ABSTRACT ALGEBRA, DIFFERENTIAL EQUATION & FOURIER SERIES

UNIT-I

1. Group Automorphism, Inner Automorphism, Group of Automorphisms 1-22

Introduction 1; Homomorphism of Group 1; Types of Homomorphism 1; Kernel of a Homomorphism 3; Some Theorems (Properties of Group Homomorphism) 3; Isomorphism of Groups 3; Fundamental Theorem of Homomorphism of Groups 3; More Properties of Group Homomorphism 4; Automorphism of a Group 4; Inner Automorphism 8; Theorem 4; Definition of Inner Automorphism 8; Centre of a Group 9; Group of Automorphisms 12; Group of Automorphisms of a Cyclic Group 14.

2. Cayley's Theorem 23-32

Permutation Groups and Transformations 23; Equality of Two Permutations 24; Identity Permutations 24; Cayley’s Theorem for Finite Group 25; Regular Permutation Group 26; Cayley’s Theorem for Infinite Group 26.

3. Counting Principle 33-44

Conjugate Elements and Conjugacy Relation 33; Conjugate Classes 33; Conjugate Subgroups 33; Conjugate Class of a Subgroup 34; Self Conjugate Elements 34; Normalizer or Centralizer of an Element 34; Normalizer of a Subgroup of a Group 34; Self-Conjugate Subgroups 34; Counting Principle 34.

UNIT-II

4. Introduction to Rings and Subrings 45-72

Introduction 45; Ring 45; Examples of a Ring 46; Properties of a Ring 46; Types of Rings 46; Some Properties of a Ring 61; Integral Multiples of the Elements of a Ring 63; Some Special Kinds of Ring 63; Cancellation Laws in a Ring 65; Invertible Elements in a Ring with Unity 66; Division Rings or Skew Field 67; Quotient Ring or Factor Ring or Ring of Residue Classes 67; Subrings 69; Smallest Subring 72.

5. Integral Domain 73-84

Integral Domain 73; Sub-domain 74; Ordered Integral Domain 75; Inequalities 76; Well-ordered 76; Field 76; Some Theorems 76; The Characteristic of a Ring 78.

6. Ideals 85-100

Ideal 85; Theorem 85; Improper and Proper Ideals 86;Unit and Zero Ideals 86; Some Theorems 89; Smallest Ideal Containing a given Subset of a Ring 91; Principal Ideal 91; Principal Ideal Ring (or Principal Ideal Domain) 91; Prime Ideal 91; Maximal Ideal 92; Minimal Ideal 93; Sum of Two Ideals 93; Theorems 93; Product of Two Ideals 94; Important Theorems 94.

DIFFERENTIAL EQUATIONS & FOURIER SERIES

UNIT-III

1. Series Solutions of Differential Equations : Power Series Method 1-33

Power Series Method 1; Analytic or Regular or Holomorphic Function 1; Singular Point of the Differential Equation 1; Power Series 2; General Method for Solving a Differential Equation by Power Series Method 2; Frobenius Method 9; When Two Roots of Indicial Equation are Unequal and Differ by a Quantity not an Integer 10; Roots of the Indicial Equation Unequal and Differing by an Integer 17; When Roots of Indicial Equation are Equal 23; Series Solution Near an Ordinary Point (Power Series Method) 28.

2. Legendre’s Equation, Legendre’s Polynomial, Generating Function, 

Recurrence Formulae and Orthogonal Legendre’s Polynomials 34-78

Legendre’s Equation 34; Solution of Legendre’s Equation 34; Legendre’s Functions and its Properties 36; Legendre’s Functions 36; Legendre’s Function of the First Kind 37; Legendre’s Function of the Second Kind 37; Another Form of Legendre’s Polynomial Pn(x) 37; General Solution of Legendre’s Equation 39; Associated Legendre’s Functions 39; Generating Function for Legendre’s Polynomial 40; Orthogonal Properties of Legendre’s Polynomials 49; Recurrence Formulae 51; Beltrami’s Result 53; Christoffel’s Expansion Formula 54; Christoffel’s Summation Formula 55; Rodrigue’s Formula Pn(x) 65; Laplace’s Integral for Pn(x) 67; Some Bounds on Pn(x) 68.

3. Bessel’s Equation, Recurrence Formula 79-106

Bessel’s Equation 79; Solution of Bessel’s Equation 79, Bessel’s Functions 81; Bessel’s Function of the First Kind of Order n (or Index n) 81; Bessel’s Function of the Second Kind of Order n (or Neumann's Function) 82; General Solution of Bessel’s Equation 82; Integration of Bessel’s Equation for n = 0 and Bessel’s Functions of Zeroeth Order 82; Linear Dependence of Bessel Functions Jn(x) and J-n(x) 84; Recurrence Relations for Jn(x) 84; Elementary Functions 90.

UNIT-IV

4. Fourier Series 107-134

Introduction 107; Periodic Function 107; Even and Odd Functions 107; Fourier Series for Even and Odd Functions 108; Euler’s Formulae 109; Orthogonal Functions 109; Important Definite Integrals 110; To Determine the Fourier Coefficients a0, an and bn 110; Dirichlet Conditions 122; Fourier Series for Discontinuous Functions 122; Change of Interval 127.

5. Half Range Fourier Sine and Cosine Series 135-152

Half Range Series : Fourier Sine and Cosine Series 135; Parseval’s Theorem 143; Complex Form of Fourier Series 146.