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Rainville E. D. — Intermediate Course in Differential Equations
Rainville E. D. — Intermediate Course in Differential Equations

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Название: Intermediate Course in Differential Equations

Автор: Rainville E. D.


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.
I have written this book with three main aims. The book is intended as a bridge for the gap between elementary courses and really advanced courses. It furnishes the student with further tools for constructing computable solutions for specific differential equations, in particular for linear equations in the neighborhood of regular singular points. Finally, it affords an introduction to several topics of importance in the classical theory.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1943

Количество страниц: 213

Добавлена в каталог: 13.11.2012

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Предметный указатель
$Bei_{n}x$      183
$Ber_{n}x$      183
$Kei_{n}x$      183
$Ker_{n}x$      183
Abel      48 133
Abel's formula      48
Adjoint      9
Adjoint, invariant of      10
Analytic continuation      62
Analytic function      26 54
applications      192—209
Bateman      177
Bernoulli numbers      61
Bernoulli, Jacob      61
bessel      96
Bessel functions of index zero, application      200
Bessel functions of index zero, orthogonality of      201
Bessel functions of the second kind, Neumann's      181
Bessel — Clifford equation      183
Bessel's equation of index n      180
Bessel's equation of index n, applications of      197 198
Bessel's equation of index n, obtained by confluence from Fuchsian equation      167
Bessel's equation of index two      105
Bessel's modified equation of index n      182
Bessel's modified equation of index zero      96
Bilinear transformation      18 65
Boundary-value problem      199 206
Cauchy      54
Cauchy product of series      72
Cauchy — Riemann equations      54
Change of both variables      19
Change of dependent variable      2
Change of independent variable      12
Circular plate, deformation of      193
Confluence of Fuchsian equation into Bessel's equation      167
Confluence of Fuchsian equation with three singular points      165
Confluence of hypergeometric equation into Bessel's equation of index zero      163
Confluence of singularities      163
Confluent hypergeometric equation, Pochhammer — Barnes      190
Confluent hypergeometric equation, Whittaker's      169 171
Conformal mapping      63
Contiguous functions      159
Crelle      133
Cross-ratio group      46 153
Cross-ratio of solutions of Riccati equation      45
Differential operator, factorization of      21 30 43
Differential operator, proper divisor      44
Equivalence of two second-order equations      7
Equivalence of two second-order equations, under change of both variables      19
Essential singularity of a function      56
Euler      96 145
Euler's constant      100
Existence theorem on solution near a regular singular point      90
Existence theorem on solution near an ordinary point      83
Exponents of an equation at a regular singular point      87
Fuchs      130 132 133
Fuchsian equation      130
Fuchsian equation of order n      133
Fuchsian equation with (m+1) singularities      132 134
Fuchsian equation with 3 singularities      135
Fuchsian equation with 4 singularities      137
Fuchsian invariant      135
Gauss      136 145 159
Generating function for Hermite polynomials      187
Generating function for Laguerre polynomials      190 209
Heat equation      199 209
Hermite polynomials      187
Hermite polynomials, generating function for      187
Hermite polynomials, orthogonality of      189
Holomorphic      54
Hutchinson      53 56
Hypergeometric equation      142 144
Hypergeometric equation, general solution of      148
Hypergeometric equation, Kummer's 24 solutions of      157
Hypergeometric function, derivative of      162
Hypergeometric function, special cases of      160
Hypergeometric series      145
In variance of Schwarzian Derivative      17
Indicial equation      87
Integrating factor      9
Invariant in normal form of second-order equation      6
Invariant, Fuchsian      135
Invariants under change of independent variable      29 30
Involution      10
Isolated essential singularity      56
Kelvin      183
Kelvin's ber and bei functions      183
Kelvin's ber and bei functions, application of      193
Kelvin's ber and bei functions, equation satisfied by      185
Kummer      158
Kummer's 24 solutions of the hypergeometric equation      157
Kummer's first formula      175
Laguerre polynomials      189 191 209
Laguerre polynomials, generalized      189
Laguerre polynomials, generating function for      190 209
Laguerre polynomials, orthogonality of      190
Laplace's equation      206 209
Laurent series      59
legendre      160
Legendre polynomials      160 185
Legendre polynomials, orthogonality of      186
Legendre's equation      161 185 207
Legendre's equation, associated      186
Linear equation of second order with only one singularity      135
Linear fractional transformation      18 65
Liouville      57
Liouville's theorem      57
Mascheroni's constant      100
Modified Bessel equation of index n      182
Modified Bessel equation of index zero      96
Multiplication of series      72
Natural boundary      63
Neumann Bessel function of the second kind      181
Neumann sphere      56
Normal form of second-order equation      5
Operators, factorization of      21 30 43
Operators, factorization of, proper divisor of      44
Ordinary points of second-order equation      79
Ordinary points of second-order equation, solution valid near      80
Ordinary points of second-order equation, solution valid near, existence theorem      83
Orthogonality of Bessel functions      201
Orthogonality of Hermite polynomials      189
Orthogonality of Laguerre polynomials      190
Orthogonality of Legendre polynomials      186
Papperitz      139
Penstock design, a detail in      192
Pochhammer — Barnes confluent hypergeometric function      190
Pochhammer — Barnes confluent hypergeometric function, application of      195
Pole      56
Polynomial solutions of partial differential equations      208
Polynomial solutions of Riccati equations      36
Power series      58
Power series, inequality on coefficients in      62
Power series, multiplication of      72
Power series, radius of convergence of      58
Regular singular point      85
Regular singular point, solution valid near      86
Regular singular point, solution valid near, existence theorem      90
Riccati      31
Riccati equation      31
Riccati equation with (n+1) polynomial solutions      51
Riccati equation with 5 polynomial solutions      42
Riccati equation with 6 polynomial solutions      52
Riccati equation with given general solution      51
Riccati equation, application of      197
Riccati equation, form of general solution      34
Riccati equation, polynomial solutions of      36
Riccati equation, relation to second-order equation      32
Riccati equation, solution, when 3 particular solutions are known      47
Riccati equation, solution, when a particular solution is known      32
Riccati problem      31
Riemann      54 159
Riemann P-function      140
Riemann — Papperitz equation      137
Schwarzian derivative      16
Self-adjoint      11
Single-valued function      53
Singularities of functions      55
Singularities of second-order equation      83
Singularities of second-order equation, regular      85
Singularities of second-order equation, regular, solution valid near      86
Singularities of solutions of a differential equation      84
Temperatures in a cylinder      198
Transformation of both variables      19
Transformation of dependent variable      2
Transformation of independent variable      12
Transformation, bilinear or linear fractional      18 65
Whitehead      184
Whittaker      171
Whittaker's confluent hypergeometric function      176
Whittaker's equation      169 171
Whittaker's equation, solutions of      174
Wronski      47
wronskian      47
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