ANALYSIS

1. METRIC SPACE 1-42

Metric and Metric Space 1; Quasi-Metric Space 5; Pseudo-Metric Space 5; Distance between Point and Set 6; Distance between Two Sets 6; Diameter of a Set 7; Some Important Inequalities 7; Product 11; Finite Product in General 12; Product of the Metric Spaces 13; Open Sphere 18; Open Disk (in Real Plane) 18; Open Disk (in Complex Plane) 18; Neighbourhood of a Point 18; Limit Point of a Set 18; Derived Set 19; Interior Point 19; Open Set 19; Closed Sphere 21; Closed Disk (in Real Plane) 21; Closed Disk (in Complex Plane) 21; Open and Closed Balls in RK 21; Convexity in RK 21; Closed Set 22; Closure of a Set 26; Interior of a Set 29; Exterior of a Set 29; Boundary Points 31; Subspace of a Metric Space 32; Relative Open Set 32; Convergence of a Sequence in a Metric Space 33; Cauchy Sequence 33; Bounded Set and Bounded Sequence 34; Complete Metric Space 36; Completeness 37; Nested Sequence 38; Contraction Mapping 39; Contraction Principle or Banach Fixed Point Theorem 40.

2. COMPACTNESS 43-57

Cover 43; Subcover 43; Finite Subcover 43; Open Cover 43; Compact Set and Compact Space 43; Some Theorems 44; Bolzano-Weierstrass Property 46; Sequential Compactness 46; Theorems 47; Heine-Borel Theorem 49; e-Net 50; Totally Bounded 50; Some Theorems 50; Lebesgue Number 52; Lebesgue Covering Lemma 52; Theorem 52; Theorem 53; Finite Intersection Property 53; Some Theorems 53.

Unit-2

3. RIEMANN INTEGRAL 58-91

Introduction 58; Definition 58; Upper and Lower Riemann Sums 59; Some Important Theorems 59; Upper and Lower Riemann Integrals 62; Darboux Theorem 63; Riemann Integral 64; Oscillatory Sum 64; Integrability of Continuous Function 75; Integrability of Monotonic Function 76; Properties of Riemann Integral 76; Continuity and Differentiability of Integral Function 82; Second Fundamental Theorem 83; Mean Value Theorems 84.

Unit-3

4. COMPLEX INTEGRATION 92-143

Complex Integration 92; Some Definitions 92; Rieman's Definition of Integration or Line Integral or Definite Integral or Complex Line Integral 96; Relation between Real and Complex Line Integrals 97; Some Properties of Line Integrals 97; Evaluation of the Integrals with the Help of the Direct Definition 97; Complex Integral as the Sum of Two Real Line Integrals 99; An Upper Bound for a Complex Integral 112; Cauchy's Fundamental Theorem or Cauchy's Original Theorem or Cauchy's Integral Theorem 113; Cauchy-Goursat Theorem or Cauchy's Integral Theorem (Revised Form) 114; Corollary 117; Cross-Cut or Cut 117; A More General Form of Cauchy's Integral Theorem 117; Extension of Chauch's Theorem Multi-Connected Region 118; Cauchy's Integral Formula 118; Extension of Cauchy's Integral Formula to Multiply Connected Regions 120; Cauchy's Integral Formula for the Derivative of an Analytic Function 120; Analytic Character of Higher Order Derivatives of an Analytic Function 121; Corollary 122; Cauchy's Inequality Theorem 123; Integral Functions or Entire Function 123; Converse of Cauchy's Theorem or Morera's Theorem 123; Indefinite Integrals or Primitives 124; Theorem 124; Fundamental Theorem of Integral Calculus for Complex Functions 125; Liouville's Theorem 125; Maximum Modulus Theorem or Maximum Modulus Principle 126; Minimum Modulus Principle or Minimum Modulus Theorem 127.

5. SINGULARITY 144-169

The Zeroes of an Analytic Function 144; Zeroes are Isolated 144; Singularities of an Analytic Function 145; Different Types of Singularities 145; Meromorphic Functions 149; Theorem 149; Theorem 150; Theorem 150; Entire Function or Integral Function 150; Theorem 150; Theorem (Due to Riemann) 151; Theorem (Weierstrass Theorem) 151; Theorem 152; The Point at Infinity 153; Limit Point of Zeroes 153; Limit Point of Poles 154; Identity Theorem 154; Theorem 154; Theorem 154; Theorem 155; Theorem 155; Theorem 156; Theorem 157; Detection of Singularity 157; Rouche's Theorem 158; Fundamental Theorem of Algebra 160.

6. RESIDUE THEOREM 170-274

Definition of the Residue at a Pole 170; Residue of f(z) at a Simple Pole z = a 170; Theorem 171; Residue of f(z) at a Pole of Order m 171; Rule of Finding the Residue of f(z) at a Pole z = a of any Order 172; Theorem 172; Definition of Residue at Infinity 173; Theorem 174; Theorem 175; Cauchy's Theorem of Residues or Cauchy's Residue Theorem 175; Theorem 176; Liouville's Theorem 177; Evaluation of Real Definite Integrals by Contour Integration 195; Theorem 195; Theorem 196; Jordan's Inequality 196; Jordan's Lemma 197; Integration Round the Unit Circle 197; Evaluation of the Integral ò-¥+¥ f(x) dx 218; Poles on the Real Axis 247; Evaluation of Integrals whose Integrands Involve many Valued Functions 257; Integration along Contour other than Circle or Semi-circle 262.

Unit-4

7. FOURIER TRANSFORMS 275-283

Periodic Function 275; Even and Odd Functions 275; Dirichlet Conditions 275; Fourier Series 276; Fourier Integral Theorem 276; Fourier Sine and Cosine Integrals 276; Complex Form of Fourier Integral 277; Finite Fourier Transform of f (x) or Finite Fourier Sine/Cosine Transform of f (x) 277; Determination of f (x) 282.

8. INFINITE FOURIER TRANSFORMS 284-297

Definition 284; Properties 284; Inverse Fourier Transform 291; Fourier Transform or Complex Fourier Transform 294.

9. PROPERTIES OF FOURIER TRANSFORMS AND 

PARTIAL DIFFERENTIAL EQUATIONS 298-304

Theorems 298; Fourier Transform of Partial Derivatives 300.