RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

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16.2 ABEL’S extension to the complex rectangle . . . . . . . . . . . . . . . . . . . 305 18.1 CARL GUSTAV JACOB JACOBI (1804–1851) . . . . . . . . . . . . . . . . . . . . 332 19.1 GEORG FRIEDRICH BERNHARD RIEMANN (1826–1866) . . . . . . . . . . . . 374 21.1 The equality of the concepts of explicit algebraic expressions and ABEL’s normal form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 21.2 Delineating the border between a concept and its super-concept. . . . . . . . 396 x
List of Boxes 1 The algebraic reduction of the cubic equation . . . . . . . . . . . . . . . . . . 60 2 Proof of Lehrsatz I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 3 A modern reconstruction of DIRICHLET’s proof. . . . . . . . . . . . . . . . . 239 4 ABEL’s Lehrsatz V derived from DU BOIS-REYMOND’s theorem . . . . . . . . 249 5 Proof that the trigonometric coefficients cannot approach zero . . . . . . . . 256 6 Rectification of the ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7 Rectification of the lemniscate . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 8 A.-M. LEGENDRE’S (1752–1833) reduction of elliptic integrals . . . . . . . . 295 9 An important lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 10 Linear interdependence of q0, . . . , qn−1 . . . . . . . . . . . . . . . . . . . . . . 359 xi
Page 1 and 2: RePoSS: Research Publications on Sc
Page 3 and 4: The Mathematics of NIELS HENRIK ABE
Page 5 and 6: The Mathematics of NIELS HENRIK ABE
Page 7 and 8: Contents Contents i List of Tables
Page 9 and 10: 8.3 Refocusing on the equation . .
Page 11: V ABEL’s mathematics and the rise
Page 15: List of Figures 2.1 NIELS HENRIK AB
Page 21 and 22: Summary The present PhD dissertatio
Page 23 and 24: Preface to the 2004 edition For thi
Page 25: in connection with the Abel centenn
Page 28 and 29: items published in the same year ar
Page 30 and 31: the Mittag-Leffler archives in Djur
Page 33 and 34: Chapter 1 Introduction In the after
Page 35 and 36: 1.2. The mathematical topics involv
Page 37 and 38: 1.3. Themes from early nineteenth-c
Page 39 and 40: 1.4. Reflections on methodology 9 l
Page 41 and 42: 1.4. Reflections on methodology 11
Page 43 and 44: 1.4. Reflections on methodology 13
Page 45: 1.4. Reflections on methodology 15
Page 48 and 49: 18 Chapter 2. Biography of NIELS HE
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36 Chapter 2. Biography of NIELS HE
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Chapter 3 Historical background The
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3.2. ABEL’s position in mathemati
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3.3. The state of mathematics 43 tr
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3.4. ABEL’s legacy 45 As can be s
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Chapter 4 The position and role of
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4.1. Outline of ABEL’s results an
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4.2. Mathematical change as a histo
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4.2. Mathematical change as a histo
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58 Chapter 5. Towards unsolvable eq
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98 Chapter 6. Algebraic insolubilit
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100 Chapter 6. Algebraic insolubili
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Chapter 7 Particular classes of equ
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7.1. Solubility of Abelian equation
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7.2. Elliptic functions 153 Figure
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7.2. Elliptic functions 155 7.2.1 T
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7.3. The concept of irreducibility
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7.3. The concept of irreducibility
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7.4. Enlarging the class of solvabl
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164 Chapter 8. A grand theory in sp
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Part III Interlude: ABEL and the
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192 Chapter 9. The nineteenth-centu
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194 Chapter 10. Toward rigorization
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208 Chapter 11. CAUCHY’s new foun
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Chapter 12 ABEL’s reading of CAUC
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12.1. ABEL’s critical attitude 22
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12.3. Convergence 229 12.3 Converge
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12.3. Convergence 231 and {εm} was
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12.4. Continuity 233 it followed th
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12.4. Continuity 235 Actually, if t
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12.4. Continuity 237 DIRICHLET intr
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12.5. ABEL’s “exception” 239
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12.6. A curious reaction: Lehrsatz
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12.7. From power series to absolute
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12.7. From power series to absolute
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12.8. Product theorems of infinite
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12.8. Product theorems of infinite
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12.9. ABEL’s proof of the binomia
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12.10. Aspects of ABEL’s binomial
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12.10. Aspects of ABEL’s binomial
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266 Chapter 13. ABEL and OLIVIER on
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Chapter 14 Reception of ABEL’s co
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14.1. Reception of ABEL’s rigoriz
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14.2. Conclusion 281 of basic notio
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Part IV Elliptic functions and the
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286 Chapter 15. Elliptic integrals
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300 Chapter 16. The idea of inverti
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Chapter 17 Steps in the process of
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17.1. Infinite representations 323
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17.2. Elliptic functions as ratios
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17.2. Elliptic functions as ratios
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17.3. Characterization of ABEL’s
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Chapter 18 Tools in ABEL’s resear
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18.1. Transformation theory 333 The
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18.1. Transformation theory 335 Obv
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18.1. Transformation theory 337 Sum
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18.2. Integration in logarithmic te
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18.3. Conclusion 345 Summary. ABEL
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Chapter 19 The Paris memoir N. H. A
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19.1. ABEL’s approach to the Pari
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19.2. The contents of ABEL’s Pari
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19.3. Additional, tentative remarks
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19.3. Additional, tentative remarks
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19.4. The fate of the Paris memoir
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19.6. Conclusion 377 hyperelliptic
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Chapter 20 General approaches to el
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20.2. Other ways of introducing ell
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20.3. Conclusion 383 All these four
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Chapter 21 ABEL’s mathematics and
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21.1. From formulae to concepts 389
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21.1. From formulae to concepts 391
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21.2. Concepts and classes enter ma
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21.3. The role of counter examples
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21.4. Conclusion 401 It is beyond t
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404 Appendix A. ABEL’s correspond
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Appendix B ABEL’s manuscripts Man
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1. Précis d’une théorie des fon
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Bibliography Abel (MS:351:A). “M
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Bibliography 413 in: Journal für d
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Bibliography 415 — (1990). “Geo
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Bibliography 417 — (1830). “Mé
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Bibliography 419 — (1768). “Ins
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Bibliography 421 Grattan-Guinness,
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Bibliography 423 Jahnke, H. N. (198
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Bibliography 425 Legendre, A. M. (1
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Bibliography 427 Rosen, M. (1981).
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Bibliography 429 Toti Rigatelli, L.
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432 Index of names Frobenius, Georg
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Index École Normale, 40, 187 Écol
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Index 437 Lagrange interpolation, 3
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