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“Jana Prosason” (Public Administration)( “জনপ্রশাসন”) is authored by Prangovinda Das.(NCBA)
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1. Real Analysis
This section establishes the foundational theory of calculus on the real line, dealing with rigorous proofs involving real numbers and functions:
- The Real Number System: Field axioms, order properties, upper/lower bounds, completeness axiom, and the Archimedean property.
- Point Set Topology: Countable and uncountable sets, neighborhoods, open and closed sets, limit points, Bolzano-Weierstrass theorem, and compact sets.
- Sequences and Series: Convergence, Cauchy sequences, monotone sequences, subsequences, tests for convergence of infinite series, and absolute vs. conditional convergence.
- Continuity and Limits: Rigorous \(\epsilon \)-\(\delta \) definitions of function limits, continuous functions, uniform continuity, and properties of functions continuous on compact sets.
- Differentiation: Definition of derivatives, Mean Value Theorems (Rolle’s, Lagrange’s, Cauchy’s), and Taylor’s Theorem with remainder forms.
- Integration: Riemann integration theory, upper and lower sums, integrability conditions, Fundamental Theorem of Calculus, and improper integrals.
2. Complex Analysis
This part introduces function theory over the complex plane, which generalizes calculus to complex-valued variables:
- Complex Variables: Geometry of the complex plane, stereographic projection, point sets, sequences, and limits of complex variables.
- Analytic Functions: Continuity, differentiability, Cauchy-Riemann equations (Cartesian and polar forms), harmonic functions, and orthogonal trajectories.
- Complex Integration: Line integrals in the complex plane, Cauchy’s Integral Theorem, Cauchy’s Integral Formula for derivative evaluations, and Liouville’s Theorem.
- Power Series: Regions of convergence, Taylor’s series, and Laurent series expansions for complex functions.
- Theory of Residues: Singularities (isolated, essential, poles), Residue Theorem, and its application in evaluating complex contours and definite real integrals.
- Conformal Mapping: Geometric transformations, bilinear/Möbius transformations, and fixed points.
3. Metric Spaces
This abstract section generalizes the concept of “distance” to broader geometric settings:
- Basic Definitions: Definition of a metric, standard metrics (Euclidean, discrete, taxicab, max metric), and bounded metric spaces.
- Topology in Metric Spaces: Open and closed balls, interior, exterior, boundary points, dense sets, and subspace metrics.
- Convergence and Completeness: Cauchy sequences in metric spaces, completeness, Cantor’s Intersection Theorem, and Banach Fixed Point Theorem.
- Compactness and Connectedness: Compact metric spaces, Heine-Borel theorem analogue, sequentially compact spaces, and connected vs. disconnected metric spaces.
- Continuity: Continuous mappings between metric spaces and homeomorphisms.
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