas and Determinants in 2D, Gradients, Curl and
Divergence, Volumes and Determinants in space. Differential equations of first order and first
degree. Linear ordinary differential equations of higher order with constant coefficients. Elements
of Partial Differential Equation (PDE).
Unit 4 [7.5 hours]
Analytic function of complex variable, CR Equation, harmonic functions, Laplace equation,
applications.
Unit 5 [9 hours]
Integration of a function of a complex variable, M-L inequalities. Cauchy’s Integral Theorem.
Cauchy’s Integral formula. Taylor’s and Laurent Expansion, Poles and Essential Singularities,
Residues, Cauchy’s residue theorem, Simple contour integrals.
A project related to the above syllabus will be done by students to be submitted by the
end of the semester.