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Cayley A. — An elementary treatise on elliptic functions
Cayley A. — An elementary treatise on elliptic functions



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Название: An elementary treatise on elliptic functions

Автор: Cayley A.

Язык: en

Рубрика: Математика/Анализ/Специальные функции/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1895

Количество страниц: 386

Добавлена в каталог: 09.04.2005

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Предметный указатель
Abel, linear and quadric transformations      322 325 375 376
Abel, linear and quadric transformations, irrational transformation      382
Addition      see “Function (2)...(9)”
Arc of curve, determination of curves the arc of which represents elliptic integral of first kind      352 (Chap. xv.)
Arc of curve, representing integrals $E(k, \phi)$, $F(k, \phi)$ (ellipse, hyperbola, lemniscate)      35
Arithmetico-geometrical mean      332
Baehr’s formuls for multiplication of sn, en, dn      79
Circular functions, illustration by reference to      10
Definitions and Notations      1 (Chap. I.)
Definitions and Notations, amplitude, modulus, complementary modulus, parameter      3
Definitions and Notations, complete functions $F_1k$, $E_1k$      4
Definitions and Notations, elliptic functions am, sin am, cos am, $\Delta$ am, or sn, en, dn      8
Definitions and Notations, elliptic integrals $F(k, \phi)$, $E(k, \phi)$, $\Pi(n,k, \phi)$      3
Definitions and Notations, Eu, Zu, $\Pi(u, a)$      15
Definitions and Notations, K, K'      12
Definitions and Notations, \Theta u$, Hu      16
Differential equation of third order, satisfied by modulus in transformation of nth order      222
Differential equation, addition-equation      5 21
Differential equation, equation $dx/\sqrt{X} = dy/\sqrt{Y}$      339 (Chap XIV.)
Differential equation, partial, satisfied by $\Theta$, H, &c, and the numerators and denominators in multiplication and transformation of sn, en, dn      226 (Chap. IX.).
Differential equation, satisfied by $F(k, \phi)$, $E(k, \phi)$, or by $F_1k$, $E_1k$      47 50
Differential equation, satisfied by multiplier      220
Differential equation, special equation      250
Dimidiation      72 see
Doubly infinite products      see “Factorial”
Duplication      71 see
Duplication by two quadratic transformations      181
Euler, partition formula      290
Factorial formule      92 97
Factorial formule, doubly infinite product forms      101 300
Function, (1), $\dfrac{Rdx}{\sqry{X}}$ reduction to standard form 1      311 (Chap. xii.)
Function, (10), $\Theta u$, Hu      142 (Ch. VI.) 226
Function, (10), $\Theta$, h expressed as q-functions      291
Function, (10), connexion with sn, en, dn, Zu and $\Pi(u, a)$      155
Function, (10), development in q-series      297
Function, (10), double factorial expressions      300
Function, (10), Hu introduced      156
Function, (10), multiplication      159
Function, (10), partial differential equations      229
Function, (10), properties of $\Theta u$      150 152
Function, (10), transformation      306
Function, (11), connexion with $\Theta u$, Hu      161
Function, (11), connexion with functions $\Theta$, H      309
Function, (11), numerators and denominator in multiplication and transformation of sn, en, dn      226 (Chap. rx.)
Function, (11), partial differential equations      235
Function, (11), verification for cubic transformation      245
Function, (12), $\Theta$, H expressed as q-formulae      291
Function, (12), derivation of the q-formulas      282
Function, (12), q-functions      282 (Chap. xi.)
Function, (13), further theory      342
Function, (13), Lagrange’s integration      339
Function, (13), The general equation $dx/\sqrt{X} = dy/\sqrt{Y}$      339 (Chap. XIV.)
Function, (2), addition - equation $\dfrac{d\phi}{\Delta \phi}+\dfrac{d\psi}{\Delta \psi} = 0$      21 (Chap. II.)
Function, (3), $F(k, \phi)$, march of function      41
Function, (3), addition      103
Function, (3), imaginary modulus $\sin(\alpha + i\beta)$      367
Function, (3), properties as function of k      46
Function, (3), quadric transformation, &c      327
Function, (4), $E(k, \phi)$      41 (see also “Infra march
Function, (4), addition      104
Function, (4), properties as function of k      46
Function, (4), quadric transformation, &c      327
Function, (5), $\Pi(n, k, \phi)$      see also “(9) infra”
Function, (5), addition      104
Function, (5), addition of parameters, and reduction to standard form      117
Function, (5), interchange of amplitude and parameter      133
Function, (5), outline of further theory      108
Function, (5), reduction of parameter to form $sn(\alpha + i\beta)$      114
Function, (6), gd u, sg u, eg u., addition and other properties      56
Function, (7), addition and subtraction formulae      63
Function, (7), anticipation of doubly infinite product forms      101
Function, (7), connexion with $\Theta u$, Hu      155 156
Function, (7), dimidiation      72
Function, (7), duplication      71
Function, (7), factorial formulae      92
Function, (7), functions of u + (0, 1, 2, 3) K + (0, 1, 2, 3) iK'      69
Function, (7), imaginary transformation      68
Function, (7), multiplication      78
Function, (7), n-thic transformation      251 (Ch. x.).
Function, (7), new form of same      97
Function, (7), periods 4K, 4iK'      66
Function, (7), quadric transformations      183
Function, (7), sn u, en u, dn u      63 (Chap. IV.)
Function, (7), triplication      77
Function, (8), addition      107
Function, (8), connexion with $\Theta u$      113
Function, (8), Em, Zu, connexion with $E(k, \phi)$      107
Function, (8), further theory      147
Function, (9), $\Pi(u, a)$      142 (Chap. VI.).
Function, (9), $\Pi(u, a)$ expressed in terms of Qu      151
Function, (9), addition of amplitudes      157
Function, (9), addition of parameters      157
Function, (9), connexion with $\Pi(n, k, \phi)$      107
Function, (9), connexion with 6m      113
Function, (9), interchange of amplitude and parameter      159
Function, (9), values of $\Pi(u+a, a)$ for $a=\dfrac12 iK', \dfrac12 K, \dfrac12 K+\dfrac12 iK'$      144
Gauss, the arithmetico - geometrical mean      331 338
Glaisher, proof of Legendre’s relation between complete functions $F_1k$, &c      49
Glaisher, tables of theta functions      155
Gordan, integral reducible to elliptic functions      369
Gudermann      44 58
Gudermannian      58 (see also “Function (6)”)
Imaginary transformation, Jacobi’s      68
Imaginary, reduction of given, to form $sn(\alpha + i\beta)$      114
Integrals involving root of a quintic function and reducible to elliptic integrals      360 (Chap. XVI.)
Integrals involving root of a quintic function and reducible to elliptic integrals, involving root of a sextic function      369
Integration, general theorem of      354
jacobi      16 18 19 23 28 66 150 176 220 223 224 227 323
Kummer, formula in hypergeometric series      55
Lagrange, integration of differential equation $dx/\sqrt{X} = dy/\sqrt{Y}$      339
Landen’s theorem      30 327
Legendre, his relation between complete functions, $E_1$, $E_1'$, $F_1$, $F_1'$, formula relating to third kind of elliptic integral      119 121 134
Legendre, proof of addition-equation by spherical triangle      27
Legendre, reduction of differential expression to standard form      314
Liouville, formula for arc of curve      40
Modular equation or relations, cubic transformation      188 195 206
Modular equation or relations, linear transformation      322
Modular equation or relations, odd-prime transformation      45 200 275 278
Modular equation or relations, quadric transformation      45 179
Modular equation or relations, quintic transformation      192 195
Modular equation or relations, septic transformation      194
Modular equation, differential equation of third order satisfied by the transformed modulus      222
Modular equation, properties of      200
Modular equation, verification for quadric transformation      223
Multiplication from two transformations      201 280
Multiplication of sn, en, dn      78 86
Multiplication tables      80 81
Multiplier in cubic, &c. transformations      203
Multiplier, differential equation satisfied by      220
Multiplier, Jacobi’s relation $nM^2=\dfrac{\lambda \lambda'^2}{kk'^2}\dfrac{dk}{d\lambda}$      218
Multiplier, relation between M, K, $\Lambda$, E, G      224
Notation explained      5
Numerical instance for complete functions $E_1$, $F_1$, and for incomplete F      337
Richelot, representation of given imaginary quantity in form $sn(\alpha + i\beta)$      117
Roberts, M., integral reducible to elliptic functions      369
Serret, formula for arc of curve      40
Tables of the theta functions      155
Transformation of functions, H, $\Theta$      306
Transformation, combined, and irrational      379
Transformation, cubic      188 195 206 216 245
Transformation, general outline      164 (Chap. VII.)
Transformation, Landen’s theorem      30 327
Transformation, linear transformation of integral      312 319 370
Transformation, multiplication formulae obtained from two transformations      280
Transformation, odd or odd-prime, by the n-division of the complete functions      251 (Chap. x.)
Transformation, quadric      179 (Chap viii.) 323 376
Transformation, quintic      191 197
Transformation, septic      194
Transformation, two transformations leading to multiplication      201
Triplication      77 (see also “Function (7)”)
Walton, proof of addition-equation      22
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