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Baker H.F. — Abel's Theorem and the Allied Theory Including The Theory of the Theta Functions
Baker H.F. — Abel's Theorem and the Allied Theory Including The Theory of the Theta Functions

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Название: Abel's Theorem and the Allied Theory Including The Theory of the Theta Functions

Автор: Baker H.F.

Аннотация:

Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincaré, Clebsch, Jacobi and other outstanding mathematicians of the last century, was mainly an analytical theory. In our century the methods and ideas of topology, commutative algebra and Grothendieck's schemes enriched it and seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This classic book, written in 1897, covers the whole of algebraic geometry and associated theories. Baker discusses the subject in terms of transcendental functions, and theta functions in particular. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1897

Количество страниц: 684

Добавлена в каталог: 25.03.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\wp$ Function      292 324 333 516
Abel      90 173 205 206 207ff. 221 225ff. 231 243 246 377 397 657
Abelian functions      236 600
Abelian integrals      see Integrals;
Abelian matrix      669
Abel’s Abel’s proof of      219 220
Abel’s converse of      222
Abel’s differential equations      225 ff.
Abel’s for curves in space      231
Abel’s for factorial functions      397
Abel’s for radical functions      377
Abel’s number of independent equations given by      222ff.
Abel’s proof of      213
Abel’s statement of      210 214
Abel’s theorem      207 ff.
Addition equation for hyperelliptie theta functions, deduced algebraically      331 ff.
Addition for theta functions in general      457 461 472 476 481 513 521
Adjoint expression of rational function by      127 see Integrals Sets Lots.
Adjoint number of terms in      128
Adjoint polynomial (or curve), definition of      121
Appell      200 392
Argument and parameter, interchange of      16 185 187 189 191 194 206
Associated, associated system of factorial functions      397
Associated, Forms associated with fundamental integral functions      62
Associated, integrals of second kind associated with integrals of the first kind      193 195 198 532
Automorphic connection with factorial functions      439ff.
Automorphic functions, simple ease of      352 ff.
Azygetic characteristics      487 497
Azygetic transformation of      542 547
Bacharach      141
Bacharaeh’s modification of Cayley’s theorem for plane curves      141
Bachmann      666
Bam      57 112
Benoist      153 156 222 647
Bertini      137
Biermann      663
Biquadratic      see Gopel.
Birational transformation of a Riemann surface , by $\phi$-polynomials      142 152
Birational transformation of a Riemann surface , does not affect the theory      3 7
Birational transformation of a Riemann surface , for hyperelliptie surface      152 85
Birational transformation of a Riemann surface , number of invariants in      9 144 148 150
Birational transformation of a Riemann surface , of plane curves      11
Birational transformation of a Riemann surface , of surface into itself      658 see Invariants and
Birational transformation of a Riemann surface , when $p = 1$, or $0$      153
Bitangents of a plane curve      381 390 644 646
Bobek      647
Bolza      177 294 296 815 329 842 436 663
Borahardt      340 342 468
Bouquet      90
Branch places      see Places.
Braunmiihl      486 521
Brill      12 29 134 137 145 149 436 647
Brioschi      296 311 342 448 516 526 668 666
Briot      90
Broch      221
Burkhardt      43 429 436 555 623 663
Burnside      345 373 663
Canonical curve discussed by Klein      159
Canonical equation for a Riemann surface      83 91 103 143 145 152
Canonical integral of the third kind      168 185 189 194 195
Cantor      239
Caspary      474 486 525
Castelnuovo      647 665
Cauchy      600
Cayley      12 137 141 145 165 168 198 222 230 283 296 340 342 374 387 459 540 646 647 650 664 666 668
Cayley’s theorem for plane curves      141
Characteristics, expression of any half-integer characteristic by means of a fundamental system      301 500 502
Characteristics, G$\ddot{o}$pel groups and systems of      489 490 494 ff.
Characteristics, general theory of      486 ff.
Characteristics, of a theta function, number of odd and even      251
Characteristics, of radical functions      380 564
Characteristics, period characteristics and theta characteristics      543 564
Characteristics, syzygetic, azygetic      487
Characteristics, tables of half-integer characteristics for $p = 2$, $p = 3$      303 305
Characteristics, transformation of      536 542 547 564 568
Characteristics, Weirstrass’s number notation for      570 887 303
ChasUs      187 647
Christoffel      666
Clebsch      131 142 153 156 165 168 183 222 241 244 288 295 392 423 448 544 545 556 578 623 647
Coincidences of a correspondence      645
Column and row      see Matrices.
Column of periods      571
Complex multiplication of theta functions      629 ff. 639 660
Composition of transformations of theta functions      551
Condition of dimensions      49
Conformal representation      343 356 372
Congruence, meanings of sign of      236 256 261 264 487
Constants, in any transformation of theta functions      620 622
Constants, in linear transformation of theta functions      555 559
Constants, invariant in rational transformation      9 88 144 148 150
Contact curves      see Curves and
Convergence of an automorphic series      350
Convergence of transformed theta function      538
Coresidual sets of places on a Riemann surface      135 ff. 213
Coresidual sets of places on a Riemann surface, are equivalent sets      136
Coresidual sets of places on a Riemann surface, enter in statement of Abel’s theorem      210
Correspondence of Riemann surfaces      3 ff. 81 639 642 647 648 649 654 662
Covariant      see Invariant.
Cubic surface associated with a plane quartic curve      382 389
Curves, Abel’s theorem for      281
Curves, adjoint curves      121 129
Curves, as alternative interpretation of fundamental algebraic equation      11
Curves, bitangents of a plane quartic curve      384
Curves, Cayley’s theorem for      141
Curves, contact curves      381
Curves, coresidual and equivalent sets upon      134 136
Curves, correspondence of      see Correspondence;
Curves, curves in space      157 ff. 166 664
Curves, general form of Pl$\ddot{u}$cker’s equations for      124
Curves, generalisation      40
Curves, inflexions and bitangents in connection with theory of correspondence      644 646
Curves, inflexions of a plane quartic in connection with the gap theorem      36
Curves, special sets upon      146 ff.
Curves, transformation of      see Birational Invariants and
Curves, Weierstrass’s canonical equation for      93 103
Cusps      11 114
Darboux      666
De Jonquieres      137
Dedekind      57
Deficiency of a Riemann surface      7 55 60
Defining relation for theta functions      443
Definition equation of theta functions of general order      448
Degenerate Abelian integrals      657
Dependence of the poles of a rational function      27
Dersch      647
Differential equations of inversion problem      225 ff.
Differential of theta functions      see Addenda
Differentials of integrals of first kind      25 62 67 127 169
Dimension of an integral function      48 ff. 55
Dimension, condition of dimensions      49
Dini      239
Dirichlet      246 600
Discriminant of a fundamental set of integral functions      74 101 124
Dissection of the Riemann surface      26 529 253 257 569 297 550 560
Double points of a Riemann surface (or curve)      1 2 3 11 114
Double tangents of a plane curve      644 646
Eisenstein      246 674
Elementary integrals      see Integrals.
Epstein      342
Equivalence, meanings of sign of      236 256 261 264 487
Equivalent sets of places on a Riemann surface      134 ff. 136 213
Essential factor of the discriminant      60 74 124
Euler      159
Existence references      14
Existence theorems, algebraically deducible      78
Expression of any rational function      77 176 212
Expression of fundamental integral functions      105 ff.
Expression of half-integer characteristic by means of a fundamental system      301 487 500 502
Factorial functions      392 ff.
Factorial functions, connection with automorphic functions      439 ff.
Factorial functions, definition of      396
Factorial functions, expressed by factorial integrals      403
Factorial functions, expressed by fundamental factorial function      413
Factorial functions, used to express theta functions      423 426
Factorial functions, which are everywhere finite      399
Factorial functions, with fewest poles      406
Factorial integrals      898
Factorial integrals, expression of factorial function by means of that Factorial integrals, integral      412
Factorial integrals, fundamental, having only poles      408
Factorial integrals, simplified form of that integral      411
Factorial integrals, which are everywhere finite      399
Fagnano      638
Forsyth      2 3 7 9 10 13 14 15 16 21 24 25 29 39 90 114 122 123 144 150 198 212 233 296 327 373 395 421 439 441 442 445 459 531 577 578
Frahm      383
Fricke      639 648
Frobenius      342 387 447 474 486 491 500 516 517 521 522 525 586 588 589 598 628 629 630 632 633 666 668 674 676
Frost      389
Fuchs      206 566
Function, $\wp$ function      292 324 338 516
Function, $\zeta$ function      287 292 320
Function, automorphic      352 ff. 439 ff.
Function, factorial      see Factorial;
Function, integral      see Rational and
Function, prime      172 205 360 363 428
Function, radical      374 390 565
Function, rational      see Rational;
Function, theta      see Theta functions and
Fundamental algebraical equation      10 113
Fundamental rational function, Weierstrass’s      171 175 177 178 ff. 182
Fundamental set for the expression of rational integral functions      48 ff. 55 56 57 105 ff.
Fundamental system of theta characteristics      301 487 500 502
G$\ddot{o}$pel biquadratic relation      338 340 465 468
G$\ddot{o}$pel group and system      see Characteristics.
G$\ddot{u}$nther      174 189 200
Gap theorem      32 34 93 174
Gasorati      579
Gauss      559 600 674
Geiser      383
Geometrical investigations      113 see Curves.
Gopel      246 338 339
Gordan      131 142 168 183 241 244 255 288 295 892 423 448 544 545 556 578 623
Goursat      663
Grade, of a polynomial      120
Grassman      137
Greenhill      639
Group, Gopel      see Characteristics.
Halphen      124 165 364 370 421 473 474 639 665
Hamburger      2
Hancock      296 326
Harkness      2 10 14 15 16 21 24 25 79 101 124 239 342
Harnack      222
Hensel      57 64 78 118
Hensel’s determination of fundamental integral functions      105 ff.
hermite      238 246 448 452 538 577 600 615 632 689 663 666 674
Hettner      177 654
hilbert      665
Homogeneous variables      118 441
Homographic behaviour of differentials of integrals of first kind      26
Humbert      222 255 340 486
Hurwitz      41 392 689 648 651 653 654
Hyperelliptie surfaces      80 ff. 152 158 373
Independence of the $2^{2p}$ theta functions with half-integer characteristics      446 447
Independence of the poles of a rational function      27
Index at the infinite place of Weierstrass’s canonical surface      129
Index of a place on a Riemann surface      122 123 124
Infinitesimal on a Riemann surface      1 2 3
Infinitesimal periods      238 573
Infinities of rational function      27 ff.
Infinity, the places at infinity on a Riemann surface, algebraic treatment of      118
Inflexions of a plane curve      36 40 646
Integral functions      see Rational and Transcendental.
Integrals, algebraic expression of      65 ff. 127 131 163 185 189 194
Integrals, all derivable from integral of third kind      22
Integrals, degenerate      657
Integrals, factorial      see Factorial ;
Integrals, formulae connecting with logarithmic differential coeffcients of theta functions      289 290 320
Integrals, hyperelliptie      195
Integrals, Riemann’s, normal elementary      15
Interchange of argument and parameter      16 185 187 189 191 194 206
Interchange of period loops      see Transformation.
Invariants in birational transformation, for transformation of the Invariants in birational transformation, dependent variable      74 124
Invariants in birational transformation, the $3p-3$ moduli      9 144 148 150
Invariants in birational transformation, the $g$-places      38 653
Invariants in birational transformation, the contact $\phi$- polynomials      281 427
Invariants in birational transformation, the number $p$      7
Invariants in birational transformation, the ratios of $\phi$-poly- nomials      26 153
Inversion theorem, by radical functions      390
Inversion theorem, in the hyperelliptic ease      317 324
Inversion theorem, Jacobi’s      285 ff. 270
Inversion theorem, solution of      239 242 244 275
J$\ddot{u}$rgensen      221
jacobi      165 206 221 230 235 237 246 360 577 600 639 657
Jacobian functions, their expression by theta functions      588 594
Jacobian functions, their periods, are generalisation of theta functions      579 588
Jacobian functions, there exists a homogeneous polynomial relation connecting any $p + 2$ Jacobian functions of same periods and parameter      594
Jacobi’s inversion theorem      see Inversion.
JORDAN      248 392 549 623 639 666 669
Joubert      639
K$\ddot{o}$nigsberger      837 342 448 459 477 529 600 607 615 628 658 661 663
Kiepert      688 639
Klein      9 25 156 159 169 177 342 343 360 373 378 383 392 429 430 431 436 438 439 544 639 648 658 654
Klein, prime form      360 427 430 433
Kohn      387
Kowalevski      658 659 662 663
Krause      296 342 468 486 600 623 674
Krazer      477 486 555 600 627
kronecker      56 79 124 445 577 600 629 631 639 666 669
Kummer      340
Lagrange      280
Laguerre      632 666
Laurent’s theorem, for $p$ variables      444
Left side of period loop      529
Liiroth      239
Lindemann      153 156 222 647
Linear transformation      see Transformation.
Linearly independent $\phi$-products of order $\mu$      154
Linearly independent columns of periods      575
Linearly independent Jacobian functions      594
Linearly independent theta functions      446 447
Loops, period loops on a Riemann surface      21 529
Lots, of sets of places on an algebraic curve, or Riemann surface      135
Malet      663
Mathews      165
Matrices      248 283 580 666 669
Minding      221
Mittag — Leffler’s theorem for uniform function on a Riemann surface      202
Mittag — Lejler      202
Moduli of periodicity      see Periods.
Moduli, for the hyperelliptie equation      88
Moduli, of the algebraic equation, are $3p - 3$ in number      9 144 148 150
Morley      2 10 14 15 16 21 24 25 79 101 124 239 342
Multiplication, by an integer, for theta functions      527
Multiplication, complex, of theta functions      629 ff.
Multiply-periodic      236 see Inversion.
Netto      20 90
Neumann      14 17 169 296 531
Noether      12 29 32 124 131 134 137 142 145 149 156 165 168 180 272 292 295 390 392 430 486 522 544 566 654 665
Noether’s (Kraus’s) $\phi$-curve in space      156 157
Normal equation for a Riemann surface      83 91 103 143 145 152
Normal integrals (Riemann’s)      see Integrals.
Number of independent products of $\mu$ $\phi$-polynomials      154
Number of Jacobian functions      594
Number of odd and even theta functions      251
Number of theta functions of general order      452 463
Order of a theta function      448
Order of small quantity on a Riemann surface      2
Parameter, interchange of argument and parameter      see Interchange.
Parameters, in the algebraic equation      see Constants.
Period characteristics      see Characteristics.
Period loop      see Loops.
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