Intermediate Differential Equations

✍ Scribed by Earl D. Rainville

• Publisher: Chelsea Publishing Co.
• Year: 1972
• Tongue: English
• Leaves: 320
• Edition: 2
• Category: Library

✦ Synopsis

The second edition of this book contains much new material,
particularly on the hypergeometric and Fuchsian equations and on the
equations of Bessel, Legendre, Hermite, and Laguerre. The treatment
of solutions about ordinary points and regular singular points has been
almost entirely rewritten and greatly expanded. The chapter of
preliminary material on complex variables, which is rewritten and
lengthened, provides the student with an adequate background to use
in his study of linear differential equations.
The portion of classical analysis to which the book is devoted is the
same as in the first edition, but a perusal of the table of contents will
reveal that within this framework many new topics have been treated.
The number of exercises has been increased to about 550, of which
400 did not appear in the first edition.
The purpose of the book remains what it was when I first wrote it
and may be described by quoting three paragraphs from the preface
to the first edition.
"Courses and books on differential equations are, for the most part,
either elementary or advanced. The student who wishes to go beyond
a first course is confronted with books which either are essentially the
same as the one he has already studied or are far too advanced for
his needs.

✦ Table of Contents

Title Page
Preface
Table of Contents
CHAPTER 1 - Linear Equations of Order Two
1. Elementary methods
2. Change of dependent variable
3. The normal form
4. The adjoint
5. Change of independent variable
6. The Schwartzian Derivative
7. Change of both variables
CHAPTER 2 - Complex Variables
8. Single-valued analytic functions
9. Singularities
10. Absolute value
11. Mapping
12. The point at infinity
13. Circles and straight lines
14. Linear fractional transformations
15. Elementary functions
16. Location and classification of singular points
17. Mapping near a singular point
18. Power series expansions
19. Cauchy product of power series
CHAPTER 3 - Solution about an Ordinary Point
20. Ordinary points of second-order linear equations
21. Singularities of second order linear equations
22. Solution valid near an ordinary point
23. Explicit solution by power series
CHAPTER 4 - Regular Singular Points
24. Solution near a regular singular point
25. Form and validity of solutions
26. The indicial equation
27. Difference of roots not an integer
28. Differentiation of a product of functions
29. Indicial equation with equal roots
30. Difference of roots a positive integer; nonlogarithmic case
31. The factorial function
32. Difference of roots a positive integer; logarithmic case
33. Summary
CHAPTER 5 - Additional Topics on Power Series Solutions
34. Many-term recurrence relations; ordinary points
35. Many-term recurrence relations; regular singular points
36. Bounding the error in computations
37. An equation with an irregular singular point
38. The role of the divergent series
CHAPTER 6 - Fuchsian Equations
39. Equation with all singular points regular
40. Elementary Fuchsian equations
41. Equation with three or more singular points
42. The Fuchsian invariant
43. Equation with three singular points
44. The Riemann-Papperitz equation
45. The Riemann P-symbol
46. A canonical form
47. Reduction to canonical form
CHAPTER 7 - The Hypergeometric Equation
48. A solution about the origin
49. A second solution
50. Kummer's twenty-four solutions
51. Regions of validity
52. Linearly independent pairs
53. Solution of specific equations
54. An alternative method
CHAPTER 8 - Logarithmic Solutions
55. Equal exponents
56. Difference of exponents a positive integer
57. Exceptional cases
CHAPTER 9 - Hypergeometric Functions
58. A fundamental identity
59. Another identity
60. The Gamma function
61. Other properties of F(alpha, beta; gamma; x)
62. Special hypergeometric functions
63. Generalized hypergeometric notation
CHAPTER 10 - Confluence of Singularities
64. The concept of confluence
65. A canonical form: Whittaker's equation
CHAPTER 11 - Whittaker's Equation
66. Solution with a three-term relation
67. Explicit solutions
68. Other solutions
69. Relations among solutions
70. Kummer's first formula
71. Solution of a three-term recurrence relation
72. Whittaker's confluent hypergeometric function
CHAPTER 12 - Bessel's Equation
73. Solutions of Bessel's equation
74. Hankel functions
75. The modified Bessel equation
76. Kelvin's ber and bei functions
77. Recurrence relations for Jn
78. Hypergeometric form of Jn
79. Integrals
80. Orthogonality
CHAPTER 13 - Legendre's Equation
81. Solution of Legendre's equation
82. Solution about an ordinary point
83. Legendre polynomials; orthogonality
84. Generating functions for Pn
85. Properties of Legendre polynomials
86. The associated Legendre equation
CHAPTER 14 - Equations of Hermite and Laguerre
87. Hermite polynomials
88. Hermite's differential equation
89. A Rodrigues formula for Hn
90. Laguerre polynomials
91. Properties of Laguerre polynomials
CHAPTER 15 - Riccati Equations
92. Relation to linear equations
93. General solution when a particular solution is known
94. Polynomial solutions
95. The cross-ratio theorem
96. Other proofs of Theorem 40
CHAPTER 16 - Applications
97. Introduction
98. Temperature in a solid cylinder
99. Remarks on other boundary value problems
100. A detail in the design of large pipes
101. Deformation of circular plates
102. Another choice for y(x)
103. Some equations drawn from applications
104. Factorization of differential operators
Index