Abelian functions: Abel's theorem and the allied theory of theta functions
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
1995
|
| Ausgabe: | 1. published 1897, reissued |
| Schriftenreihe: | Cambridge Mathematical Library
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Originalausgabe mit dem Sachtitel: Abel's theorem and the allied theory including the theory of theta functions.- Cambridge: Cambridge University Press, 1897 |
| Beschreibung: | XXXV, 684 Seiten graph. Darst. |
| ISBN: | 0521498775 |
Internformat
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| 100 | 1 | |a Baker, Henry Frederick |d 1866-1956 |e Verfasser |0 (DE-588)116041315 |4 aut | |
| 245 | 1 | 0 | |a Abelian functions |b Abel's theorem and the allied theory of theta functions |c H. F. Baker, St. John's College, Cambridge |
| 250 | |a 1. published 1897, reissued | ||
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 1995 | |
| 300 | |a XXXV, 684 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Cambridge Mathematical Library | |
| 500 | |a Originalausgabe mit dem Sachtitel: Abel's theorem and the allied theory including the theory of theta functions.- Cambridge: Cambridge University Press, 1897 | ||
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Datensatz im Suchindex
| _version_ | 1804125488027795456 |
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| adam_text | ABELIAN FUNCTIONS ABEL S THEOREM AND THE ALLIED THEORY OF THETA
FUNCTIONS H. F BAKER ST JOHN S COLLEGE, CAMBRIDGE CAMBRIDGE UNIVERSITY
PRESS CONTENTS. CHAPTER I. THE SUBJECT OF INVESTIGATION. §§ PAGES I
FUNDAMENTAL ALGEBRAIC IRRATIONALITY 1 2, 3 THE PLACES AND INFINITESIMAL
ON A RIEMANN SURFACE . . . 1, 2 4, 5 THE THEORY UNALTERED BY RATIONAL
TRANSFORMATION . . . 3*6 6 THE INVARIANCE OF THE DEFICIENCY IN RATIONAL
TRANSFORMATION; IF A RATIONAL FUNCTION EXISTS OF ORDER 1, THE SURFACE IS
OF ZERO DEFICIENCY 7, 8 7, 8 THE GREATEST NUMBER OF IRREMOVEABLE
PARAMETERS IS 3P - 3 . . 9, 10 9, 10 THE GEOMETRICAL STATEMENT OF THE
THEORY 11, 12 II GENERALITY OF RIEMANN S METHODS 12, 13 CHAPTEE II. THE
FUNDAMENTAL FUNCTIONS ON A RIEMANN SURFACE. 12 RIEMANN S EXISTENCE
THEOREM PROVISIONALLY REGARDED AS FUNDAMENTAL 14 13 NOTATION FOR NORMAL
ELEMENTARY INTEGRAL OF SECOND KIND . . 15 14 NOTATION FOR NORMAL
ELEMENTARY INTEGRAL OF THIRD KIND . . . 15 15 CHOICE OF NORMAL INTEGRALS
OF THE FIRST KIND 16 16 MEANING OF THE WORD PERIOD. GENERAL REMARKS . .
. . 16, 17 17 EXAMPLES OF THE INTEGRALS, AND OF THE PLACES OF THE
SURFACE . 18*20 18 PERIODS OF THE NORMAL ELEMENTARY INTEGRALS OF THE
SECOND KIND . 21 19 THE INTEGRAL OF THE SECOND KIND ARISES BY
DIFFERENTIATION FROM THE INTEGRAL OF THE THIRD KIND 22, 23 20 EXPRESSION
OF A RATIONAL FUNCTION BY INTEGRALS OF THE SECOND KIND . 24 21 SPECIAL
RATIONAL FUNCTIONS, WHICH ARE INVARIANT IN RATIONAL TRANS- FORMATION 25,
26 22 RIEMANN NORMAL INTEGRALS DEPEND ON MODE OF DISSECTION OF THE
SURFACE 26 CHAPTER III. THE INFINITIES OF RATIONAL FUNCTIONS. 23 THE
INTERDEPENDENCE OF THE POLES OF A RATIONAL FUNCTION . . 27 24, 25
CONDITION THAT SPECIFIED PLACES BE THE POLES OF A RATIONAL FUNCTION .
28*30 26 GENERAL FORM OF WEIERSTRASS S GAP THEOREM 31, 32 27 PROVISIONAL
STATEMENT OF THE RIEMANN-ROCH THEOREM . . . 33, 34 VI CONTENTS. §§ PAGES
28, 29 CASES WHEN THE POLES COALESCE; THE P CRITICAL INTEGERS . . 34, 35
30 SIMPLE ANTICIPATORY GEOMETRICAL ILLUSTRATION 36, 37 31*33 THE
(P-L)P(P + 1) PLACES WHICH ARE THE POLES OF RATIONAL FUNCTIONS OF ORDER
LESS THAN P + 1 38*40 34*36 THERE ARE AT LEAST 2P + 2 SUCH PLACES WHICH
ARE DISTINCT . . 41*44 37 STATEMENT OF THE RIEMANN-ROCH THEOREM, WITH
EXAMPLES . . 44*46 CHAPTER IV. SPECIFICATION OF A GENERAL FORM OF
RIEMANN S INTEGRALS. 38 EXPLANATIONS IN REGARD TO INTEGRAL RATIONAL
FUNCTIONS . . 47, 48 39 DEFINITION OF DIMENSION ; FUNDAMENTAL SET OF
FUNCTIONS FOR THE EXPRESSION OF RATIONAL FUNCTIONS 48*52 40 ILLUSTRATIVE
EXAMPLE FOR A SURFACE OF FOUR SHEETS . . . . 53, 54 41 THE SUM OF THE
DIMENSIONS OF THE FUNDAMENTAL SET OF FUNCTIONS IS P + N-1 54, 55 42
FUNDAMENTAL SET FOR THE EXPRESSION OF INTEGRAL FUNCTIONS . . 55, 56 43
PRINCIPAL PROPERTIES OF THE FUNDAMENTAL SET OF INTEGRAL FUNCTIONS .
57*60 44 DEFINITION OF DERIVED SET OF SPECIAL FUNCTIONS 0 , TF U ...,
*_, . 61*64 45 ALGEBRAICAL FORM OF ELEMENTARY INTEGRAL OF THE THIRD
KIND, WHOSE INFINITIES ARE ORDINARY PLACES; AND OF INTEGRALS OF THE
FIRST KIND 65*68 46 ALGEBRAICAL FORM OF ELEMENTARY INTEGRAL OF THE THIRD
KIND IN GENERAL 68*70 47 ALGEBRAICAL FORM OF INTEGRAL OF THE SECOND
KIND, INDEPENDENTLY DEDUCED 71*73 48 THE DISCRIMINANT OF THE FUNDAMENTAL
SET OF INTEGRAL FUNCTIONS . 74 49 DEDUCTION OF THE EXPRESSION OF A
CERTAIN FUNDAMENTAL RATIONAL FUNCTION IN THE GENERAL CASE 75*77 50 THE
ALGEBRAICAL RESULTS OF THIA CHAPTER ARE SUFFICIENT TO REPLACE RIEMANN S
EXISTENCE THEOREM 78, 79 CHAPTER V. CERTAIN FORMS OF THE FUNDAMENTAL
EQUATION OF THE RIEMANN SURFACE. 51 CONTENTS OF THE CHAPTER 80 52 WHEN
P L, EXISTENCE OF RATIONAL FUNCTION OF THE SECOND ORDER INVOLVES A (1,
1) CORRESPONDENCE 81 53*55- EXISTENCE OF RATIONAL FUNCTION OF THE SECOND
ORDER INVOLVES THE HYPERELLIPTIC EQUATION 81*84 56, 57 FUNDAMENTAL
INTEGRAL FUNCTIONS AND INTEGRALS OF THE FIRST KIND . 85*86 58 EXAMPLES
87 59 NUMBER OF IRREMOVEABLE PARAMETERS IN THE HYPERELLIPTIC EQUATION ;
TRANSFORMATION TO THE CANONICAL FORM 88*89 60*63 WEIERSTRASS S CANONICAL
EQUATION FOR ANY DEFICIENCY . . . 90*92 CONTENTS. VLL §§ PAGES 64*66
ACTUAL FORMATION OF THE EQUATION 93*98 67, 68 ILLUSTRATIONS OF THE
THEORY OF INTEGRAL FUNCTIONS FOR WEIERSTRASS S CANONICAL EQUATION 99*101
69*71 THE METHOD CAN BE CONSIDERABLY GENERALISED 102*104 72*79 HENSEL S
DETERMINATION OF THE FUNDAMENTAL INTEGRAL FUNCTIONS . 105*112 CHAPTER
VI. GEOMETRICAL INVESTIGATIONS. 80 COMPARISON OF THE THEORY OF RATIONAL
FUNCTIONS WITH THE THEORY OF INTERSECTIONS OF CURVES 113 81*83
INTRODUCTORY INDICATIONS OF ELEMENTARY FORM OF THEORY . . . 113*116 84
THE METHOD TO BE FOLLOWED IN THIS CHAPTER 117 85 TREATMENT OF INFINITY.
HOMOGENEOUS VARIABLES MIGHT BE USED . 118, 119 86 GRADE OF AN INTEGRAL
POLYNOMIAL; NUMBER OF TERMS ; GENERALISED ZEROS 120, 121 87 ADJOINT
POLYNOMIALS ; DEFINITION OF THE INDEX OF A SINGULAR PLACE . 122 88
PLIICKER S EQUATIONS; CONNECTION WITH THEORY OF DISCRIMINANT . 123, 124
89, 90 EXPRESSION OF RATIONAL FUNCTIONS BY ADJOINT POLYNOMIALS . . .
125, 126 91 EXPRESSION OF INTEGRAL OF THE FIRST KIND 127 92 NUMBER OF
TERMS IN AN ADJOINT POLYNOMIAL; DETERMINATION OF ELEMENTARY INTEGRAL OF
THE THIRD KIND 128*132 93 LINEAR SYSTEMS OF ADJOINT POLYNOMIALS ;
RECIPROCAL THEOREM . . 133, 134 94, 95 DEFINITIONS OF SET, LOT, SEQUENT,
EQUIVALENT SETS, CORESIDUAL SETS . 135 96, 97 THEOREM OF CORESIDUAL
SETS; ALGEBRAIC BASIS OF THE THEOREM . . 136 * 98 A RATIONAL FUNCTION OF
ORDER LESS THAN P+ IS EXPRESSIBLE BY POLYNOMIALS 137 99, 100 CRITICISM
OF THE THEORY; CAYLEY S THEOREM 138*141 101*104 RATIONAL TRANSFORMATION
BY MEANS OF ^-POLYNOMIALS . . . 142*146 105*108 APPLICATION OF SPECIAL
SETS 147*151 109 THE HYPERELLIPTIC SURFACE; TRANSFORMATION TO CANONICAL
FORM . 152 110*114 WHOLE RATIONAL THEORY CAN BE REPRESENTED BY MEANS OF
THE INVARI- ANT RATIOS OF (^-POLYNOMIALS ; NUMBER OF RELATIONS
CONNECTING THESE 153*159 115*119 ELEMENTARY CONSIDERATIONS IN REGARD TO
CURVES IN SPACE . . 160*167 CHAPTER VII. COORDINATION OF SIMPLE
ELEMENTS. TRANSCENDENTAL UNIFORM FUNCTIONS. 120 SCOPE OF THE CHAPTER 168
121 NOTATION FOR INTEGRALS OF THE FIRST KIND . . . . . 169 122, 123 THE
FUNCTION *$*(#, A; Z, C LT ...,C P ) EXPRESSED BY RIEMANN INTEGRALS 170,
171 124 DERIVATION OF A CERTAIN PRIME FUNCTION 172 125 APPLICATIONS OF
THIS FUNCTION TO RATIONAL FUNCTIONS AND INTEGRALS 173 VIII CONTENTS. §§
PAGES 126*128 THE FUNCTION |C (%, A; Z, C) ; ITS UTILITY FOR THE
EXPRESSION OF RATIONAL FUNCTIONS 174*176 129 THE DERIVED PRIME FUNCTION
E{X, Z); USED TO EXPRESS RATIONAL FUNCTIONS 177 130, 131 ALGEBRAIC
EXPRESSION OF THE FUNCTIONS TY (X, A; Z, C U ...,C P ), ^{X,A; 2, E) . .
. 177, 178 132 EXAMPLES OF THESE FUNCTIONS; THEY DETERMINE ALGEBRAIC
EXPRES- SIONS FOR THE ELEMENTARY INTEGRALS 179*182 133, 134 DERIVATION
OF A CANONICAL INTEGRAL OF THE THIRD KIND; FOR WHICH INTERCHANGE OF
ARGUMENT AND PARAMETER HOLDS; ITS ALGEBRAIC EXPRESSION ; ITS RELATION
WITH RIEMANN S ELEMENTARY NORMAL INTEGRAL 182*185 135 ALGEBRAIC THEOREM
EQUIVALENT TO INTERCHANGE OF ARGUMENT AND PARAMETER 185 136 ELEMENTARY
CANONICAL INTEGRAL OF THE SECOND KIND . . . 186, 187 137 APPLICATIONS.
CANONICAL INTEGRAL OF THE THIRD KIND DEDUCED FROM THE FUNCTION YFR(X,A;
Z,C X , ...,O P ). MODIFICATION FOR THE FUNC- TION ^ (X, A ; Z, 0)
188*192 138 ASSOCIATED INTEGRALS OF FIRST AND SECOND KIND. FURTHER
CANONICAL INTEGRALS. THE ALGEBRAIC THEORY OF THE HYPERELLIPTIC INTEGRALS
IN ONE FORMULA 193, 194 139, 140 DEDUCTION OF WEIERSTRASS S AND
RIEMANN S RELATIONS FOR PERIODS OF INTEGRALS OF THE FIRST AND SECOND
KIND . . . . 195*197 141 EITHER FORM IS EQUIVALENT TO THE OTHER 198 142
ALTERNATIVE PROOFS OF WEIERSTRASS S AND RIEMANN S PERIOD RELATIONS 199,
200 143 EXPRESSION OF UNIFORM TRANSCENDENTAL FUNCTION BY THE FUNCTION
TY{X, A; 2, C) 201 144, 145 MITTAG-LEFFLER S THEOREM 202*204 146
EXPRESSION OF UNIFORM TRANSCENDENTAL FUNCTION IN PRIME FACTORS 205 147
GENERAL FORM OF INTERCHANGE OF ARGUMENT AND PARAMETER, AFTER ABEL 206
CHAPTER VIII. ABEL S THEOREM. ABEL S DIFFERENTIAL EQUATIONS. 148*150
APPROXIMATIVE DESCRIPTION OF ABEL S THEOREM . . . . 207*210 151
ENUNCIATION OF THE THEOREM 210 152 THE GENERAL THEOREM REDUCED TO TWO
SIMPLER THEOREMS . . 211, 212 153, 154 PROOF AND ANALYTICAL STATEMENT OF
THE THEOREM . . . . 212*214 155 REMARK; STATEMENT IN TERMS OF
POLYNOMIALS . . . . 215 156 THE DISAPPEARANCE OF THE LOGARITHM ON THE
RIGHT SIDE OF THE EQUATION 216 157 APPLICATIONS OF THE THEOREM. ABEL S
OWN PROOF . . . . 217*222 158, 159 THE NUMBER OF ALGEBRAICALLY
INDEPENDENT EQUATIONS GIVEN BY THE THEOREM. INVERSE OF ABEL S THEOREM
160, 161 INTEGRATION OF ABEL S DIFFERENTIAL EQUATIONS 162 ABEL S THEOREM
PROVED QUITE SIMILARLY FOR CURVES IN SPACE . CONTENTS. IX CHAPTER IX.
JACOBI S INVERSION PROBLEM. §§ PAGES 163 STATEMENT OF THE PROBLEM 235
164 UNIQUENESS OF ANY SOLUTION 236 165 THE NECESSITY OF USING
CONGRUENCES AND NOT EQUATIONS . . 237 166, 167 AVOIDANCE OF FUNCTIONS
WITH INFINITESIMAL PERIODS . . . 238, 239 168, 169 PROOF OF THE
EXISTENCE OF A SOLUTION 239*241 170*172 FORMATION OF FUNCTIONS WITH
WHICH TO EXPRESS THE SOLUTION; CONNECTION WITH THETA FUNCTIONS 242*245
CHAPTER X. RIEMANN S-THETA FUNCTIONS. GENERAL THEORY. 173 SKETCH OF THE
HISTORY OF THE INTRODUCTION OF THETA FUNCTIONS . 246 174 CONVERGENCE.
NOTATION. INTRODUCTION OF MATRICES . . . 247, 248 175, 176 PERIODICITY
OF THE THETA FUNCTIONS. ODD AND EVEN FUNCTIONS . 249*251 177 NUMBER OF
ZEROS IS P 252 178 POSITION OF THE ZEROS IN THE SIMPLE CASE 253, 254 179
THE PLACES M^, ..., M P 255 180 POSITION OF THE ZEROS IN GENERAL 256,
257 181 IDENTICAL VANISHING OF THE THETA FUNCTIONS 258, 259 182, 183
FUNDAMENTAL PROPERTIES. GEOMETRICAL INTERPRETATION OF THE PLACES M 1
,...,M P . 259*267 184*186 GEOMETRICAL DEVELOPMENTS; SPECIAL INVERSION
PROBLEM; CONTACT CURVES 268*273 187 SOLUTION OF JACOBI S INVERSION
PROBLEM BY QUOTIENTS OF THETA FUNCTIONS 274, 275 188 THEORY OF THE
IDENTICAL VANISHING OF THE THETA FUNCTION. EX- PRESSION OF
(^-POLYNOMIALS BY THETA FUNCTIONS . . . 276*282 189*191 GENERAL FORM OF
THETA FUNCTION. FUNDAMENTAL FORMULAE. PERIODICITY 283*286 192
INTRODUCTION OF THE F FUNCTIONS. GENERALISATION OF AN ELLIPTIC FORMULA
287 193 DIFFERENCE OF TWO F FUNCTIONS EXPRESSED BY ALGEBRAIC INTEGRALS
AND RATIONAL FUNCTIONS 288 194*196 DEVELOPMENT. EXPRESSION OF SINGLE F
FUNCTION BY ALGEBRAIC INTEGRALS 289*292 197, 198 INTRODUCTION OF THE 0
FUNCTIONS. EXPRESSION BY RATIONAL FUNCTIONS 292*295 CHAPTER XI. THE
HYPERELLIPTIC CASE OF RIEMANN S THETA FUNCTIONS. 199 HYPERELLIPTIC CASE
ILLUSTRATES THE GENERAL THEORY . . . . 296 200 -* THE PLACES M L ,...,M
P . THE RULE FOR HALF PERIODS . . . 297, 298 201, 202 FUNDAMENTAL SET OF
CHARACTERISTICS DEFINED BY BRANCH PLACES . 299*301 X CONTENTS. §§ PAGES
203 NOTATION. GENERAL THEOREMS TO BE ILLUSTRATED . . . . 302 204, 205
TABLES IN ILLUSTRATION OF THE GENERAL THEORY 303*309 206*213 ALGEBRAIC
EXPRESSION OF QUOTIENTS OF HYPERELLIPTIC THETA FUNCTIONS. SOLUTION OF
HYPERELLIPTIC INVERSION PROBLEM . . . . 309*317 214, 215 SINGLE
FUNCTION EXPRESSED BY ALGEBRAICAL INTEGRALS AND RATIONAL FUNCTIONS
318*323 216 RATIONAL EXPRESSION OF (J? FUNCTION. RELATION TO QUOTIENTS
OF THETA FUNCTIONS. SOLUTION OF INVERSION PROBLEM BY FP FUNCTION . .
323*327 217- RATIONAL EXPRESSION OF G FUNCTION 327*330 218*220
ALGEBRAIC DEDUCTION OF ADDITION EQUATION FOR THETA FUNCTIONS WHEN JO=2;
GENERALISATION OF THE EQUATION = O%.O- 2 2 .(^-J?K) 330*337 221 EXAMPLES
FOR THE CASE P = % GOPEL S BIQUADRATIC RELATION . . 337*342 CHAPTER XII.
A PARTICULAR FORM OF FUNDAMENTAL SURFACE. 222 CHAPTER INTRODUCED AS A
CHANGE OF INDEPENDENT VARIABLE, AND AS INTRODUCING A PARTICULAR PRIME
FUNCTION . . . . 223*225 DEFINITION OF A GROUP OF SUBSTITUTIONS;
FUNDAMENTAL PROPERTIES . 226, 227 CONVERGENCE OF A SERIES; FUNCTIONS
ASSOCIATED WITH THE GROUP . 228*232 THE FUNDAMENTAL FUNCTIONS.
COMPARISON WITH FOREGOING THEORY OF THIS VOLUME 233*235 DEFINITION AND
PERIODICITY OF THE SCHOTTKY PRIME FUNCTION . 236, 237 ITS CONNECTION
WITH THE THETA FUNCTIONS 238 A FURTHER FUNCTION CONNECTED THEREWITH 239
THE HYPERELLIPTIC CASE 343 343*348 349*352 CHAPTER XIII. RADICAL
FUNCTIONS. 240 INTRODUCTORY . 241, 242 EXPRESSION OF ANY RADICAL
FUNCTION BY RIEMANN S INTEGRALS, AND BY THETA FUNCTIONS 243 RADICAL
FUNCTIONS ARE A GENERALISATION OF RATIONAL FUNCTIONS 244, 245
CHARACTERISTICS OF RADICAL FUNCTIONS 246*249 BITANGENTS OF A PLANE
QUARTIC CURVE 250, 251 SOLUTION OF THE INVERSION PROBLEM BY RADICAL
FUNCTIONS 374 CHAPTER XIV. FACTORIAL FUNCTIONS. 252 STATEMENT OF RESULTS
OBTAINED. NOTATIONS 253 NECESSARY DISSECTION OF THE RIEMANN SURFACE . .
. . 254 DEFINITION OF A FACTORIAL FUNCTION (INCLUDING RADICAL FUNCTION).
PRIMARY AND ASSOCIATED SYSTEMS OF FACTORIAL FUNCTIONS . CONTENTS. XI §§
PAGES 255 FACTORIAL INTEGRALS OF THE PRIMARY AND ASSOCIATED SYSTEMS . .
397, 398 256 FACTORIAL INTEGRALS WHICH ARE EVERYWHERE FINITE, SAVE AT
THE FIXED INFINITIES. INTRODUCTION OF THE NUMBERS OR, O- + L . . . 399
257 WHEN O-+L 0, THERE ARE EVERYWHERE FINITE FACTORIAL FUNCTIONS OF THE
ASSOCIATED SYSTEM 400 258 ALTERNATIVE INVESTIGATION OF EVERYWHERE FINITE
FACTORIAL FUNCTIONS OF THE ASSOCIATED SYSTEM. THEORY DIVISIBLE ACCORDING
TO THE VALUES OF A+1 AND O- + L 401, 402 259 EXPRESSION OF THESE
FUNCTIONS BY EVERYWHERE FINITE INTEGRALS . 403 260 GENERAL CONSIDERATION
OF THE PERIODS OF THE FACTORIAL INTEGRALS . 404 261, 262 RIEMANN-ROCH
THEOREM FOR FACTORIAL FUNCTIONS. WHEN CR + L=O, LEAST NUMBER OF
ARBITRARY POLES FOR FUNCTION OF THE PRIMARY SYSTEM IS *AI + L 405, 406
263 CONSTRUCTION OF FACTORIAL FUNCTION OF THE PRIMARY SYSTEM WITH *SR +
L ARBITRARY POLES 406, 407 264, 265 CONSTRUCTION OF A FACTORIAL INTEGRAL
HAVING ONLY POLES. LEAST NUMBER OF SUCH POLES, FOR AN INTEGRAL OF THE
PRIMARY SYSTEM, IS O - + 2 407, 410 266 THIS FACTORIAL INTEGRAL CAN BE
SIMPLIFIED, IN ANALOGY WITH RIEMANN S ELEMENTARY INTEGRAL OF THE SECOND
KIND 411 267 EXPRESSION OF THE FACTORIAL FUNCTION WITH TA- +1 POLES IN
TERMS OF THE FACTORIAL INTEGRAL WITH O- + 2 POLES. THE FACTORIAL
FUNCTION IN ANALOGY WITH THE FUNCTION ^ {X, A; Z, C LT ..., C P ). . .
411*413 268 THE THEORY TESTED BY EXAMINATION OF A VERY PARTICULAR CASE .
413*419 269 THE RADICAL FUNCTIONS AS A PARTICULAR CASE OF FACTORIAL
FUNCTIONS 419, 420 270 FACTORIAL FUNCTIONS WHOSE FACTORS ARE ANY
CONSTANTS, HAVING NO ESSENTIAL SINGULARITIES 421 271, 272 INVESTIGATION
OF A GENERAL FORMULA CONNECTING FACTORIAL FUNCTIONS AND THETA FUNCTIONS
422*426 273 INTRODUCTION OF THE SCHOTTKY-KLEIN PRIME FORM, IN A CERTAIN
SHAPE 427*430 274 EXPRESSION OF A THETA FUNCTION IN TERMS OF RADICAL
FUNCTIONS, AS A PARTICULAR CASE OF § 272 430 275, 276 THE FORMULA OF §
272 FOR THE CASE OF RATIONAL FUNCTIONS . . 431*433 277 THE FORMULA OF §
272 APPLIED TO DEFINE ALGEBRAICALLY THE HYPER- ELLIPTIC THETA FUNCTION,
AND ITS THETA CHARACTERISTIC . . 433*437 278 EXPRESSION OF ANY FACTORIAL
FUNCTION BY SIMPLE THETA FUNCTIONS; EXAMPLES 437, 438 279 CONNECTION OF
THEORY OF FACTORIAL FUNCTIONS WITH THEORY OF AUTO- MORPHIC FORMS 439*442
CHAPTER XV. RELATIONS CONNECTING PRODUCTS OF THETA
FUNCTIONS*INTRODUCTORY. 280 PLAN OF THIS AND THE TWO FOLLOWING CHAPTERS
. . . . 443 281 A SINGLE-VALUED INTEGRAL ANALYTICAL FUNCTION OF P
VARIABLES, WHICH IS PERIODIC IN EACH VARIABLE ALONE, CAN BE REPRESENTED
BY A SERIES OF EXPONENTIALS 443*445 CONTENTS. §§ 282, 283 284, 285 286
286, 288 289 PROOF THAT THE 2* THETA FUNCTIONS WITH HALF-INTEGER
CHARACTER- ISTICS ARE LINEARLY INDEPENDENT DEFINITION OF GENERAL THETA
FUNCTION OF ORDER R ; ITS LINEAR EXPRES- SION BY T* THETA FUNCTIONS.
ANY P+2 THETA FUNCTIONS OF SAME ORDER, PERIODS, AND CHARACTERISTIC
CONNECTED BY A HOMO- GENEOUS POLYNOMIAL RELATION ADDITION THEOREM FOR
HYPERELLIPTIC THETA FUNCTIONS, OR FOR THE GENERAL CASE WHEN P NUMBER OF
LINEARLY INDEPENDENT THETA FUNCTIONS OF ORDER R WHICH ARE ALL OF THE
SAME PARITY EXAMPLES. THE GOPEL BIQUADRATIC RELATION PAGES 446*447
447*455 456*461 461*464 465*470 CHAPTER XVI. A DIRECT METHOD OF
OBTAINING THE EQUATIONS CONNECTING THETA PRODUCTS. 290 CONTENTS OF THIS
CHAPTER 471 291 AN ADDITION THEOREM OBTAINED BY MULTIPLYING TWO THETA
FUNCTIONS. 471*474 292 AN ADDITION THEOREM OBTAINED BY MULTIPLYING FOUR
THETA FUNCTIONS 474*477 293 THE GENERAL FORMULA OBTAINED BY MULTIPLYING
ANY NUMBER OF THETA FUNCTIONS 477*485 CHAPTER XVII. THETA RELATIONS
ASSOCIATED WITH CERTAIN GROUPS OF CHARACTERISTICS. 294 ABBREVIATIONS.
DEFINITION OF SYZYGETIC AND AZYGETIC. REFERENCES TO LITERATURE (SEE ALSO
P. 296) 486, 487 295 A PRELIMINARY LEMMA 488 296 DETERMINATION OF A
GOPEL GROUP OF CHARACTERISTICS . . . 489, 490 297 DETERMINATION OF A
G6PEL SYSTEM OF CHARACTERISTICS . . . 490, 491 298, 299 DETERMINATION
AND NUMBER OF GSPEL SYSTEMS OF THE SAME PARITY 492*494 300*303
DETERMINATION OF A FUNDAMENTAL SET OF GOPEL SYSTEMS . . 494*501 304, 305
STATEMENT OF RESULTS OBTAINED, WITH THE SIMPLER APPLICATIONS . 502*504
306*308 NUMBER OF LINEARLY INDEPENDENT THETA FUNCTIONS OF THE SECOND
ORDER OF A PARTICULAR KIND. EXPLICIT MENTION OF AN IMPORT- ANT IDENTITY
505*510 309*311 THE MOST IMPORTANT FORMULAE OF THE CHAPTER. A GENERAL
ADDI- TION THEOREM. THE |JP FUNCTION EXPRESSED BY QUOTIENTS OF THETA
FUNCTIONS 510*516 312*317 OTHER APPLICATIONS OF THE PRINCIPLES OF THE
CHAPTER. THE EXPRES- SION OF A FUNCTION $ (NV) AS AN INTEGRAL POLYNOMIAL
OF ORDER N 2 IN 2P FUNCTIONS ${V) 517*527 CONTENTS. XM CHAPTER XVIII.
TRANSFORMATION OF PERIODS, ESPECIALLY LINEAR TRANSFORMATION. §§ PAGES
318 BEARINGS OF THE THEORY OF TRANSFORMATION 528, 529 319*323 THE
GENERAL THEORY OF THE MODIFICATION OF THE PERIOD LOOPS ON A RIEMANN
SURFACE 529*534 324 ANALYTICAL THEORY OF TRANSFORMATION OF PERIODS AND
CHARACTERISTIC OF A THETA FUNCTION 534*538 325 CONVERGENCE OF THE
TRANSFORMED FUNCTION 538 326 SPECIALISATION OF THE FORMULAE, FOR LINEAR
TRANSFORMATION . . 539, 540 327 TRANSFORMATION OF THETA CHARACTERISTICS
; OF EVEN CHARACTERISTICS; OF SYZYGETIC CHARACTERISTICS 541, 542 328
PERIOD CHARACTERISTICS AND THETA CHARACTERISTICS . . . . 543 329
DETERMINATION OF A LINEAR TRANSFORMATION TO TRANSFORM ANY EVEN
CHARACTERISTIC INTO THE ZERO CHARACTERISTIC . . . . 544, 545 330, 331
LINEAR TRANSFORMATION OF TWO AZYGETIC SYSTEMS OF THETA CHARAC- TERISTICS
INTO ONE ANOTHER 546*550 332 COMPOSITION OF TWO TRANSFORMATIONS OF
DIFFERENT ORDERS; SUPPLE- MENTARY TRANSFORMATIONS 551, 552 333, 334
FORMATION OF P + 2 ELEMENTARY LINEAR TRANSFORMATIONS BY THE COMPOSITION
OF WHICH EVERY LINEAR TRANSFORMATION CAN BE FORMED; DETERMINATION OF THE
CONSTANT FACTORS FOR EACH OF THESE 553*557 335 THE CONSTANT FACTOR FOR
ANY LINEAR TRANSFORMATION . . . 558, 559 336 ANY LINEAR TRANSFORMATION
MAY BE ASSOCIATED WITH A CHANGE OF THE PERIOD LOOPS, OF A RIEMANN
SURFACE 560, 561 337, 338 LINEAR TRANSFORMATION OF THE PLACES M^, ..., M
P . . . . 562 339 LINEAR TRANSFORMATION OF THE CHARACTERISTICS OF A
RADICAL FUNCTION 563, 564 340 DETERMINATION OF THE PLACES WIJ, ..., M P
UPON A RIEMANN SURFACE WHOSE MODE OF DISSECTION IS ASSIGNED 565*567 341
LINEAR TRANSFORMATION OF QUOTIENTS OF HYPERELLIPTIC THETA FUNCTIONS 568
342 A CONVENIENT FORM OF THE PERIOD LOOPS IN A SPECIAL HYPERELLIPTIC
CASE. WEIERSTRASS S NUMBER NOTATION FOR HALF-INTEGER CHARAC- TERISTICS
569, 570 CHAPTER XIX. ON SYSTEMS OF PERIODS AND ON GENERAL JACOBIAN
FUNCTIONS. 343 SCOPE OF THIS CHAPTER 571 344*350 COLUMNS OF PERIODS.
EXCLUSION OF INFINITESIMAL PERIODS. EXPRES- SION OF ALL PERIOD COLUMNS
BY A FINITE NUMBER OF COLUMNS, WITH INTEGER COEFFICIENTS 571*579 351*356
DEFINITION OF GENERAL JACOBIAN FUNCTION, AND COMPARISON WITH THETA
FUNCTION 579*588 357*362 EXPRESSION OF JACOBIAN FUNCTION BY MEANS OF
THETA FUNCTIONS. ANY P + 2 JACOBIAN FUNCTIONS OF SAME PERIODS AND
PARAMETER CONNECTED BY A HOMOGENEOUS POLYNOMIAL RELATION . . 588*598
CONTENTS. CHAPTER XX. TRANSFORMATION OF THETA FUNCTIONS. §§ PAGES 363
SKETCH OF THE RESULTS OBTAINED. REFERENCES TO THE LITERATURE . 599, 600
364, 365 ELEMENTARY THEORY OF TRANSFORMATION OF SECOND ORDER . . .
600*606 366, 367 INVESTIGATION OF A GENERAL FORMULA PRELIMINARY TO
TRANSFORMATION OF ODD ORDER 607*610 368, 369 THE GENERAL THEOREM FOR
TRANSFORMATION OF ODD ORDER . . . 611*616 370 THE GENERAL TREATMENT OF
TRANSFORMATION OF THE SECOND ORDER . 617*619 371 THE TWO STEPS IN THE
DETERMINATION OF THE CONSTANT COEFFICIENTS 619 372 THE FIRST STEP IN THE
DETERMINATION OF THE CONSTANT COEFFICIENTS 619*622 373 REMARKS AND
EXAMPLES IN REGARD TO THE SECOND STEP . . . 622*624 374 TRANSFORMATION
OF PERIODS WHEN THE COEFFICIENTS ARE NOT INTEGRAL 624*628 375 REFERENCE
TO THE ALGEBRAICAL APPLICATIONS OF THE THEORY . . 628 CHAPTER XXI.
COMPLEX MULTIPLICATION OF THETA FUNCTIONS. CORRESPONDENCE OF POINTS ON A
RLEMANN SURFACE. 376 SCOPE OF THE CHAPTER 629 377, 378 NECESSARY
CONDITIONS FOR A COMPLEX MULTIPLICATION, OR SPECIAL TRANSFORMATION, OF
THETA FUNCTIONS 629*632 379*382 PROOF, IN ONE CASE, THAT THESE
CONDITIONS ARE SUFFICIENT . . 632*636 383 EXAMPLE OF THE ELLIPTIC CASE
636*639 384 MEANING OF AN (R, A) CORRESPONDENCE ON A RIEMANN SURFACE .
639, 640 385 EQUATIONS NECESSARY FOR THE EXISTENCE OF SUCH A
CORRESPONDENCE 640*642 386 ALGEBRAIC DETERMINATION OF A CORRESPONDENCE
EXISTING ON A PER- FECTLY GENERAL RIEMANN SURFACE 642*645 387 THE
COINCIDENCES. EXAMPLES OF THE INFLECTIONS AND BITANGENTS OF A PLANE
CURVE 645*648 388 CONDITIONS FOR A (1, S) CORRESPONDENCE ON A SPECIAL
RIEMANN SURFACE 648, 649 389 WHEN P L A (1, 1) CORRESPONDENCE IS
NECESSARILY PERIODIC . . 649, 650 390 AND INVOLVES A SPECIAL FORM OF
FUNDAMENTAL EQUATION . . 651 391*393 WHEN P L THERE CANNOT BE AN
INFINITE NUMBER OF (1, 1) CORRE- SPONDENCES 652*654 394 EXAMPLE OF THE
CASE P = L 654*656 CHAPTER XXII. DEGENERATE ABELIAN INTEGRALS. 395
EXAMPLE OF THE PROPERTY TO BE CONSIDERED 657 396 WEIERSTRASS S THEOREM.
THE PROPERTY INVOLVES A TRANSFORMATION LEADING TO A THETA FUNCTION WHICH
BREAKS INTO FACTORS . . 657, 658 CONTENTS. XV §§ PAGES 3I)7
WEIERSTRASS S AND PICARD S THEOREM. THE PROPERTY INVOLVES A LINEAR
TRANSFORMATION LEADING TO T^ 2 = L/R 658, 659 31)8 EXISTENCE OF ONE
DEGENERATE INTEGRAL INVOLVES ANOTHER Q» = 2) . 659 39(J, 400 CONNECTION
WITH THEORY OF SPECIAL TRANSFORMATION, WHEN P = 2 . 660, 661 401*403
DETERMINATION OF NECESSARY FORM OF FUNDAMENTAL EQUATION. REFERENCES
661*663 APPENDIX 1. ON ALGEBRAIC CURVES IN SPACE. 404 FORMAL PROOF THAT
AN ALGEBRAIC CURVE IN SPACE IS AN INTERPRETA- TION OF THE RELATIONS
CONNECTING THREE RATIONAL FUNCTIONS ON A RIEMANN SURFACE (CF. § 162)
664, 665 APPENDIX II. ON MATRICES. 405*410 INTRODUCTORY EXPLANATIONS
666*669 411*415 DECOMPOSITION OF AN ABELIAU MATRIX INTO SIMPLER ONES . .
669*674 416 A PARTICULAR RESULT 674 417, 418 LEMMAS 675 419, 420 PROOF
OF RESULTS ASSUMED IN §§ 396, 397 675, 676 INDEX OF AUTHORS QUOTED 677,
678 TABLE OF SOME FUNCTIONAL SYMBOLS . . . 679 SUBJECT INDEX 680*684
|
| any_adam_object | 1 |
| author | Baker, Henry Frederick 1866-1956 |
| author_GND | (DE-588)116041315 |
| author_facet | Baker, Henry Frederick 1866-1956 |
| author_role | aut |
| author_sort | Baker, Henry Frederick 1866-1956 |
| author_variant | h f b hf hfb |
| building | Verbundindex |
| bvnumber | BV010997676 |
| classification_rvk | SK 780 SK 750 |
| classification_tum | MAT 140f |
| ctrlnum | (OCoLC)440835293 (DE-599)BVBBV010997676 |
| discipline | Mathematik |
| edition | 1. published 1897, reissued |
| format | Book |
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| id | DE-604.BV010997676 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:02:21Z |
| institution | BVB |
| isbn | 0521498775 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007362061 |
| oclc_num | 440835293 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-739 DE-703 DE-634 DE-83 DE-188 |
| owner_facet | DE-91 DE-BY-TUM DE-739 DE-703 DE-634 DE-83 DE-188 |
| physical | XXXV, 684 Seiten graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Cambridge University Press |
| record_format | marc |
| series2 | Cambridge Mathematical Library |
| spelling | Baker, Henry Frederick 1866-1956 Verfasser (DE-588)116041315 aut Abelian functions Abel's theorem and the allied theory of theta functions H. F. Baker, St. John's College, Cambridge 1. published 1897, reissued Cambridge Cambridge University Press 1995 XXXV, 684 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge Mathematical Library Originalausgabe mit dem Sachtitel: Abel's theorem and the allied theory including the theory of theta functions.- Cambridge: Cambridge University Press, 1897 Abelsche Funktion (DE-588)4140987-5 gnd rswk-swf Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Abelsche Funktion (DE-588)4140987-5 s DE-604 Algebraische Geometrie (DE-588)4001161-6 s GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007362061&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Baker, Henry Frederick 1866-1956 Abelian functions Abel's theorem and the allied theory of theta functions Abelsche Funktion (DE-588)4140987-5 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| subject_GND | (DE-588)4140987-5 (DE-588)4001161-6 |
| title | Abelian functions Abel's theorem and the allied theory of theta functions |
| title_auth | Abelian functions Abel's theorem and the allied theory of theta functions |
| title_exact_search | Abelian functions Abel's theorem and the allied theory of theta functions |
| title_full | Abelian functions Abel's theorem and the allied theory of theta functions H. F. Baker, St. John's College, Cambridge |
| title_fullStr | Abelian functions Abel's theorem and the allied theory of theta functions H. F. Baker, St. John's College, Cambridge |
| title_full_unstemmed | Abelian functions Abel's theorem and the allied theory of theta functions H. F. Baker, St. John's College, Cambridge |
| title_short | Abelian functions |
| title_sort | abelian functions abel s theorem and the allied theory of theta functions |
| title_sub | Abel's theorem and the allied theory of theta functions |
| topic | Abelsche Funktion (DE-588)4140987-5 gnd Algebraische Geometrie (DE-588)4001161-6 gnd |
| topic_facet | Abelsche Funktion Algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007362061&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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