Dipankar De, PhD is an contractual professor and guest lecturer with more than 40 years of experience. He has published several research papers in various reputed journals in the fields of fuzzy mathematics and differential geometry.

• Preface xiiiIntroduction xvAbout the Book xviiPart I: Ordinary Differential Equations 11 Preliminaries I 31.1 Introduction 31.2 Formation of a Differential Equation 41.3 Family of Curves Represented by Ordinary Differential Equations 81.4 Equation of the First Order and First Degree 111.5 Equations of the First Order and Higher Degree 241.6 Linear Differential Equation 301.7 Other Methods of Finding P.I. 401.8 Differential Equation of Other Types 441.9 Orthogonal Trajectories 511.10 Examples 521.11 Exercise 632 Existence Theorems 672.1 Introduction 672.2 Initial Value Problems and Boundary Value Problems 692.3 Picard's Method of Successive Approximation 702.4 Lipschitz Condition 782.5 Picard's Theorem: Existence and Uniqueness Theorem 802.6 Singular Solutions 902.7 Clairaut Equation 922.8 Examples 952.9 Exercise 993 System of Linear Differential Equations-I 1013.1 Introduction 1013.2 Matrix Form of a Linear System 1023.3 Reduction of an nth-Order Equation 1043.4 Matrix Preliminaries 1073.5 Fundamental Set of Solutions 1093.6 Solution of Non-Homogeneous Linear Systems 1263.7 Linear System with Constant Coefficients 1303.8 Exercise 1384 Systems of Linear Differential Equations-II 1414.1 Introduction 1414.2 Linearly Dependent and Independent Functions 1444.3 The Second-Order Homogeneous Equation 1514.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters 1574.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients 1634.6 Examples 1744.7 Exercise 1775 Adjoint Equation 1815.1 Introduction 1815.2 Adjoint Equation 1815.3 Green's Formula 1945.4 Examples 1965.5 Exercise 2016 Boundary Value Problem 2036.1 Introduction 2036.2 Green's Function 2076.3 Examples 2106.4 Exercise 2157 Strum Liouville Problem 2177.1 Introduction 2177.2 Strum–Liouville Equation 2177.3 Orthogonality of Eigen Functions 2207.4 Orthonormal Set of Functions 2227.5 Gram–Schmidt Process of Orthonormalization 2227.6 Reality of Eigenvalues 2257.7 Examples 2297.8 Exercise 233Part II: Special Functions 2358 Preliminaries II 2378.1 Introduction 2378.2 Infinite Series 2378.3 Infinite Integrals 2428.4 Infinite Products 2458.5 Some Theorems on Functions of Complex Variables 2468.6 Exercise 2509 Series Solution of Differential Equations 2539.1 Introduction 2539.2 Power Series 2549.3 Power Series Solution Near the Ordinary Point x = x0 2569.4 Series Solution About Regular Singular Point x = 0: Frobenius Method 2619.5 Examples 2699.6 Exercise 30710 Hypergeometric Functions 30910.1 Introduction 30910.2 Differentiation of Hypergeometric Functions 31410.3 An Integral Formula for a Hypergeometric Function 31610.4 Transformation of F (α,β ,γ ;x) 32410.5 Hypergeometric Equation 32810.6 Confluent Hypergeometric Series 33510.7 Contiguous Hypergeometric Functions 34210.8 Generalized Hypergeometric Series 34410.9 Integrals Involving Generalized Hypergeometric Functions 35510.10 Some Special Generalized Hypergeometric Functions 35710.11 Barnes Type Contour Integrals 36510.12 Example 36710.13 Exercise 37011 Bessel Functions 37311.1 Introduction 37311.2 Bessel's Equation 37411.3 Recurrence Formulae for Jn(x) 37811.4 Expansion of J0 , J1 ,and J1/2 38211.5 Generating Function for Jn(x) 39911.6 Modified Bessel Functions 41111.7 Equations Reducible to Bessel Equation 41611.8 Orthogonality of Bessel Functions 41811.9 Zeros of Bessel Functions 42411.10 Ber and Bei Functions 42711.11 Exercise 42812 Legendre Polynomials 43112.1 Introduction 43112.2 Legendre's Equation 43212.3 Another Form of Legendre's Polynomial Pn(x) 43512.4 Generating Function for Legendre's Polynomials 43812.5 Various Forms of Pn(x) 44212.6 Recurrence Formulae for Pn(x) 44612.7 Christoffel's Summation Formula 44812.8 Orthogonality of Legendre Polynomials 45112.9 Fourier–Legendre's Expansion of f (x) 45312.10 Associated Legendre's Functions 46812.11 Legendre's Functions of the Second Kind—Qn(x) 48112.12 Examples 48812.13 Exercise 49413 Hermite Polynomials 49713.1 Introduction 49713.2 Hermite Equation and Its Solution 49713.3 Generating Function for Hermite Polynomials 50213.4 Recurrence Relations 50613.5 Orthogonal Property 51113.6 Expansion of Polynomials 51213.7 More Generating Functions 51513.8 Examples 51713.9 Exercise 52214 Laguerre Polynomials 52514.1 Introduction 52514.2 Laguerre's Equation and Its Solution 52514.3 Generating Function of Laguerre Polynomials 52814.4 Orthogonality Properties of Laguerre Polynomials 53114.5 Recurrence Relations 53314.6 Expansion of Laguerre Polynomials 53714.7 Properties of Laguerre Polynomials 53914.8 Generalized Laguerre Polynomial 54114.9 Examples 55114.10 Exercise 55815 Jacobi Polynomials 56115.1 Introduction 56115.2 Jacobi Polynomial 56215.3 Generating Functions 56415.4 Rodrigues' Formula 56715.5 Orthogonality of Jacobi Polynomial 56815.6 Recurrence Relations 57215.7 Expansions 58015.8 Examples 58215.9 Exercise 58516 Chebyshev Polynomials 58716.1 Introduction 58716.2 Chebyshev Polynomials 58716.3 Orthogonality Property 59116.4 Recurrence Relations 59316.5 Identities of Chebyshev Polynomials 59416.6 Expansions 59516.7 Generating Function 59816.8 Rodrigues Formula of Chebyshev Polynomials 59916.9 Exercise 601Appendix A: Answer to Even-Numbered Exercises 603References 611Index 613

Dipankar De, PhD is an contractual professor and guest lecturer with more than 40 years of experience. He has published several research papers in various reputed journals in the fields of fuzzy mathematics and differential geometry.

• Preface xiiiIntroduction xvAbout the Book xviiPart I: Ordinary Differential Equations 11 Preliminaries I 31.1 Introduction 31.2 Formation of a Differential Equation 41.3 Family of Curves Represented by Ordinary Differential Equations 81.4 Equation of the First Order and First Degree 111.5 Equations of the First Order and Higher Degree 241.6 Linear Differential Equation 301.7 Other Methods of Finding P.I. 401.8 Differential Equation of Other Types 441.9 Orthogonal Trajectories 511.10 Examples 521.11 Exercise 632 Existence Theorems 672.1 Introduction 672.2 Initial Value Problems and Boundary Value Problems 692.3 Picard's Method of Successive Approximation 702.4 Lipschitz Condition 782.5 Picard's Theorem: Existence and Uniqueness Theorem 802.6 Singular Solutions 902.7 Clairaut Equation 922.8 Examples 952.9 Exercise 993 System of Linear Differential Equations-I 1013.1 Introduction 1013.2 Matrix Form of a Linear System 1023.3 Reduction of an nth-Order Equation 1043.4 Matrix Preliminaries 1073.5 Fundamental Set of Solutions 1093.6 Solution of Non-Homogeneous Linear Systems 1263.7 Linear System with Constant Coefficients 1303.8 Exercise 1384 Systems of Linear Differential Equations-II 1414.1 Introduction 1414.2 Linearly Dependent and Independent Functions 1444.3 The Second-Order Homogeneous Equation 1514.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters 1574.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients 1634.6 Examples 1744.7 Exercise 1775 Adjoint Equation 1815.1 Introduction 1815.2 Adjoint Equation 1815.3 Green's Formula 1945.4 Examples 1965.5 Exercise 2016 Boundary Value Problem 2036.1 Introduction 2036.2 Green's Function 2076.3 Examples 2106.4 Exercise 2157 Strum Liouville Problem 2177.1 Introduction 2177.2 Strum–Liouville Equation 2177.3 Orthogonality of Eigen Functions 2207.4 Orthonormal Set of Functions 2227.5 Gram–Schmidt Process of Orthonormalization 2227.6 Reality of Eigenvalues 2257.7 Examples 2297.8 Exercise 233Part II: Special Functions 2358 Preliminaries II 2378.1 Introduction 2378.2 Infinite Series 2378.3 Infinite Integrals 2428.4 Infinite Products 2458.5 Some Theorems on Functions of Complex Variables 2468.6 Exercise 2509 Series Solution of Differential Equations 2539.1 Introduction 2539.2 Power Series 2549.3 Power Series Solution Near the Ordinary Point x = x0 2569.4 Series Solution About Regular Singular Point x = 0: Frobenius Method 2619.5 Examples 2699.6 Exercise 30710 Hypergeometric Functions 30910.1 Introduction 30910.2 Differentiation of Hypergeometric Functions 31410.3 An Integral Formula for a Hypergeometric Function 31610.4 Transformation of F (α,β ,γ ;x) 32410.5 Hypergeometric Equation 32810.6 Confluent Hypergeometric Series 33510.7 Contiguous Hypergeometric Functions 34210.8 Generalized Hypergeometric Series 34410.9 Integrals Involving Generalized Hypergeometric Functions 35510.10 Some Special Generalized Hypergeometric Functions 35710.11 Barnes Type Contour Integrals 36510.12 Example 36710.13 Exercise 37011 Bessel Functions 37311.1 Introduction 37311.2 Bessel's Equation 37411.3 Recurrence Formulae for Jn(x) 37811.4 Expansion of J0 , J1 ,and J1/2 38211.5 Generating Function for Jn(x) 39911.6 Modified Bessel Functions 41111.7 Equations Reducible to Bessel Equation 41611.8 Orthogonality of Bessel Functions 41811.9 Zeros of Bessel Functions 42411.10 Ber and Bei Functions 42711.11 Exercise 42812 Legendre Polynomials 43112.1 Introduction 43112.2 Legendre's Equation 43212.3 Another Form of Legendre's Polynomial Pn(x) 43512.4 Generating Function for Legendre's Polynomials 43812.5 Various Forms of Pn(x) 44212.6 Recurrence Formulae for Pn(x) 44612.7 Christoffel's Summation Formula 44812.8 Orthogonality of Legendre Polynomials 45112.9 Fourier–Legendre's Expansion of f (x) 45312.10 Associated Legendre's Functions 46812.11 Legendre's Functions of the Second Kind—Qn(x) 48112.12 Examples 48812.13 Exercise 49413 Hermite Polynomials 49713.1 Introduction 49713.2 Hermite Equation and Its Solution 49713.3 Generating Function for Hermite Polynomials 50213.4 Recurrence Relations 50613.5 Orthogonal Property 51113.6 Expansion of Polynomials 51213.7 More Generating Functions 51513.8 Examples 51713.9 Exercise 52214 Laguerre Polynomials 52514.1 Introduction 52514.2 Laguerre's Equation and Its Solution 52514.3 Generating Function of Laguerre Polynomials 52814.4 Orthogonality Properties of Laguerre Polynomials 53114.5 Recurrence Relations 53314.6 Expansion of Laguerre Polynomials 53714.7 Properties of Laguerre Polynomials 53914.8 Generalized Laguerre Polynomial 54114.9 Examples 55114.10 Exercise 55815 Jacobi Polynomials 56115.1 Introduction 56115.2 Jacobi Polynomial 56215.3 Generating Functions 56415.4 Rodrigues' Formula 56715.5 Orthogonality of Jacobi Polynomial 56815.6 Recurrence Relations 57215.7 Expansions 58015.8 Examples 58215.9 Exercise 58516 Chebyshev Polynomials 58716.1 Introduction 58716.2 Chebyshev Polynomials 58716.3 Orthogonality Property 59116.4 Recurrence Relations 59316.5 Identities of Chebyshev Polynomials 59416.6 Expansions 59516.7 Generating Function 59816.8 Rodrigues Formula of Chebyshev Polynomials 59916.9 Exercise 601Appendix A: Answer to Even-Numbered Exercises 603References 611Index 613

Dipankar De, PhD is an contractual professor and guest lecturer with more than 40 years of experience. He has published several research papers in various reputed journals in the fields of fuzzy mathematics and differential geometry.

• Preface xiiiIntroduction xvAbout the Book xviiPart I: Ordinary Differential Equations 11 Preliminaries I 31.1 Introduction 31.2 Formation of a Differential Equation 41.3 Family of Curves Represented by Ordinary Differential Equations 81.4 Equation of the First Order and First Degree 111.5 Equations of the First Order and Higher Degree 241.6 Linear Differential Equation 301.7 Other Methods of Finding P.I. 401.8 Differential Equation of Other Types 441.9 Orthogonal Trajectories 511.10 Examples 521.11 Exercise 632 Existence Theorems 672.1 Introduction 672.2 Initial Value Problems and Boundary Value Problems 692.3 Picard's Method of Successive Approximation 702.4 Lipschitz Condition 782.5 Picard's Theorem: Existence and Uniqueness Theorem 802.6 Singular Solutions 902.7 Clairaut Equation 922.8 Examples 952.9 Exercise 993 System of Linear Differential Equations-I 1013.1 Introduction 1013.2 Matrix Form of a Linear System 1023.3 Reduction of an nth-Order Equation 1043.4 Matrix Preliminaries 1073.5 Fundamental Set of Solutions 1093.6 Solution of Non-Homogeneous Linear Systems 1263.7 Linear System with Constant Coefficients 1303.8 Exercise 1384 Systems of Linear Differential Equations-II 1414.1 Introduction 1414.2 Linearly Dependent and Independent Functions 1444.3 The Second-Order Homogeneous Equation 1514.4 Non-Homogeneous Equation of Second-Order: Method of Variation of Parameters 1574.5 Higher-Order Homogeneous Linear Differential Equations with Constant Coefficients 1634.6 Examples 1744.7 Exercise 1775 Adjoint Equation 1815.1 Introduction 1815.2 Adjoint Equation 1815.3 Green's Formula 1945.4 Examples 1965.5 Exercise 2016 Boundary Value Problem 2036.1 Introduction 2036.2 Green's Function 2076.3 Examples 2106.4 Exercise 2157 Strum Liouville Problem 2177.1 Introduction 2177.2 Strum–Liouville Equation 2177.3 Orthogonality of Eigen Functions 2207.4 Orthonormal Set of Functions 2227.5 Gram–Schmidt Process of Orthonormalization 2227.6 Reality of Eigenvalues 2257.7 Examples 2297.8 Exercise 233Part II: Special Functions 2358 Preliminaries II 2378.1 Introduction 2378.2 Infinite Series 2378.3 Infinite Integrals 2428.4 Infinite Products 2458.5 Some Theorems on Functions of Complex Variables 2468.6 Exercise 2509 Series Solution of Differential Equations 2539.1 Introduction 2539.2 Power Series 2549.3 Power Series Solution Near the Ordinary Point x = x0 2569.4 Series Solution About Regular Singular Point x = 0: Frobenius Method 2619.5 Examples 2699.6 Exercise 30710 Hypergeometric Functions 30910.1 Introduction 30910.2 Differentiation of Hypergeometric Functions 31410.3 An Integral Formula for a Hypergeometric Function 31610.4 Transformation of F (α,β ,γ ;x) 32410.5 Hypergeometric Equation 32810.6 Confluent Hypergeometric Series 33510.7 Contiguous Hypergeometric Functions 34210.8 Generalized Hypergeometric Series 34410.9 Integrals Involving Generalized Hypergeometric Functions 35510.10 Some Special Generalized Hypergeometric Functions 35710.11 Barnes Type Contour Integrals 36510.12 Example 36710.13 Exercise 37011 Bessel Functions 37311.1 Introduction 37311.2 Bessel's Equation 37411.3 Recurrence Formulae for Jn(x) 37811.4 Expansion of J0 , J1 ,and J1/2 38211.5 Generating Function for Jn(x) 39911.6 Modified Bessel Functions 41111.7 Equations Reducible to Bessel Equation 41611.8 Orthogonality of Bessel Functions 41811.9 Zeros of Bessel Functions 42411.10 Ber and Bei Functions 42711.11 Exercise 42812 Legendre Polynomials 43112.1 Introduction 43112.2 Legendre's Equation 43212.3 Another Form of Legendre's Polynomial Pn(x) 43512.4 Generating Function for Legendre's Polynomials 43812.5 Various Forms of Pn(x) 44212.6 Recurrence Formulae for Pn(x) 44612.7 Christoffel's Summation Formula 44812.8 Orthogonality of Legendre Polynomials 45112.9 Fourier–Legendre's Expansion of f (x) 45312.10 Associated Legendre's Functions 46812.11 Legendre's Functions of the Second Kind—Qn(x) 48112.12 Examples 48812.13 Exercise 49413 Hermite Polynomials 49713.1 Introduction 49713.2 Hermite Equation and Its Solution 49713.3 Generating Function for Hermite Polynomials 50213.4 Recurrence Relations 50613.5 Orthogonal Property 51113.6 Expansion of Polynomials 51213.7 More Generating Functions 51513.8 Examples 51713.9 Exercise 52214 Laguerre Polynomials 52514.1 Introduction 52514.2 Laguerre's Equation and Its Solution 52514.3 Generating Function of Laguerre Polynomials 52814.4 Orthogonality Properties of Laguerre Polynomials 53114.5 Recurrence Relations 53314.6 Expansion of Laguerre Polynomials 53714.7 Properties of Laguerre Polynomials 53914.8 Generalized Laguerre Polynomial 54114.9 Examples 55114.10 Exercise 55815 Jacobi Polynomials 56115.1 Introduction 56115.2 Jacobi Polynomial 56215.3 Generating Functions 56415.4 Rodrigues' Formula 56715.5 Orthogonality of Jacobi Polynomial 56815.6 Recurrence Relations 57215.7 Expansions 58015.8 Examples 58215.9 Exercise 58516 Chebyshev Polynomials 58716.1 Introduction 58716.2 Chebyshev Polynomials 58716.3 Orthogonality Property 59116.4 Recurrence Relations 59316.5 Identities of Chebyshev Polynomials 59416.6 Expansions 59516.7 Generating Function 59816.8 Rodrigues Formula of Chebyshev Polynomials 59916.9 Exercise 601Appendix A: Answer to Even-Numbered Exercises 603References 611Index 613