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Inhalt
I. The Career of a Mathematician.- I. Youth (1809-1830).- Early Interests in Mathematics.- Student at the École Polytechnique.- The Ponts et Chausées.- The Independent Researcher.- II. Climbing the Academic Ladder (1830-1840).- The Public School System.- The École Centrale; Colladon and Sturm.- Scientific Societies; the Société Philomatique.- The Creation of New Fields.- The Creation of Liouville's Journal.- Defeats at the Académie and at the École Polytechnique.- Magnanimity toward Sturm.- Success at the Collège de France and at the École Polytechnique.- A Vacancy in the Astronomy Section of the Académie.- Opposition from Libri.- Liouville and Dirichlet against Libri.- Election to the Bureau des Longitudes.- III. Professor, Academician, and Editor (1840-1848).- Setting the Stage.- The École Polytechnique.- Administrative Duties.- Liouville's "Cours d'Analyse et de Mécanique".- Rigor.- Notes in Navier's Résumé.- Other Related Works; Transcendental Numbers.- Influence.- Collège de France.- Inspiring Courses.- A Scandalous Election; Liouville, Cauchy, and Libri.- Académie des Sciences.- The Active Examiner.- Fermat's Last Theorem.- International Contacts.- Prize Competitions.- The Bureau des Longitudes.- Cauchy's Membership?.- Presentation of New Ideas.- Journal Editor.- Guiding Young Talents.- Le Verrier (Catalan and Delaunay).- From Irresolution to Authority.- Quarrel with Pontécoulant.- The Name Neptune.- Hermite, Bertrand, and Serret.- Two Reports.- Last Clash with Libri.- Hermite and Doubly Periodic Functions.- J. A. Serret; "Elliptic Curves".- Galois Theory.- Foreign Visitors.- Steiner, The Dublin School, Geometry.- William Thomson-Lord Kelvin.- A Coherent Mathematical Universe.- IV. The Second Republic (1848-1852).- Banquets Rèformistes (1840).- Political Opposition (1840-1848).- The 1848 Revolution.- Candidate for the Constituting Assembly.- Member of the Constituting Assembly.- The Bitter Defeat (1849).- Reduced Mathematical Activity.- The Second Election at the College de France; Retirement from the École Polytechnique.- Lectures at the Collège de France.- The Disappointing Outcome of the Second Republic.- V. The Last Flash of Genius (1852-1862).- Imperial Politics; Its Influence on Liouville's Family.- Ernest Liouville, Statistics, Bienaymè (1852-1853).- Liouville Opposing Le Verrier (1852-1854).- Friendship with Dirichlet.- Mathematical Production (1852-1857).- Sturm's Death.- Chasles's Substitute?.- Competition with J. A. Serret.- Professor of Mechanics at the Facultè des Sciences.- Courses at the Collège de France.- Chebyshev.- Scandinavian Students.- Great Teaching Load-No Research.- Bad Health.- Liouville's Final Opinion of Cauchy.- Liouville Commerating Dirichlet.- "I myself, who only like my hole".- The Quarrels with Le Verrier Continued.- Declining Influence in the Académie; Bour.- Official Honors.- VI. Old Age (1862-1882).- Mathematical Work.- Lecturer and Promoter.- Public Life.- The Franco-Prussian War and the Commune.- Foreign Member of the Berlin Academy.- The Last Courses.- Domestic Life.- The Stay in Toul, Summer and Autumn 1876.- Longing for Death.- Posthumous Reputation.- II. Mathematical Work.- VII. Juvenile Work.- Electrodynamics.- Ampère's Electrodynamics.- Liouville's Contributions.- Theory of Heat.- Laplace, Fourier, and Poisson on the Heat Equation.- Liouville's Contribution.- Differential Equations.- VIII. Differentiation of Arbitrary Order.- Applications, the Source of Interest.- Foundations.- Fractional Differential Equations.- Rigor.- Concluding Remarks.- IX. Integration in Finite Terms.- Historical Background.- Abel's Contributions.- Integration in Algebraic Terms.- Integration in Finite Terms.- Solution of Differential Equations in Finite Terms.- Further Developments.- Conclusion.- X. Sturm-Liouville Theory.- The Roots of Sturm-Liouville Theory.- The physical origins.- D'Alembert's Contribution.- Fourier's Contribution.- Poisson's Contribution.- Sturm's First Memoir.- Origins.- Sturm's Second Memoir.- Liouville's Youthful Work on Heat Conduction.- Liouville's Mature Papers on Second-Order Differential Equations. Expansion in Fourier Series.- Convergence of Fourier Series.- Determination of the Sum of the Fourier Series.- Minor Results.- Liouville's Generalization of Sturm-Liouville Theory to Higher-Order Equations.- Third Degree; Constant Coefficients.- Higher Degree; Variable Coefficients.- Further Generalizations.- Concluding Remarks.- XI. Figures of Equilibrium of a Rotating Mass of Fluid.- Prehistory; Maclaurin Ellipsoids.- Jacobi Ellipsoids.- Stability of Equilibrium Figures.- Sources.- Results on Stability.- Successors.- Methods and Proofs.- Formulas for the Force Vive.- Liquid on an Almost Spherical Kernel.- Lame Functions.- Stability of Fluid Ellipsoids.- A Corrected and an Uncorrected Error.- Concluding Remarks.- XII. Transcendental Numbers.- Historical Background.- Liouville on the Transcendence of e (1840).- Construction of Transcendental Numbers (1844).- The Impact of Liouville's Discovery.- XIII. Doubly Periodic Functions.- General Introduction; Diffusion of Liouville's Ideas.- Division of the Lemniscate.- The Discovery of Liouville's Theorem.- The Gradual Development of the General Theory.- The Final Form of Liouville's Theory.- Conclusion.- XIV. Galois Theory.- The "Avertissement".- Galois's Friends?.- Galois Theory According to Galois.- Liouville's Commentaries.- Proposition II.- Proposition VI-VIII.- Liouville's Publication of the Works of Galois.- Liouville's Understanding of Galois Theory.- Liouville's Impact.- XV. Potential Theory.- The Genesis of Potential Theory.- Liouville's Published Contributions.- Liouville on Potentials of Ellipsoids.- Liouville's Unpublished Notes on Spectral Theory of Integral Operators in Potential Theory.- Interpretation of the Major Result.- A Posteriori Motivation.- Reconstruction of the A Priori Proof.- The Paper on the General Spectral Theory of Symmetric Integral Operators.- Unsolved Problems, Alternative Methods.- Anticipation of Weierstrass's Criticism of the Dirichlet Principle.- Are Liouville's Results Correct?.- Poincaré's Fundamental Functions.- Later Developments.- XVI. Mechanics.- The Theory of Perturbation by Variation of the Arbitrary Constants.- Celestial Mechanics.- Two Publications by Liouville on Planetary Perturbations.- The Chronological Development of Liouville's Ideas.- Atmospherical Refraction.- Conclusion.- Liouville's Theorem "on the Volume in Phase Space".- Rational Mechanics.- The Hamilton-Jacobi Formalism.- Liouville's Theorem and a Precursor.- Poisson on Liouville's Theorem.- The Publication of Jacobi's Ideas.- Jacobi on Liouville's Theorem.- Parisian Developments Prompting Liouville's Publication of His Theorem.- Liouville's Lecture on Mechanics.- The Geometrization of the Principle of Least Action.- Liouville's Unpublished Notes; Generalized Poisson Brackets.- The "Liouvillian" Integrable Systems.- Concluding Remarks.- XVII. Geometry.- Analytic versus Synthetic Geometry; Chasles's Influence.- Relations with Mechanics and Elliptic and Abelian Functions; Jacobi's Influence.- Inversion in Spheres; William Thomson's Influence.- Contributions to Gaussian Differential Geometry.- Appendix I. Liouville on Ampère's Force Law.- Appendix II. Liouville's Notes on Galois Theory.- Notes.- Unpublished Manuscripts and Other Archival Material.- List of J. Liouville's Published Works.- Other References.